Numerical Predictions of Turbine Cascade Secondary Flows and Heat Transfer With Inflow Turbulence

The presence of endwalls in turbine cascade passages (either blades or vanes) generates a complex three-dimensional flow often referred to as secondary flows. These flow patterns alter the heat transfer and film cooling. Many reports exist in the literature that characterize and study the secondary flows [1–3]. Despite the differences in the flow parameters and cascade geometry, common features of these secondary flows are presented as vortex patterns describing the dominant vortices formed inside the passage, their respective locations, and their evolution through the passage. Sieverding [4] and Langston [3] reviewed early turbine passage secondary flow studies and provided a good summary of understanding on the vortex patterns and evolution. Wang et al. [5] provided the most detailed understanding of the multi-vortex secondary flow patterns as shown in Fig. 1. Their model identifies eight dominant flow structures. The first two are the legs of the horseshoe vortex (or leading edge vortex) that form near the stagnation region. One of the legs extends over the pressure side while the other moves toward the suction side of the adjacent passage. They also speculated on the presence of two leading edge corner vortices based on prior surface flow visualizations. Their smoke detection method, however, did not reveal these vortices because of their small sizes. These vortices are formed beneath the horseshoe vortex and extend on either side of the vane. Similarly, though not observed in their smoke detection method, they suggested a formation of two corner vortices on the junction of the endwall with suction and pressure surface. The passage vortex is the most dominant flow structure inside the passage and is a multi-vortex system, as described by Wang et al. [5]. According to their description, the pressure side leg of the horseshoe vortex is a major part of the passage vortex; however, both inlet boundary layer and the suction side leg contribute to it. The passage vortex grows in size as it entrains the endwall boundary layer and the main flow and moves away from the endwall. They also distinguished their model from the earlier passage vortex system descriptions by observing that the suction side leg of the horseshoe vortex wraps around the passage vortex.

Graziani et al. [6] provided one of the early heat transfer measurements on the endwall, suction, and pressure surfaces of turbine cascade. Their measurements indicate that more than 40% of the suction side heat transfer near the trailing edge is affected by the passage vortex and exhibits higher heat transfer rates. Similar observations on the effect of turbine passage secondary flows on turbine heat transfer have been made by Wang et al. [5], Goldstein and Spores [7], Giel et al. [8], Han and Goldstein [9], and Goldstein et al. [10]. Most of the mentioned studies were conducted with a limited range of freestream turbulence levels. More recently, Varty et al. [11] measured the Stanton number distribution over the suction surface of an aft loaded vane for five different turbulence levels (from 0.7% to 17.4%) and three different Reynolds numbers (5 × 10⁵ to 2 × 10⁶) using an infrared camera thermography method and provided a complete picture on the combined effects of freestream turbulence and secondary flows on turbine heat transfer.

Several attempts have been made to accurately predict the turbine passage heat transfer distribution. Giel et al. [12] used Reynolds-averaged Navier–Stokes (RANS) models to predict the turbine passage heat transfer which shows the lack of accuracy in the heat transfer predictions using RANS models although the qualitative effects of the secondary flows on heat transfer were evident. More recently, Papa et al. [13,14] evaluated the capability of RANS models with and without transitional models in the mass transfer predictions (which is analogous to heat transfer) under the influence of turbine passage secondary flows. Contrary to the regular shear stress transport (SST) model, the Re_θ-γ transition model predictions were in good agreement with the measurements especially in the prediction of the secondary flow region and the corresponding mass transfer rates. However, the midspan predictions did not capture the transition to turbulence near the trailing edge of the suction side.

Perhaps because of the huge computational cost and the lack of a complete data set [15], there are only a few higher fidelity simulations of the secondary flow in a turbine cascade passage [15–18]. In a recent study, Pichler et al. [18] performed large eddy simulation (LES) of a full turbine blade cascade with turbulence intensity of 5% and studied the secondary flows and the loss coefficients. The accuracy of their simulations are validated by comparing the predicted pressure coefficient distribution and the pitch average loss coefficient with the experiments.

