## Abstract

High Reynolds flow over a nozzle guide-vane with elevated inflow turbulence was simulated using wall-resolved large eddy simulation (LES). The simulations were undertaken at an exit Reynolds number of 0.5 × 106 and inflow turbulence levels of 0.7% and 7.9% and for uniform heat-flux boundary conditions corresponding to the measurements of Varty and Ames (2016, “Experimental Heat Transfer Distributions Over an Aft Loaded Vane With a Large Leading Edge at Very High Turbulence Levels,” ASME Paper No. IMECE2016-67029). The predicted heat transfer distribution over the vane is in excellent agreement with measurements. At higher freestream turbulence, the simulations accurately capture the laminar heat transfer augmentation on the pressure surface and the transition to turbulence on the suction surface. The bypass transition on the suction surface is preceded by boundary layer streaks formed under the external forcing of freestream disturbances which breakdown to turbulence through inner-mode secondary instabilities. Underneath the locally formed turbulent spot, heat transfer coefficient spikes and generally follows the same pattern as the turbulent spot. The details of the flow and temperature fields on the suction side are characterized, and first- and second-order statistics are documented. The turbulent Prandtl number in the boundary layer is generally in the range of 0.7–1, but decays rapidly near the wall.

## Introduction

The flow and heat transfer in the first stage of the turbine vane passage are affected by the high freestream turbulence exiting the combustor chamber which influences the cooling of the turbine component [14]. An accurate prediction of the turbine airfoil heat transfer is important for an effective cooling design that is tailored according to the heat load and minimizes the use of coolant and the performance penalties associated with coolant use [5]. Flow around the turbine airfoils is characterized by variable pressure gradient profiles, curved surface, secondary flows, and freestream turbulence. Therefore, acquiring a detailed knowledge and understanding of the fluid flow and heat transfer in the cascade geometry are challenging either through experiments or numerical simulations. There are a very limited number of experimental studies that have characterized and documented the details of the fluid flow (e.g., boundary layer profiles, second-order statistics, temperature-velocity cross-correlations) in vane passages under high freestream turbulence [610]. The limitations of measurement techniques have constrained the vast majority of the studies to only report the surface characteristics such as the heat transfer coefficient and skin friction over the airfoil that does not reveal the underlying physics of the flow and heat transfer [3,1114].

Since the flow is transitional, the accurate prediction of the flow and heat transfer through numerical simulations is extremely challenging. Solutions of the Reynolds-averaged Navier–Stokes (RANS) equations are associated with significant errors due to inaccurate turbulent modeling and transition onset prediction correlations [10,1520]. Advances in computing power have enabled researchers to utilize higher-fidelity approaches such as large eddy simulation (LES) and direct numerical simulation (DNS) in turbine passages. However, several previous studies indicate challenges associated with the accurate prediction of the transition onset and heat transfer in turbine passages with high inflow turbulence even with LES and DNS [17,2127]. In the following section, we only discuss the high-fidelity simulations of the heat transfer in turbine passage with elevated freestream turbulence that is directly related to the current study; more information on the RANS solutions and the experimental studies can be found in the references cited above.

Bhaskaran and Lele [21,22] conducted LES to predict the heat transfer on the high-pressure turbine cascade with exit Reynolds number of 0.58 × 106 and freestream turbulence of 4% corresponding to an experiment by Arts et al. [28]. They used an overset grid that was composed of a body-fitted O-grid and a background H-grid with a total of 8.3 × 107 grid points with periodic boundary conditions applied in the span of the vane separated by 0.17 of the chord. The pressure and suction side heat transfer agreed well with the experiments; however, the onset of the transition to turbulence occurred slightly downstream when compared with the measurements. The pressure side heat transfer augmentation under freestream turbulence was also captured very well. The challenge of accurate prediction of the heat transfer was compounded by the fact that the measured turbulence intensities were reported only in one location upstream without reporting the length scale or the dissipation rate. Thus, the accurate turbulence level near the leading edge of the vane was not known.

Collado Morata et al. [23] also conducted structured and unstructured LES study corresponding to the measurements of Arts et al. [28] for exit Reynolds number of 1.15 × 106 and freestream turbulence of 1% and 6% with a numerical domain containing 2.97 × 107 grid points with a spanwise size of 15% of the chord. For the lower turbulence case, an earlier onset of transition is predicted but the pressure side heat transfer agrees very well with the experiments. For the higher freestream turbulence of 6%, the onset of the transition is in good agreement with the experiments but the heat transfer coefficient in the transition region and over the pressure surface is underpredicted. A similar LES study was reported by Gourdain and Gicquel [17] for the same cases as well as the exit Reynolds number of 2.1 × 106 with 6% turbulence intensity which indicates better agreement with the experiments for the lower Reynolds number. However, for the higher Reynolds number, the transition region on the suction side is much shorter than the experiments, and the transition to turbulence on the pressure side was not captured which indicates the sensitivity of the grid resolution in higher Reynolds number flows.

Wheeler et al. [26] performed DNS of the same geometry at Reynolds number of 0.58 × 106 with 1%, 4%, and 6% freestream turbulence on a 6.4 × 108 grid point mesh. The onset of the transition was slightly delayed when compared with the experiments, and the heat transfer augmentation over the pressure side was not captured properly. The spanwise size of the numerical domain was limited to the 10% of the axial chord (concerning computational cost) which enforced a limitation on the maximum turbulent length scale that could be used. Thus, the freestream turbulence decayed much faster in the numerical simulation which is believed to be the main factor affecting the errors in the predictions.