This paper presents high-fidelity numerical simulations of fluid flow and heat transfer in the vane cascade, including the endwall secondary flows and heat transfer under high freestream turbulence. In this study, numerical simulations of flow and heat transfer of the first stage vane with a large leading edge are conducted using wall-resolved LESs. The vane geometry is identical to the measurements of Varty et al. [11]. Calculations are done at Reynolds number of 5 × 10⁵ based on the exit velocity and the true chord length L and for two inflow turbulence conditions of low (i.e., no disturbances) and high (⁠$Tu=7.9%$⁠) intensities. Kanani et al. [19,20] presented the effect of freestream turbulence on the vane midspan boundary layer and heat transfer. They found that despite the high levels of inflow turbulence (up to 12.4%), the boundary layer remains laminar on the pressure surface without transitioning to turbulence [19]. The leading edge structures perturb the boundary layer forming high- and low-speed streaks which kept their characteristics downstream up to the trailing edge. These streaky flow structures augment the heat transfer on the pressure surface. Conversely, the laminar boundary layer on the suction side is bypassed to turbulent for higher inflow turbulence levels [20,21]. They showed that the inner mode bypass transition is the route to turbulence and documented the first- and second-order statistics [20].

This report focuses on the near endwall flow and heat transfer characteristics including (a) the structure and evolution of the secondary flows and (b) the effect of the secondary flows on the vane and endwall heat transfer for two levels of inflow turbulence. Therefore, the current computational domain includes the vane as well as the endwalls. The numerical approach and the grid resolution are identical to the numerical simulations reported in Ref. [20].

Current calculations with 7.9% inflow turbulence captures all flow and heat transfer features present in the experiments including the laminar heat transfer augmentation on the pressure surface, transition to turbulence, and the regions affected by the secondary flows on the suction side. First, the numerical details are summarized and the accuracy of the predictions are evaluated by comparing the vane suction surface Stanton number distribution to the measurements. Furthermore, flow visualizations are presented to provide a complete picture of formation and development of the secondary flows inside the vane passage. The unique characteristics of the predicted secondary flows are discussed and compared with the secondary flow models available in the literature. In the remaining portion of the paper, the influence of the secondary flows on the heat transfer is evaluated and the flow and temperature fields for both inflow conditions are documented. This information is useful to enhance our knowledge on the formation and evolution of the secondary flows in vane passages with large leading edges and provides insights to cooling system designers.

The current linear vane cascade geometry is identical to the earlier reports by Kanani et al. [19,20,22], Varty and Ames [21], and Varty et al. [11]. Computations only include one vane and the periodicity in the linear cascade is imposed in the pitch direction (Fig. 2). The periodic boundaries are separated by the pitch P = 0.773L where L is the true chord length and is selected as the length scale. The inflow velocity is in the x-direction of the global xyz-coordinate and the vane span is aligned with the z-direction (see Fig. 2). A local vane coordinate system is defined such that the axes are tangent and normal to the vane cross section. The origin of the vane tangent axis s (i.e., surface distance) is at the leading edge with positive values along the suction side. The normal axis of the local coordinate d is normal to the vane surface with an origin at the vane surface, i.e., d = 0. Both coordinate systems share the z-axis with an origin at the endwall (i.e., z = 0).

For the current computations, the Reynolds number based on the exit velocity and the true chord length L is Re = 5 × 10⁵ with an incoming flow with zero and 7.9% turbulence intensity. This corresponds to an axial chord exit Reynolds number of Re = 3 × 10⁵. As reported by Kanani et al. [19,20], the effect of 0.7% inflow turbulence on the vane midspan heat transfer is negligible which justifies imposing the inflow velocity with no disturbances at the low freestream condition. The reported streamwise turbulence length scale for the higher turbulence inflow is 0.04L [21]. In the current calculations, turbulence length scales of 0.04L are imposed in all directions. The inlet boundary is at x = −0.51L. To capture flow and heat transfer features including pressure side heat transfer augmentation, suction side transition to turbulence and more importantly, the effects of the secondary flows on vane heat transfer, the numerical domain includes the endwalls on either ends of the vane (z-direction in Fig. 2) with a total span of 0.5L.

Without flow disturbances at the inflow, the midspan of the full vane is a plane of symmetry for both instantaneous and time-averaged fields. Thus, for this case, the numerical domain of the low inflow turbulence simulation only covers a half of the total span same as the LES conducted by Lynch [15]. However, the full vane with both endwalls is considered for the numerical simulations of the higher freestream turbulence case. This is because the presence of the turbulence inside the vane passage eliminates the validity of the symmetry assumption for the instantaneous fields at the vane midspan. To elaborate, one may impose a symmetry boundary condition at the midspan plane of the cascade flow with high freestream turbulence only if the time-averaged quantities are being solved (i.e., in RANS equations). In contrast, solving time-dependent equations (i.e., LES) with the presence of normal components of velocity fluctuations across the midspan plane invalidates a symmetric plane assumption. Moreover, preliminary simulations of a half-span domain with higher freestream turbulence (not shown here) indicate formations of spurious flows near the midspan which alters the flow and heat transfer near the symmetry plane.

The endwall boundary layer characteristics reported by Ames et al. [23] corresponding to similar experimental conditions are utilized to impose a proper boundary layer profile at the inlet of the numerical domain. These measurements are taken upstream of the vane and are the closest information available to estimate and reproduce the state of the boundary layer at the inlet. For the low freestream turbulence case, a Blasius boundary layer with Re_x = 1.58 × 10⁵ and δ₉₉ = 0.015L is imposed at the inlet normal to the endwall. This boundary layer is thin and only covers about 3% of the total vane span on each side. For the higher inflow turbulence case, a turbulent boundary layer with Re_θ = 1400 is imposed at the inlet of the numerical domain [24]. To generate the inflow turbulence for the LESs, the synthetic eddy method [25] described earlier by Kanani et al. [19,20] is used. To reproduce the endwall turbulent boundary layer statistics, the reported statistics by Schlatter et al. [24] corresponding to this Reynolds number are used as the reference. Table 1 summarizes the endwall boundary layer characteristics. Note that the effect of the boundary layer thickness and the freestream turbulence may not be distinguished in the current measurements and simulations as both parameters vary at these inflow conditions.

LESs are performed by solving the incompressible Navier–Stokes and temperature equations (as a passive scalar) with a laminar Prandtl number of 0.711 using OpenFOAM. The outlet of the numerical domain is 0.4L away from the trailing edge and homogeneous Neumann boundary condition is applied to velocity, pressure, and temperature fields. No-slip boundary condition is imposed to the endwalls and the vane. The vane is fully heated with a uniform heat flux boundary condition. The endwall heating starts at 0.153L upstream of the leading edge with an unheated (adiabatic) section upstream. Further details on the numerical solution can be found in Ref. [20].

The total number of points are 5.4 × 10⁷ and 1.78 × 10⁸ for the low and high inflow turbulence cases which resolve the boundary layer both on the endwall and the vane [20]. The half-span domain (low turbulence case) is discretized with 206 points in the spanwise direction clustered near the endwall with the expansion ratio of 1.02. There are 10 points inside the endwall boundary layer at the inlet of the domain. The full vane comprises 678 points in the spanwise direction and expansion ratio of 1.01 is used to cluster grids near both endwalls which results in 45 points inside either endwall inlet boundary layers. In the streamwise direction, the domain has three different zones with different grid numbers. The number of cells doubles by approaching the vane and further quadruples closer to the vane such that finest grids enclose the boundary layer. This results in over 1800 points on the suction surface. As shown in an earlier report [20], this method helps to achieve near direct numerical simulation grid resolution requirements inside the boundary layer while taking the advantage of the LES farther away from the wall that reduces the computational power requirements. The grid distribution in regions away from the endwalls (i.e., midspan region) is identical to Ref. [20] and have enough resolution to capture the transition to turbulence at the high inflow turbulence condition. The computations are conducted on Stampede2 supercomputer using 72 SKX computing nodes (i.e., 3456 CPUs). All quantities are time-averaged for 11 and 29 flow-through times L_x/u_in for low and high turbulence cases after at least six flow-through times of initialization. These simulations took 48 and 624 h for low and high turbulence cases which correspond to 1.65 × 10⁵ and 2.65 × 10⁵ CPU-h, respectively.