Jee et al. [27] performed LES on the identical geometry at exit Reynolds number of 1 × 106 with 5% and 7.5% inflow turbulence with a coarser grid of 5 × 107 points and a finer grid of 1.4 × 108. Both grids perform similarly and the onset of transition to turbulence is predicted slightly downstream, but the pressure side heat transfer is predicted accurately when compared with a 4% inflow turbulence case in the measurements. Very earlier transition to turbulence is reported in the experiments for the inflow turbulence of 6% which is not captured numerically even for the inflow turbulence of 7.5%.

The most recent study is reported by Garai et al. [24], and the spectral element discontinuous Galerkin method is used to simulate the flow and heat transfer over the same geometry mentioned above at exit Reynolds number of 1 × 106 with low, moderate (7%), and high inflow turbulence of 20%. Predictions of the onset of the transition on the suction side for the inflow turbulence levels of 7% and 20% seem to agree with the reported measurements at 4% and 6% turbulence intensity, respectively. For the pressure side, the predictions of the 7% inflow turbulence agree well with the corresponding measurements but with 4% inflow turbulence and even the highest simulated turbulence level of 20% is not capable of predicting further heat transfer augmentation on the pressure side observed for 6% inflow turbulence.

The predicted location of the transition to turbulence in most of these studies lies in the adverse pressure gradient region near the suction side surface exit. As reported by Jee et al. [27], it is very challenging to capture an earlier onset of transition corresponding to higher inflow turbulences. To capture the observed upstream shift in the onset of transition to turbulence under 6% inflow turbulence, Garai et al. [24] had to increase the inflow turbulence level up to 20%.

The literature cited above shows the promise of the high-fidelity simulations in capturing the transitional behavior and the heat transfer over turbine vanes under elevated freestream turbulence. In the current study, we use wall-resolved LES to study the effect of elevated freestream turbulence on the flow and heat transfer behavior (including boundary layer transition) on an aft-loaded vane at a very low Mach number of 0.045 studied experimentally by Varty et al. [3] and Varty and Ames [11]. The experiments report the measurements of midspan and full surface Stanton number distributions over the vane at three Reynolds numbers of 0.5 × 106, 1 × 106, and 2 × 106 under freestream levels of up to 17.4% and different length scales [3,11]. The measurements reveal the effect of the freestream turbulence on the heat transfer augmentation in the laminar boundary layer over the pressure surface and on the transition to turbulence on the suction side. In an earlier paper, Kanani et al. [29] report the LES results on the pressure side that detail the mechanism of the laminar heat transfer augmentation on the pressure side of the vane at Re = 0.5 × 106 with 0.7%, 7.9%, and 12.4% inflow turbulence. In the current study (Re = 0.5 × 106, Tux = 7.9%), attention is focused on the suction surface, and highly resolved calculations are undertaken to ensure accurate prediction of the transition on the suction surface. In order to achieve the resolution needed, computations were limited to the midspan region covering 47% of the full vane span (24% of the chord). The computational domain chosen in the vane-span direction is larger than high-fidelity simulations in the literature.

The resolved simulations are used to study the mechanism of the transition including the evolution and transport of turbulent spots [3032]. Further, the first- and second-order statistics including mean skin friction and Stanton number distribution, mean velocity, and temperature profiles, components of Reynolds stress tensor, turbulent normal heat flux and turbulent Prandtl number on the suction surface are documented and discussed in detail.

## Computation Details

The selected vane geometry in the linear cascade is identical to the previous experimental and numerical studies conducted by Varty and Ames [11] and Kanani et al. [29]. The current numerical domain includes only one single vane; the periodic boundary conditions separated by 0.773L in the pitch direction enforce the periodicity of the flow in the linear cascade (Fig. 1). Here, L represents the vane true chord and is used as the primary length scale unless otherwise stated. The total endwall-to-endwall span of the vane in the reference experiments is 0.51L [11]. The secondary flows on the suction surface only extend to about 30% of the span from either endwall [3] and hence have minimal effects on the midspan characteristics. To capture the heat transfer distribution in the mid-regions of the 2D extruded vane [11], periodic boundary conditions are applied in the vane-span direction (z-direction in Fig. 1) and are separated by 0.24L (i.e., 47% of the span in the full vane). Over this region, endwall effects are not important (verified by the experimental results in Ref. [3] and a full-span calculation [29]), and the enforcement of periodicity is justified. This approach will be further verified by comparing the current predictions with the reported midspan measurements. The domain length in the span direction is six times larger than the reported turbulent length scale of 0.04L in the experiments.

Fig. 1
Fig. 1

The Reynolds number based on the exit conditions and the chord length is 0.5 × 106 and the exit to the inlet velocity ratio of the current vane geometry is 3.98. The time-averaged velocity at the inlet of the domain is u0 parallel to the global x-coordinate, and the inlet boundary is placed 0.51L upstream of the leading edge. The inflow turbulence level of 7.9% with a length scale of 0.04L is considered [11] for the current simulation.