The reported uncertainty of the Stanton number measurements is $±12%$ [11]. The root mean square error in the predictions are $13.9%$ and $10%$ for the low and high inflow turbulence, respectively. The current numerical simulations predict the higher heat transfer coefficient region associated to the secondary flows with a good accuracy. For the low inflow turbulence condition, the peak heat transfer coefficient is near the endwall in the regions affected by the secondary flows. Similar distribution is also observed for the high turbulence case. However, a dominant secondary peak also appears away from the endwall both in measurements and predictions. Predictions for this inflow condition capture the transition to turbulence over the last part of the suction surface as reported earlier in the simulations of the midspan fluid flow and heat transfer [20].

Figure 5 provides a quantitative comparison of the predicted Stanton number with the measurements at s/L = 0.75 and s/L = 1.1. The figures also include the uncertainty levels of the measurements. Predictions are within the uncertainty level of the experiments for most of the reported values. Considering the high uncertainty in the experimental measurements, current predictions agree well with the measurements which validates the accuracy of the numerical approach. Accurate predictions of the suction side heat transfer distribution are contingent to capture the dynamics of the secondary flows. Considering a good match in the suction side heat transfer distribution predictions, one can further verify the validity of the secondary flow predictions and study their characteristics inside the current turbine cascade geometry using the flow and temperature fields.

In this section, the details of the secondary flow patterns are presented which details the similarity and differences between the current secondary flow pattern with those described in the literature (e.g., Fig. 1). This is achieved by visualizing the flow fields for both simulations. Next, the effects of these flow structures on the vane and endwall heat transfer are described.

The flow field is sampled at several sampling planes (see Fig. 6) to visualize the secondary flows. These planes extend from the suction surface toward the pressure surface and cover the whole passage cross section. For convenience, these planes are named according to their corresponding surface distance from the leading edge on the suction surface, i.e., s/L value. The s/L interval of the planes shown in Fig. 6 is 0.02. The s/L of a selected planes are shown in this figure. Planes between 0.02 < s/L ≤ 0.22 are circular and selected to cover the whole passage cross section to reveal the secondary flow patterns. Planes corresponding to s/L ≥ 0.24 are planar and normal to the suction surface. The midspan flow streamlines as seen in Fig. 6 justify the choice of the sampling planes.

Figure 7 depicts the time-averaged secondary flow pattern and the corresponding endwall and vane heat transfer inside the vane passage for both low and high inflow turbulence simulations. Normal component of the time-averaged vorticity vector $ω→⋅n→$ is shown for several sampling planes where $n→$ is the local normal vector to the sampling planes. Very low values of $ω→⋅n→$ are made transparent such that the prominent features of the secondary flows are visible throughout the passage. The distribution of the Stanton number on the vane and the endwall is also shown in these figures. The top vane is removed to show the structures near the pressure side. Regions with negative $ω→⋅n→$ rotate clockwise and those with positive $ω→⋅n→$ values rotate in a counter-clockwise direction as seen in Fig. 7.

Figures 8–10 show the contours of the normal component of the vorticity $ω→⋅n→$ for both simulations. Secondary flow streamlines are also shown in these figures. For the lower turbulence case, both instantaneous and time-averaged quantities are shown. The instantaneous fields for the high inflow turbulence case are omitted as the main flow features are only evident in the time-averaged plots. The main flow direction in these figures is into the paper.

The streamline plots were obtained by calculating the in-plane components of the velocity field and then subtracting the midspan component (i.e., z/L = 0.255) at each x and y from the in-plane velocity field of the corresponding x and y [26]. The in-plane velocity components are calculated using the local normal vector to the sampling planes. Note that the notion of the local normal vector is only necessary for the non-planar sampling planes, as otherwise, the normal vector of the planar sampling planes does not change spatially.