The inflow turbulence is generated using the synthetic Eddy method (SEM) [33] at the inlet of the numerical domain. The imaginary eddy box contains a total of 1182 synthetic eddies with a length scale of 0.04L in all directions and the velocity distribution
$f={3/2(1−|ζ|)ifζ<10otherwise$
(1)
is imposed on each synthetic eddy where ζ is the distance to the eddy origin scaled with a prescribed turbulence length scale [34]. It should be noted that turbulence intensities reported by Varty and Ames [11] are at the location of the leading edge without the vane in place. Measurements indicate that the turbulence intensity at the leading edge plays the most important role on the vane heat transfer behavior since for the three cases with similar turbulence levels at the leading edge and different turbulence length scales, the Stanton number distributions are very similar (including the onset transition to turbulence). Despite the same level of turbulence at the leading edge, the upstream inflow turbulence and decay rates are different for these cases due to different length scales [35]. This emphasizes the necessity to match the level of turbulence near of the leading edge with the experiments. Therefore, a series of simulations with various inflow conditions are performed in a box domain of the identical span and pitch dimensions as the main simulation to match the experimental conditions but without vane in place. Figure 2 shows the decay of the streamwise component of the inflow turbulence such that Tux = 7.96% at x/L = 0 is achieved in the absence of the vane.
Fig. 2
Fig. 2

The homogeneous Neumann condition on velocity and pressure is imposed at the outlet of the domain which is 0.4L downstream of the trailing edge axially which allows the wake to exit the domain after passing the domain twice in the periodic pitch direction. All walls are treated with the no-slip condition for the velocity and homogeneous Neumann for the pressure. Like the reference experiments, a constant heat flux boundary condition is applied to the vane surface.

The numerical simulation methods are similar to the previously reported study by Kanani et al. [29]. The computations include solving the incompressible Navier–Stokes equations using the PISO algorithm [36] and temperature equations as a passive scalar with a laminar Prandtl number of 0.711 with the OpenFOAM flow code (version 5.0) which is a finite volume solver. All standard codes are used except the SEM inflow turbulence boundary condition which is implemented in-house. All spatial derivatives are discretized with a second-order central scheme. For the convective terms, the central difference scheme is blended with a second-order upwind scheme to ensure the stability of the numerical simulations. A limiter is applied to the central differencing scheme of the convection term in the passive scalar equation to ensure the stability and boundless of the solution. Temporal derivatives are discretized with a second-order backward scheme. The WALE subgrid-scale stress model is used in the large eddy simulations [37] with a constant subgrid-scale Prandtl number of 0.9.

As it will be shown later, the boundary layer edge velocity ue exceeds 3.75u0 on the majority part of the suction side. Considering the total surface distance of the suction surface 1.21L, the estimated time that flow requires to pass the suction surface fully is 0.32L/u0. All variables are time-averaged for about 2.65 L/u0 which is enough for flow to pass the suction side for more than eight times. This corresponds to 23.6 units of δ99,exit/uτ,exit at the suction surface exit where δ99exit is the boundary layer thickness at s/L = 1.205 and is defined as the wall-normal distance where the streamwise velocity reaches 99% of the edge velocity. The uτ,exit is also calculated at s/L = 1.205 and is defined as $uτ,exit=τw,exit/ρ$ with τw,exit being the wall shear stress and ρ as the fluid density. To further enhance the statistical sample, all variables are averaged in the periodic span of the domain. To calculate the second-order statistics, all variables are decomposed to the mean and fluctuation parts. The time-averaged quantities are denoted using italic characters and the fluctuations are specified with the prime symbol. All the vector and tensor quantities are transformed to the local tangent (i.e., streamwise) and normal (wall-normal) coordinate over the vane surface and are denoted by t and n subscripts, respectively.

To resolve the turbulent structures in the proximity of the wall, a highly refined grid needs to be used in all directions [38]. In a typical grid, the spanwise and streamwise grid resolution requirement extends to all regions of the numerical domain and significantly increases the number of grids. Different approaches have been used previously to coarsen the grid away from the wall. Bhaskaran and Lele [22] used an overset grid including a body-fitted O-grid and a background H-grid. In this study, we are using a nested grid approach where grid spacing increases away from the wall both in streamwise and spanwise directions [38,39] which roughly halves the total number of grid points in the current numerical setup. This method has been successfully used in the flat plate turbulent boundary layer simulations by Kravchenko et al. [40].

Two different numerical grids with 15 × 106 and 31 × 106 grid points are evaluated for the accuracy of the predictions. Both grids have identical grid distribution in the streamwise and wall-normal directions and grids are refined in the spanwise direction. In the spanwise direction, the domain contains three different zones with different grid numbers. The number of cells is doubled by approaching the vane and quadrupled very close to the vane such that the maximum boundary layer thickness on the suction side is enclosed in the last level of the refinement. The total span of the domain is discretized with 56 points away from the vane, 110 points in the proximity of the vane, and 220 points close to the vane surface for the 31 × 106 grid points. Similar refinements are done in the streamwise direction such that the near-wall region of the vane is discretized with more than 1800 points on the suction surface. This method helps to achieve near DNS grid resolution requirements inside the boundary layer while taking advantage of the LES capability farther away from the wall outside the boundary layer that significantly reduces the computational power requirements. In the wall-normal direction, grids are clustered toward the wall with an expansion ratio of about 1.1. The distance of the first grid point from the vane surface varies from 3.6 × 10−5L around the leading edge to 2 × 10−5L over both pressure and suction sides, and there are more than 48 grid points inside the boundary layer at the suction surface exit. The near-wall region grid size distributions in the wall units in the wall-normal d+, streamwise s+, and spanwise z+ directions are shown in Fig. 3 for the refined grid. The value of d+ is maximum at the leading edge with d+ 0.4 and decreases further downstream of the suction surface with a minimum value of less than 0.14. The streamwise and spanwise grid sizes are 9 < s+ < 20 and 17 < z+ < 28 throughout the suction surface, respectively. Overall, the grid provides a DNS-like resolution inside the boundary layer, especially in the wall-normal and spanwise directions. The spanwise grid size is about two times of a typical DNS of the turbulent boundary layer [41]. The current near-wall grid resolution is more refined than the LES of Bhaskaran [42], Gourdain [17], and Collado Morata et al. [23].

Fig. 3
Fig. 3

## Validation of the Numerical Solution

Figure 4 shows the predicted and measured time-averaged Stanton number distribution St over the vane as a function of the surface distance s/L.
$St=h/(ρ|uoutflow|cp)$
(2)
$h=qw/(Tw−T0)$
(3)
where qw is the surface heat flux, Tw represents the wall temperature, T0 is the inflow temperature, ρ is the fluid density, and cp is the heat capacity. The stagnation point is located at s/L = 0, and the positive and negative s/L values correspond to suction and pressure surfaces, respectively. The average exit velocity |uoutflow| is calculated by averaging the magnitude of the velocity at the outflow plane. Figure 4 also includes the predictions [29] and measurements [11] for the low turbulence case Tux = 0.7% as a reference for comparison.
Fig. 4
Fig. 4

The Stanton number predictions with 31 × 106 grid point mesh agree very well with the measurements throughout the vane surface and hence the following results reported in this paper corresponds to this grid. Current simulation accurately predicts the observed transition to turbulence in the last portion of the suction side which indicates the accuracy of the current numerical predictions and validates the various aspects of the numerical approach used (numerical schemes, boundary conditions, domain size, mesh resolution, inflow turbulence generation, etc.). The details of the flow and temperature fields including the mean profiles and second-order statistics are presented in the following sections.

## Results and Discussion

In this paper, we focus our attention on the suction surface results and the boundary layer transition behavior. The details of the pressure side flow and heat transfer have been presented in an earlier paper by Kanani et al. [29] where the laminar heat transfer augmentation by the freestream turbulence is discussed. They observed that the wrapped vortex tubes around the leading edge perturb the boundary layer and form high- and low-speed streaks which retain their characteristics throughout the pressure surface and enhance the heat transfer coefficient. However, the flow was not transitional or turbulent, and the time-averaged quantities were characteristics of a laminar boundary layer flow. Although the pressure side heat transfer predictions agreed very well with the measurements, the numerical grid used in that study [29] was not fine enough to accurately capture the location of the transition to turbulence on the suction side in the experiments. As noted earlier, to alleviate the high computational cost required to simulate the full vane passage with the refined grid needed, 47% of the vane-span centered around the midspan with periodic boundaries in the span is used in this study based on the assumption that the endwall effects are limited to the corner regions. The validation of the numerical simulations in Fig. 4 confirms this approach. Further analysis is then performed to characterize and document the flow and temperature fields over the suction side featuring the transition to turbulence.

### Pressure and Edge Velocity Distributions.

To characterize the pressure gradient, variation of the pressure coefficient Cp ≡ (pmax−p)/(0.5ρ|uoutflow|2) with the surface distance is shown in Fig. 5. There is a strong favorable pressure gradient from the leading edge to s/L ≈ 0.3 on the suction side and, further downstream, the magnitude of the favorable pressure gradient decreases until it vanishes completely at s/L ≈ 0.6. For the surface distance of s/L > 0.6, there is only a mild adverse pressure gradient on the boundary layer starting at s/L ≈ 0.9 and extending up to the trailing edge.

Fig. 5
Fig. 5

Figure 6 shows the variation of the boundary layer edge velocity ue/u0 with the surface distance. As expected, the edge velocity distribution follows the pressure coefficient over the vane. Flow accelerates with a greater extent from the leading edge to s/L = 0.3, and afterward, there is mild flow acceleration that continues to s/L = 0.6. Close to the trailing edge, the flow decelerates due to the adverse pressure gradient beyond 0.9L.

Fig. 6
Fig. 6

### Freestream Turbulence.

As mentioned earlier, the inflow turbulence condition is adjusted to match the measured level of the turbulence at the leading edge location without vane in place (see Fig. 2). Identical inflow turbulence is imposed in the main simulation and the streamwise component of the turbulence decays similar to the a priori case up to x/L ≈ −0.1 but increases toward the stagnation point. This enhancement in the turbulence intensity is due to the stretching and tilting of the flow structures due to high strain flow near the leading edge, as observed and discussed by Bhaskaran and Lele [21] and Chowdhury and Ames [35], which enhances axial and spanwise velocity fluctuations. Figure 7 shows the energy spectra of the velocity signal at the stagnation point without vane in place. The inertial subrange is resolved in the current calculations and is shown by the −5/3 slope line.