Dominant flow structures in the current simulations are the pressure and suction sides passage circulations (PPC and SPC), leading edge horseshoe vortex (PH and SH), leading edge corner vortex (PLC and SLC), the suction surface corner vortex (SC), and the passage vortex which is a multi-vortex structure comprising the PPC, PH, SPC, SH, and SH. In the following sections, the development and evolution of these structures inside the turbine cascade passage are described.

Figures 8 and 10 indicate the presence of a pair of counter-rotating flow circulations which extends up to the midspan (figures only extend to z/L = 0.12 for clarity). The pressure side passage circulation (PPC) is rotating counter-clockwise when viewed in the main flow direction while the suction side passage circulation (SPC) rotates clockwise. This passage circulation pair forms as the inlet boundary layer turns because of the large leading edge. The lower momentum fluid inside the endwall boundary layer turns faster than the higher momentum fluid at the freestream. This is evident in Fig. 11 where the velocity streamlines parallel to the endwall are plotted at z/L = 0.0015 and z/L = 0.015. Near the endwall, a secondary flow forms which sweeps the incoming flow away from either side of the vane. As seen in Fig. 8, a stagnation point forms where the two streams meet their counterpart which lifts the fluid from the endwall and forms a pair of counter-rotating flow circulation. At s/L = 0.02, the stagnation point is at d/L ≈ 0.45 for both inflow conditions (Figs. 8(a) and 10(a)).

At the high inflow turbulence condition, the PPC and SPC are only evident in the time-averaged streamlines, but both instantaneous and time-averaged plots of the low inflow turbulence simulations clearly show both PPC and SPC. The core of the PPC and SPC are at d/L ≈ 0.65 and d/L ≈ 0.19 at s/L = 0.02 for the low inflow turbulence case. The location of the core is not evident for the higher turbulence case until s/L = 0.08. At this plane, the PPC and SPC are located at d/L ≈ 0.5 and d/L ≈ 0.1. The corresponding locations for the low inflow turbulence case at this plane are d/L ≈ 0.37 and d/L ≈ 0.15. The PPC and SPC cores are close to the endwall (i.e., z/L ≈ 0.12) with the thin and laminar endwall boundary layer (low freestream inflow condition), while they are farther away from it with the thick approach boundary layer (high inflow turbulence condition). This might be due to the thick inlet boundary layer (i.e., larger portion of low momentum fluid) which increases the thickness of endwall crossflow and pushes the core away from it.

As flow enters the passage, the upward motion (positive y-direction) induced by the large leading edge (on the suction surface) gradually diminishes up to s/L ≈ 0.22 as seen in Figs. 8(a)–8(e). Afterwards, the evolution of these circulations are determined by the pitchwise pressure gradients and the flow acceleration through the passage. The PPC migrates towards the suction surface because of the pitchwise pressure gradient and squeezes the SPC (as seen in Figs. 8 and 10). These circulations interact with other secondary vortex systems while passing through the passage. These interactions will be discussed later.

Contrary to the previous studies of the secondary flows in turbine passages, where the pressure side leg of the horseshoe vortex was identified as the main component of the passage vortex [1–4], the current simulations reveal that the PPC is the dominant component of the passage vortex for this flow configuration. This might be due to a different vane geometry in this study that consists of a large leading edge.

The endwall boundary layer separates while approaching the leading edge and forms a leading edge vortex system known as the horseshoe vortex system [27]. Depending on the boundary layer state and the Reynolds number, one or multiple vortices may form near the leading edge [5,27]. The horseshoe vortex wraps around the leading edge, one leg moves to the pressure side of the vane, PH, and the other leg moves towards the suction side of the adjacent passage, SH (Figs. 8(a) and 10(a)). For the thin laminar boundary layer, both PH and SH are closer to the endwall and are less intense compared to the case with a thick turbulent endwall boundary layer. In the case of low inflow turbulence, the PH vortices migrate towards the suction surface as flow enters the passage and moves closer to the core of the PPC as seen in Figs. 9(a)–9(g). These vortices eventually move inside the PPC one by one starting from s/L ≈ 0.18. A distinct core of the PH remains above the PPC up to s/L ≈ 0.78 (Figs. 9(g)–9(j)). The suction side leg of the horseshoe vortex SH goes through the same process, but merging with the SPC occurs in a shorter distance as the SPC is squeezed through the passage (Figs. 9(a)–9(c)). The SH cores remain distinct inside the SPC as seen in Fig. 9(d).