Fig. 7
Fig. 7
Figure 8 shows the calculated autocorrelation functions Ruu and Rvv at the stagnation point when the vane is removed. The autocorrelation functions are defined as
$Ruu(τ)=u′(t)u′(t+τ)¯u′(t)2¯$
(4)
$Rvv(τ)=v′(t)v′(t+τ)¯v′(t)2¯$
(5)
Fig. 8
Fig. 8

The integral length scale calculated by integrating the Ruu function and using the Taylor hypothesis is 0.04L which matches with the length scale measured experimentally.

Due to the variable boundary layer edge velocity, there is no unique velocity scale to calculate the local turbulence intensity at the edge of the boundary layer. The choices of the velocity scales are the inlet velocity u0 and edge velocity ue. The local turbulence intensity at the edge of the boundary layer is calculated using both scales and are plotted in Fig. 9.
$Tu0=[(ut′ut′¯+un′un′¯+w′w′¯)/3u02]0.5$
(6)
$Tue=[(ut′ut′¯+un′un′¯+w′w′¯)/3ue2]0.5$
(7)
Fig. 9
Fig. 9

The magnitude of the scaled velocity fluctuations at the edge of boundary layer ($Tue$) does not vary significantly throughout the suction side (s/L > 0.2); however, in the proximity of the leading edge, the apparent turbulence intensity is higher due to the lower edge velocities and exhibits a sharp decay till s/L of 0.2.

### Bypass Transition to Turbulence.

For the low turbulence case at Tux = 0.7% [11] (see Fig. 4) and exit chord Reynolds of 0.5 × 106, no transition to turbulence is observed on the suction surface. By increasing the freestream turbulence level to 7.9%, the measured Stanton number indicates an onset of transition to turbulence at s/L ≈ 0.8. The onset of the transition moves upstream for higher turbulence cases [11].

Bypass and separated-flow transition on the turbine vanes have been observed in several experimental studies [7,10,12,22,43,44], and different numerical studies attempted to accurately predict the onset of the turbulence [17,21,23,24,26,27]. The current vane geometry does not impose a strong adverse pressure gradient and hence separated-flow transition is not observed. For the lowest turbulence case, Kanani et al. [29] indicate a very small reverse flow starting at s/L ≈ 1.11 which explains the rise of the Stanton number at the suction surface exit for the low turbulence case. However, no separation or reverse flow is observed in the mean quantities of the current simulation at the higher freestream turbulence. Hence, the increase in the Stanton number should be attributed to the bypass transition. In the bypass transition, the perturbed boundary layer breaks down to turbulence in the form of turbulent spots. The turbulent spot formation is associated with the secondary instabilities that grow in the presence of the Klebanoff modes [30,4547].

Figure 10 shows the contours of the instantaneous wall-normal velocity at d/L = 0.002 or d/δ99exit 0.11. This corresponds to d+ ≈ 42 using the exit friction velocity uτ,exit calculated at s/L = 1.206 as the velocity scale. Five consecutive time snapshots are shown in the figure and are separated by time intervals of 2.67δ99,exit/ue,exit or 0.11δ99,exit/uτ,exit. Turbulent spots form intermittently both in space and time in the transition region and grow and merge further downstream. Various methods have been used by different researchers to discriminate the laminar and turbulent flow to calculate the intermittency function. An overview of the methods is provided by Hedley and Keffer [48]. A laminar-turbulence discrimination approach similar to Nolan and Zaki [49] is performed here on each snapshot by calculating the detector function and by sensitizing and applying an arithmetic low-pass filter [49]. The threshold is fixed for all the snapshots but is adjusted to determine the laminar-turbulent boundary at the first snapshot. The detected boundaries of the laminar and turbulent flow are shown in the same figure with white lines. Secondary instabilities are triggered at surface distances 0.6 < s/L < 0.7 and form the incipient spots close to the wall. In this range of surface distances, the edge velocity does not vary significantly, however, as it can be seen in Fig. 10, the front boundary of the turbulent spots propagates downstream faster than the rear boundary which was also observed by Jacob and Durbin [50] and indicates that the spots are propagating faster than the averaged velocity of the spot.

Fig. 10
Fig. 10

Several routes to turbulence have been reported in the literature [3032,51] and are classified into two groups of outer mode and inner-mode transition mechanisms [52]. The inner and outer refer to the location where the breakdown to turbulence occurs when compared with the boundary layer thickness. Outer mode transition mechanism (either sinuous or varicose modes) is located near the edge of the boundary layer while the breakdown occurs closer to the wall in the inner-mode transition to turbulence. The outer mode is mostly observed in zero pressure gradient boundary layers [31] and is attributed to the lift-off of the low-speed streaks close to the edge of the boundary layer where they get exposed to the higher frequency freestream turbulence [50]. The inner-mode transition mechanism was first reported by Nagarajan et al. [32] and was explored further in detail by Vaughan and Zaki [52]. The formed wave packet in the inner mode is located closer to the wall and the breakdown occurs at the intersection of the high- and low-velocity streaks. The inner-mode breakdown is observed to be a predominant route to turbulence in flows with a sufficiently adverse pressure gradient [31,45] or the presence of a large leading edge in the study of Nagarajan et al. [32]. The average wall distances of the reported inner and outer modes are 0.3 and 0.7, respectively [31]. The velocity contours reported in Fig. 10 correspond to d/δ99(s/L=0.6) 0.31 which suggests the presence of the inner-mode transition to turbulence for s/L > 0.6 even though the flow is not under adverse pressure gradient for 0.6 < s/L < 0.8. However, at s/L = 0.6 the favorable pressure gradient vanishes which suggests that the flow acceleration has a stabilizing effect.

The zoomed view of the incipient spot located at z/L = 0.055 and s/L = 0.7 (see Fig. 10, topmost contour) is shown in Fig. 11. The checkered pattern in the spanwise velocity perturbations near the wall reported by Nagarajan et al. [32] and Zaki [30] is evident in Fig. 11(b). The shape of the wall-normal velocity distribution suggests varicose inner instability which was also observed in the previous studies [31,52].

Fig. 11
Fig. 11

Figure 12 shows the distribution of the instantaneous Stanton number over the suction surface. The turbulent spot boundaries are also included in this figure. The effect of the incipient spots on the surface heat transfer is not evident in Fig. 12(a). However, the zoomed view of the turbulent spot located at the z/L ≈ 0.09 and s/L ≈ 0.8 indicates the elevated heat transfer on the surface but over a smaller portion (Fig. 12(b)). This is expected since the boundary of the detected spot is at wall distance of d/L = 0.002. On the other hand, at z/L ≈ −0.06 and for s/L > 0.8, the elevated Stanton number is at a lower s/L value relative to the detected boundary of the turbulent spot. This is perhaps because the three-dimensional turbulent spot has already passed the regions of elevated St, but due to the lower convective velocities near the wall, there is a lag in the recovery of the surface heat transfer. The wall shear stress (Fig. 13) shows a similar pattern to the Stanton number in overlapping regions.

Fig. 12
Fig. 12
Fig. 13
Fig. 13

### Surface Skin Friction and Heat Transfer.

Figure 14 shows the distribution of the instantaneous and time-averaged skin friction. The predicted skin friction distribution by Kanani et al. [29] for a two-dimensional simulation is plotted as a reference laminar solution. Skin friction initially increases from the leading edge to s/L ≈ 0.2 and decreases afterward until the onset of the transition to turbulence at s/L = 0.64. The instantaneous skin frictions at z/L = 0.04 and z/L = −0.008 at the identical time instance as the first snapshot in Fig. 10 are also plotted in this figure. The lateral locations where the instantaneous skin friction is plotted are identified with dashed lines in Fig. 10. The instantaneous values show large excursions in excess of the averaged value in local s/L regions indicative of a turbulent spot at these locations.

Fig. 14
Fig. 14
Due to the unavailability of the local boundary layer edge velocity, the Stanton number in the experiments is calculated using the cascade exit velocity [3] (see Fig. 4). To compare the Stanton number values with available correlations for the zero-pressure gradient laminar and turbulent boundary layers, the Stanton numbers (St = Nu/Re.Pr) shown in Fig. 15 are scaled with the local Reynolds number based on the boundary layer edge velocity ue. The corresponding laminar and turbulent Stanton number over a flat plate is given by [53]
$Stlaminar=0.453Pr−0.66Res−0.5$
(8)
$Stlturbulent=0.03Pr−0.4Res−0.2$
(9)
where Res = sue and s is the surface distance from the stagnation or the leading edge. As seen in Fig. 15, the laminar Stanton correlation over the flat plate agrees reasonably well with the predictions especially for the low inflow turbulence (dashed line). The predicted St at the elevated turbulence level (7.9%) follows the trend of the laminar flat plate correlation till the transition point although the predicted values are about 20–30% higher due to the freestream turbulence. The measured and predicted Stanton number at the elevated turbulence indicates that the fully turbulent boundary layer state is not reached, and the flow is still transitional at the exit of the suction surface. The heat transfer coefficient is augmented under freestream turbulence even before the transition region which agrees with the measurements and previous studies in the literature [7,43].
Fig. 15
Fig. 15

### Transition Onset Momentum Thickness.

To identify the transition onset momentum thickness, a plot of the momentum thickness with the surface distance is shown in Fig. 16. The transition onset momentum thickness is calculated as Reθt = 325 which is higher than the predictions by the available correlation by Mayle [5] and Menter [54]. For the freestream turbulence level of 3.1% based on the local boundary layer edge velocity (see Fig. 9), Mayle and Menter’s correlations predict the transition onset momentum thickness of Reθt = 197 and Reθt = 215, respectively. The acceleration parameter of zero is considered the correlation provided by Menter since there is no significant pressure gradient at the onset of transition.

Fig. 16
Fig. 16

### Time-Averaged Velocity and Temperature Profiles.

The mean velocity profiles scaled with friction velocity as a function of inner wall coordinate are shown in Fig. 17. With increasing s/L, the shape of the mean velocity profiles changes from laminar toward the fully turbulent boundary layer profile but does not reach the fully turbulent log-law profile at the trailing edge. This observation is consistent with the Stanton number trend shown in Fig. 15. In related DNS simulations of the transition to turbulence, Wu and Moin [55] and Madavan and Rai [53] also report a similar trend.