The pressure side leg of the horseshoe vortex (PH) formed in the high inflow turbulence simulation is located at the top left corner of the PPC when viewed in the flow direction and is a part of the PPC from the early stages of the passage flow (Figs. 10(a) and 10(b)). The PH migrates towards the suction surface with the PPC but the core of the PH and PPC remains separate up to s/L ≈ 0.58 where it will be drawn to the core of the PPC as seen in Fig. 10(i). The SH vortex, however, shows the same behavior as the low inflow turbulence case and merges with the SPC at s/L ≈ 0.18 (Fig. 10(c)).

As Wang et al. [5] described in their secondary flow representation, the presence of the SH induces a very small leading edge corner vortex (SLC) at the junction of the suction side and the endwall (Figs. 10(b) and 8(e)). The corner vortex appears earlier at the high inflow turbulence condition at s/L ≈ 0.02, while its formation is delayed until s/L ≈ 0.22 for the low inflow turbulence case. This seems to be related to the strength and location of the SH with respect to the suction surface and the endwall. For both cases, the SLC is lifted from the endwall after s/L ≈ 0.28 because of the counter-clockwise motion (viewed in the flow direction) induced by the PPC when it gets closer to the suction surface. This can be seen more clearly in the zoomed view of the secondary flow motion at s/L = 0.32 in Fig. 12. The SLC continues to move away from the endwall and grow in size and thus, become a part of the passage vortex. The core of the SLC is at z/L ≈ 0.8 at the trailing edge for both cases (Figs. 8(n) and 10(n)).

On the pressure side, the PH also induces a very small leading edge corner vortex PLC (Fig. 10(a)) which persists up to the trailing edge of the pressure side (Fig. 10(h)) in the high inflow turbulence simulations. The PLC does not appear at the early stages of the passage flow for the low inflow turbulence case. Contrary to the SLC, the PLC never forms (Fig. 8) at this inflow condition as the PH moves away from the pressure side and does not induce the corner vortex in the junction of the pressure surface and the endwall.

According to Wang et al. [5], suction and pressure side corner vortices form close to the endwall inside the passage. Although a corner vortex is observed in the high inflow turbulence simulation, this vortex is in fact the PLC which persists through the passage as discussed earlier. The suction side corner vortex SC, however, is present in the current predictions for both inflow conditions and appears after the PPC starts sweeping the SLC away from the endwall at s/L ≈ 0.32. As those vortices are very small, they can be hardly seen in Figs. 8(f) and 10(f). The zoomed view of the SC for both cases at s/L = 0.32 is shown in Fig. 12. This vortex forms because of the sweeping motion of the endwall flow towards the suction surface. The SC grows in size downstream but remains attached to the corner of the endwall and suction surface.

A distinct corner vortex on the pressure side is not observed for the low inflow turbulence case. But, a trace of high Stanton number appears near the endwall on the pressure surface (see Fig. 13(a)). A closer look at the corner flow (not shown here) suggests a small separation and reattachment region on the pressure surface while the flow is swept towards the endwall by the PPC.

As shown earlier, the current simulations suggest that the main part of the passage vortex is the pressure side passage circulation (PPC) which drives the evolution of the other near wall vortex structures. This circulation initially forms upstream of the passage closer to the pressure side of the vane and rotates in the counter-clockwise direction when viewed in the flow direction. Entering the passage, the PPC moves toward the suction surface and squeezes the SPC and eventually merges with it to form a single multi-vortex system as described earlier. In this section, the differences in the dynamics of these interactions for the two different inflow conditions are discussed.

Here, the core of the PPC is located close to the endwall and migrates toward the suction surface downstream while attached to the endwall till s/L ≈ 0.22. When the core of the PPC reaches the outer part of the SPC (rotating clockwise when viewed in the flow direction), it will be lifted away from the endwall (Figs. 8(e) and 8(f)). As the PPC continues moving towards the suction surface, individual SH vortices inside the SPC wrap around the PPC and revolve around it as seen in Fig. 9(g). During this process, the SPC disappears as it fully wraps around the PPC around s/L ≈ 0.44.