Fig. 17
Fig. 17
In the zero-pressure gradient turbulent boundary layers, the temperature profile also follows a law of the wall. Analogous to the inner scaling of the velocity by the friction velocity $uτ=τw/ρ$, the temperature is scaled with the friction temperature $Tτ=q˙w/(ρcρuτ)$ and is defined as [56]:
$T+=(Tw−T)/Tτ$
(10)
Above the viscous sublayer and buffer layer, the zero-pressure gradient turbulent boundary layer temperature profile follows a log-law profile as [56]:
$T+=1KTln(y+)CT$
(11)
where KT ≈ 0.48 and CT = 3.9 for Pr = 0.71. Figure 18 shows the temperature profiles scaled with the friction temperature in the inner wall coordinate. Contrary to the velocity profiles, where transition occurred for s/L > 0.64, the temperature profiles show the characteristics of the transitional boundary layer for s/L > 0.8 which was also observed in the Stanton plots (see Figs. 4 and 15) and suggest a lag between temperature and flow in the transition region.
Fig. 18
Fig. 18

### Reynolds Stress Tensor.

Components of the Reynolds stress tensor are calculated and presented in this section. The profiles of utrms scaled with the edge velocity ue as a function of inner and outer wall coordinates are shown in Figs. 19 and 20. In the transition region (i.e., s/L > 0.64), the peak of utrms profiles increases and moves toward the wall while a secondary peak grows for larger surface distances. In the late transition region s/L > 1.0, the peak value is located at d+ = 16 which is close to the value of 17 reported in the transitional region by Madavan and Rai [53] for the case of transition to turbulence over a flat plate under freestream turbulence. For the zero pressure gradient flows under low freestream turbulence, the peak is reported to be closer to the wall at y+ ≈ 13 [41,55]. The peak of utrms exceeds the fully turbulent value (see utrms profile at s/L = 1.065 in Fig. 19) and increases to 17.3% of the edge velocity which is consistent with the experimental reports of the transitional flow under freestream turbulence where Wang et al. [57] and Kim and Simon [58] reported a value of 17.5%. The utrms decreases further downstream with increasing surface distance. The evolution of the peak of all three components of the turbulence intensity is shown in Fig. 21. The maximum value of the peak utrms is located at s/L ≈ 1.06. The excess utrms in the transition region compared to the fully turbulent boundary layer is observed in several numerical and experimental studies of the transition to turbulence [53,55,5961]. According to Kim and Simon [58], this is due to a significant change in the mean values of the velocity in the laminar/turbulent intermittent flow (see Fig. 8 in Ref. [58]) and hence the measured/calculated velocity fluctuations are greater than the actual value of the fluctuation in each type of the flow. The observed plateau in the Reynolds stress growth rate in the beginning of the transitional region (see Fig. 21, 0.64 < s/L < 0.84) is potentially due to the mismatch between the freestream turbulence length scale and the boundary layer thickness as explained by Fransson et al. [62] and is the length which is required for the adjustment, i.e., receptivity distance. Simulations of Ovchinnikov et al. [59] support this explanation since the slower growth in the Reynolds stress was more evident in the case with larger turbulence length scales.

Fig. 19
Fig. 19
Fig. 20
Fig. 20
Fig. 21
Fig. 21

Figures 22 and 23 show the profiles of unrms and wrms as a function of outer wall coordinate d/δ99 in the transition region. Contrary to the utrms, the near-wall peak of these components monotonically grows with surface distance (see Fig. 21) which is consistent with previous reports [53,55]. By increasing s/L, peaks move slightly toward the wall.

Fig. 22
Fig. 22
Fig. 23
Fig. 23

All three-components diagonal components of the Reynolds stress scaled with the friction velocity as a function of inner wall coordinate d+ are shown in Fig. 24. Unfortunately, these profiles cannot be quantitatively compared with any reported data in the literature since the boundary layer is still transitional and most of the reported profiles in the literature for the transitional flows are at different Rex values which may not directly relate to this problem due to unique acceleration profile near the leading edge. However, as discussed earlier, all the reported profiles show the characteristics of the transitional boundary layer similar to those reported in the literature.

Fig. 24
Fig. 24

Figure 25 shows the Reynolds shear stress profiles in the transition region as a function of inner wall coordinate d+. The behavior of the Reynolds shear stress is different for transition regions with or without freestream turbulence. The DNS of Moin and Wu [55] under low freestream turbulence indicates that the peak in the Reynolds shear stress profile (scaled with friction velocity) increases almost monotonically in the transition region and reaches the value of about 0.9. However, both DNS and experimental measurements in transition under freestream turbulence indicate a rapid increase in the Reynolds shear stress to values higher than one in the transition region which follows by the decrease in the peak value until it reaches to a value of slightly lower than one in the fully turbulent region [53,57,63]. Our simulation results are in line with the latter condition, and the maximum peak of the Reynolds stress profile of 1.34 occurs at d+ = 64 which, interestingly, are similar to the values obtained by Madavan and Rai through DNS (i.e., the peak of 1.4 at d+ ≈ 71). Considering the differences in the current simulation and the mentioned study, these similarities in the predictions are very interesting. However, it should be noted that although their study is conducted with the similar freestream turbulence level (∼3%), the quantities obtained here may not be compared directly with their case because it was done for a flat plate, the data is only reported at three stations in the transition region (i.e., beginning, middle and end of transition region), and finally, there is no straightforward method to identify a similarity parameter.

Fig. 25
Fig. 25

In the pre-transition region (e.g., s/L = 0.4) a negative value of the Reynolds shear stress is observed for the wall distance of d+ < 23 or d/δ99 < 0.33 with the maximum negative value of −0.02 at d+ ≈ 14 or d/δ99 0.2. A similar observation is evident in the reported measurements by Zhou and Tang [63] in the pre-transitional region. However, other DNS simulations of transition to turbulence under freestream turbulence (e.g., by Madavan and Rai [53]) do not show this behavior. The reason for this anomaly is not clear and needs further investigation to identify the possible physical mechanism.