The SLC continues moving away from the endwall and becomes a part of the passage vortex at s/L ≈ 0.4. Farther downstream, the core of the PPC continues its movement toward the suction surface. The counter-clockwise rotation of the PPC pushes away the SLC farther from the endwall. Near the trailing edge s/L ≈ 1.1 (Fig. 8(m)), the cores of the PPC and SLC are both close to the suction surface at the same wall distance.

The core of the PPC for this inflow condition is farther away from the endwall and moves towards the suction surface when it enters the passage (Fig. 10). Similar to the previous case, the SPC revolves around the PPC. The core of the SPC, however, remains distinct up to the trailing edge. Contrary to the previous flow condition, the core of the PPC does not reach the suction surface and is at d/L ≈ 0.5 when it reaches the trailing edge as seen in Fig. 8(n).

In this section, the effects of the secondary flows on the endwall and vane heat transfer are presented. Figures 13 and 14 show the heat transfer distribution (Stanton number) over the full vane surface and the endwall. The range and the color map of the contours are same in all figures. Dominant vortex systems are labeled on these figures such that the lines indicate the location of the core of each vortex. Note that s/L and z/L axes in Fig. 13 are not plotted with the same scale.

Various vortex structures and their evolution inside the vane cascade passage are discussed in the earlier sections. Near the stagnation region, suction and pressure sides of the horseshoe vortex enhance the heat transfer rate in a very small area. As the core of the PPC and SPC are close to the endwall, the enhanced heat transfer regions are also present beneath the core of these structures as seen in Fig. 14(a). Both the SPC and PPC inject fluid toward their counterparts. The trace of high Stanton number beneath the PPC core diminishes starting from s/L ≈ 0.22 which is the location where it separates from the endwall (see Fig. 8(e)). Starting from s/L ≈ 0.22, the SPC and the enclosed SH vortices move away from the endwall and no longer augment the heat transfer on the endwall. A low heat transfer coefficient band exists between the PPC and SPC. The excess of fluid mass in that band creates a velocity component normal to the endwall (as seen in Fig. 8(a)) and thickens the boundary layer.

The influence of the suction side leg corner vortex (SLC) is clear on the vane heat transfer distribution as seen in Fig. 13(a) at s/L ≈ 0.2. This vortex is lifted away from the endwall while the PPC moves toward the suction surface. Starting from s/L ≈ 0.4, the lower left corner of the PPC (when viewed in the flow direction) reaches the suction surface and enhances the heat transfer. Figure 13(a) clearly shows this heat transfer augmentation. The SLC continues moving away from the endwall and increases the augmented heat transfer area. Similar to the horseshoe vortex pattern, the SLC causes a separation and reattachment region on the suction surface (as depicted in Fig 5(b) by Baker et al. [27]). This pattern is also observed in the current simulations as can be seen in Fig. 12. A narrow band between the separation and attachment line corresponds to a lower Stanton number values while the reattachment enhances the heat transfer. This can be seen in Fig. 13(a) beyond the SLC path (larger z/L values). On the pressure side, although a distinct pressure corner vortex is not observed, a trace of higher Stanton number appears near the endwall.

The endwall heat transfer pattern for this inflow condition is different than the case with low inflow turbulence (see Fig. 14(b)). Since the PPC core is away from the endwall, no heat transfer augmentation occurs beneath the PPC on the endwall. However, a stronger horseshoe vortex on the pressure side is responsible for the higher heat transfer coefficients. Interestingly, the high Stanton number does not coincide with the core of the PH and is located slightly closer to the suction surface. This is because the main PH vortex induces some secondary vortices which impinges the flow towards the endwall. This process is depicted in a zoomed view of an instantaneous snapshot of the PH vortex in-plane s/L = 0.16 in Fig. 15. The bounding box of this figure is also depicted in Fig. 7(b). A higher Stanton number band is also observed above the PH and PPC vortex as the combination of these vortices impinge the flow toward the endwall. As the core of the PPC moves closer to the endwall, intensifies and merges with the PH (see Fig. 10), the area of the higher Stanton number above the PPC (close to the pressure surface) increases.

The stagnation region heat transfer behaves similarly to the previous inflow condition with the only difference being that the heat transfer coefficient is higher because of the stronger horseshoe vortex system. The SPC also merges with the SH and forms a single vortex at s/L ≈ 0.18. Like the previous case, the migration of the PPC toward the suction surface lifts the SPC and SLC away from the endwall and hence the high heat transfer area region near the leading edge diminishes at s/L ≈ 0.24. The migration of the SLC from the endwall is evident in Figs. 13(b) and 7 which increases the high Stanton number region because of the secondary flows. The separation and reattachment beyond the SLC vortex is much larger than the low inflow turbulence case. This might be because of the interaction of the leading edge vortices formed away from the endwall [20] with the SLC.

The lower left corner of the PPC (when viewed in the flow direction) reaches the suction surface at s/L ≈ 0.34 and further increases the Stanton number. The effect of the PLC is also evident from the Stanton number distribution on the pressure surface near the endwall (Fig. 13(b)).

In this paper, high-fidelity numerical simulations of the vane cascade flow and heat transfer at an exit chord Reynolds number of 5 × 10⁵ with low and high levels of inflow turbulence are presented. Numerical predictions are validated with the surface Stanton number measurements of Varty et al. [11]. Current predictions are mostly within the reported uncertainty of the measurements.

The velocity field inside the vane passage is visualized and the secondary flow patterns are identified for the current vane geometry for both simulations. A counter-rotating circulation observed for both cases amplifies through the passage and forms the majority of the passage vortex. Other components of the secondary flow system such as the horseshoe vortex system, leading edge corner vortex, and suction and pressure side corner vortices are identified. Although the overall nature of the secondary flows are identical for both cases, the positioning, evolution, and interactions of these flow features are quite different at these inflow conditions. As both boundary layer thickness and the inflow turbulence were different in the reference measurements, it was not possible to distinguish their individual effects. Future studies are required to shed more light in order to provide a clearer understanding of the effect of the boundary layer characteristics and the inflow turbulence separately.

The effect of each component of the secondary flows on the heat transfer coefficient is identified by analyzing the full vane and endwall heat transfer predictions. The endwall heat transfer pattern significantly varies for two different inflow conditions. For the low inflow condition case, the PPC and SPC are close to the endwall and augment the heat transfer beneath their cores. A very low heat transfer region is found between the PPC and SPC cores. On the suction surface, the passage vortex sweeps the leading edge corner vortex toward the midspan for both cases which creates a region of high Stanton number growing in size downstream. Regardless of the differences in the secondary flow patterns of the two different inflow conditions, the effect of the secondary flows on the suction surface seems very similar. The main difference is the size of the reattachment region induced by the SLC. This behavior persists in both the measurements and the predictions. Although this behavior is not understood well, it might be because of the interactions of the secondary vortices formed away from the endwall [20] and the SLC.

Computational resources were provided by the National Science Foundation’s (NSF) Extreme Science and Engineering Discovery Environment (XSEDE) via grant number ACI-1053575. Their support is gratefully acknowledged.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

d =

wall-normal distance (m)

h =

heat transfer coefficient (W/m² K)

n =

normal vector to the sampling plane

q =

heat flux (W/m²)

s =

surface distance from the leading edge (m)

x =

global x-coordinate (m)

y =

global y-coordinate (m)

z =

global z-coordinate (m)

L =

true chord length (m)

P =

pitch (m)

T =

temperature (K)

c_p =

specific heat capacity (J/kg K)

L_x =

axial chord length (m)

Pr =

Prandtl number

Re =

Reynolds number

St =

Stanton number, h/ρ|u_outflow|c_p

Tu =

turbulence intensity

⁺ =

scaled in inner wall units

δ₉₉ =

boundary layer thickness (m)

θ =

momentum thickness (m)

ρ =

density (kg/m³)

$ω→$ =

vorticity vector (s⁻¹)

0 =

inlet

w =

wall