### Turbulent Heat Flux.

The wall-normal component of the turbulent heat flux $u′nT′¯$ scaled with the wall heat flux is shown in Fig. 26. Like the Reynolds shear stress profiles, the peak of the calculated $u′nT′¯$ profile exceeds the value of one in the transition region and negative values are also observed in the pre-transitional region. This anomaly was observed in several experimental studies [6365]. Madavan and Rai [53] performed DNS simulations to examine this anomaly but no evidence of negative values of turbulent normal heat flux was observed and attributed the experimental observations of negative flux to the measurement techniques. However, in the numerical simulations of Kasagi et al. [66] for turbulent boundary layer with iso-heat flux, negative values of the turbulent normal heat flux are also observed especially for higher Pr values and are attributed to the phase lag between development the thermal sublayer and the vortical fluid motions. Also, looking closely at the data reported by Zhou and Wang [63], the negative flux values are also observed but only very close to the wall (y/δ ≈ 0.16) and for the pre-transition region (corresponds to Rex < 7.2 × 104 in their experiment). No evidence of the negative values is observed in the transition and fully turbulent regions.

Fig. 26
Fig. 26

### Turbulence Prandtl Number.

Turbulent Prandtl number defined as
$Pr=ut′un′¯/∂ut∂nun′T′¯/∂T∂n$
(12)
is shown in Fig. 27 as a function of the inner wall coordinate d+ at the exit of the suction surface s/L = 1.21. Although the behavior of the turbulent Prandtl number in the turbulent boundary layers is studied extensively [6771], there are only limited reported data on the turbulent Prandtl number distribution inside the boundary layer in the transitional region [53,63,7274]. With isothermal boundary condition, the value of Prt in the fully turbulent boundary layer approaches to a value greater than one at the wall [67,69]. However, for the isoflux boundary condition [53,67], Prt is shown to reduce rapidly near the wall. The results of our simulations show that the calculated Prt with isoflux boundary condition in the transition region approaches zero at the wall. The same behavior was observed in the pressure side of the vane as reported by Kanani et al. [29]. The value of Prt decreases away from the wall which is consistent with the calculated Prt by Madavan and Rai [53] in the transition region.
Fig. 27
Fig. 27

## Concluding Remarks

This paper presents the capability and accuracy of wall-resolved LES in the vane passage flows with high inflow turbulence at exit Reynolds number of 0.5 × 106 and inflow turbulence of 0.7% and 7.9%. The Stanton number predictions on both the suction and pressure sides of the vane were compared well with the measurements of Varty and Ames [11]. On the suction side, the onset of the transition to turbulence and its effect on the distribution of the heat transfer coefficient in the transition region are predicted accurately.

The formation and evolution and growth of the turbulent spots are captured on the suction side and shown as a time series. To identify the route to turbulence, the breakdown to turbulence in the incipient turbulent spot was studied, and the inner mode was identified as the mechanism responsible for the transition which was also previously observed in the transition to turbulence downstream of the large leading edge and adverse pressure gradient flows [31,32].

The comparison of the skin friction and Stanton number indicated a lag on the effect of the transition to turbulence such that the skin friction starts increasing after s/L = 0.643 but the Stanton number distribution shows characteristics of transition only after s/L = 0.8. The same behavior was observed in the velocity and temperature profiles. Velocity profiles scaled with friction velocity as a function of inner wall coordinate show the characteristics of transitional flow after s/L = 0.643 where this is delayed to s/L = 0.8 for temperature profiles. All the mentioned quantities indicate that the fully turbulent boundary layer is not achieved yet and the flow is still transitional at the exit of the suction surface.

The components of the Reynolds stress tensor are documented and qualitatively agree with the previous studies. The utrms and $u′tu′n¯$ profiles show a peak higher than those expected in the fully turbulent boundary layer similar to previous studies [53]. This is perhaps because the boundary layer intermittently switches between laminar and turbulent states in the transition region. If conditionally averaged, the mean velocity of the turbulent state is higher than those corresponding to the laminar state. However, a simple averaging over time leads to a time-average velocity which is between the mean value corresponding to each state. Thus, the fluctuations are overpredicted, which contributes to the excess of the quantities compared to the fully turbulent flow. The same behavior is observed for the turbulent normal heat flux where it exceeded the wall heat flux value in the transition region. Anomalous behavior in the Reynolds shear stress and turbulent heat flux is observed in the pre-transitional region where a slight region of negative values occurred near the surface. The calculated turbulent Prandtl number shows behavior similar to those reported earlier in the transitional and turbulent boundary layer with isoflux boundary conditions. It approaches zero at the wall and rises to about 1 in most of the boundary layer.

## Acknowledgment

This work was sponsored by a grant from the US Department of Energy’s University Turbine Systems Program (UTSR) with Dr. Robin Ames as the Project Monitor. Their support is gratefully acknowledged. Computational resources were provided by the National Science Foundation’s (NSF) Extreme Science and Engineering Discovery Environment (XSEDE) via a grant number ACI-1053575. The views expressed in the article are those of the authors and do not reflect the official policy or position of the Department of Energy or the U.S. Government.

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