## Abstract

The aerodynamic efficiency of turbomachinery blades is profoundly affected by the occurrence of laminar-turbulent transition in the boundary layer since skin friction and losses rise for the turbulent state. Depending on the free-stream turbulence level, we can identify different paths toward a turbulent state. The present study uses direct numerical simulation as the primary tool to investigate the flow behavior of the low-pressure turbine blade. In the simulations, the flow past only one blade is computed, with periodic boundary conditions in the cross-flow directions to account for the cascade. Isotropic homogeneous free-stream turbulence is prescribed at the inlet. The free-stream turbulence is prescribed as a superposition of Fourier modes with a random phase shift. Two levels of the free-stream turbulence intensity were simulated ($Tu=0.19%$ and $5.2%$), with the integral length scale being 0.167c, at the leading edge. We observed that in the case of low free-stream turbulence on the suction side, the Kelvin–Helmholz instability dominated the transition process and full-span vortices were shed from the separation bubble. Transition on the suction side proceeded more rapidly in the high-turbulence case, where streaks broke down into turbulent spots and caused bypass transition. On the pressure side, we have identified the appearance of longitudinal vortical structures, where increasing the turbulence level gives rise to more longitudinal structures. We note that these vortical structures are not produced by Görtler instability.

## Introduction

Numerical simulations of fluid flow have become a necessary part of the turbomachinery design process. An essential aspect of turbine aerodynamic design is understanding the flow transition process. Although the flow around turbine blades is highly turbulent and unsteady, the flow next to the surfaces may be either laminar or turbulent. The transition process in turbomachinery flows can be triggered in many different ways due to different geometrical and flow parameters like the level of free-stream turbulence intensities, the wide range of Reynolds numbers, particular pressure gradient distributions, curvature effects and interactions with incoming wake or flow separation. The location of the onset and the extension of laminar-turbulent transition are crucial for the overall efficiency and performance of devices since they determine drag and lift forces. Relative portions of the blade surface covered by the laminar, turbulent, and separated flow are directly correlated to the losses. Especially in the low-pressure turbine applications, understanding of the transition process is of significant practical interest since the behavior of the boundary layer largely determines the overall efficiency of a low-pressure turbine.

Investigation of transitional flows in the low-pressure turbine has been performed by several researchers, both in experimental and numerical works. A clear understanding of multiple effects during the real operation of low-pressure turbine blades, such as the elevated level of the free-stream turbulence or wake passage, demands more in-depth knowledge of the dynamics characterizing both the separated flow and the bypass type of transition processes. A detailed characterization of the coherent structures that can penetrate inside the boundary layer upstream of the separation position may increase the knowledge and understanding of the complicated transition mechanisms. Studies have shown that in the case of laminar boundary-layer separation, velocity fluctuations are amplified at the inflection point of the velocity profile, which in turn induce the formation of roll-up vortices [1–4].

A large number of works can be found in the literature describing the dynamics of streaky structures in the Blasius boundary layer, see, e.g., Refs. [5,6]. In work [7], large-eddy simulations (LES) have been carried out with different free-stream turbulence (FST) levels, where the authors clearly show the formation of the streaky structures at elevated FST, suppressing the laminar separation appearing in the case of low FST. They also show that only low-frequency disturbances can penetrate inside the boundary layer. Velocity fluctuations, once generated inside the boundary layer, grow in a transient way causing the formation of energetic low- and high-speed streaks. These structures affect the pre-transitional (still laminar) part of the boundary layer and trigger transition only after their breakdown. Their role in the transition process in turbine-like conditions, for both steady and unsteady calculations, is described in detail in Ref. [8]. The breakdown of the streaky structures occurs as a consequence of sinuous and/or varicose instability processes. Experimental and numerical works such as Refs. [9–13] give a statistical representation of these phenomena in terms of streak spacing, the wavelength at the breakdown, amplitude threshold level of the streak as well as self-similarity properties. However, data available in the literature for adverse pressure gradient conditions, characterizing the real operation of low-pressure turbine blades, are very scarce. Michálek et al. [14] report examples of detailed experiments on low-pressure turbines. There are recent high-fidelity computations that focus on data analysis of loss quantification. These recent works [15–17] highlighted some typical developments on characteristics of the boundary layer structures. However, since physics is highly complex further efforts are needed to understand it better. Detailed knowledge of the effects induced by different FST levels on the generation of coherent structures inside the boundary layer of low-pressure blades may further contribute to understanding their role under unsteady inflow conditions.

Here, we investigate the role of free-stream turbulence on the dominant transition mechanism on a low-pressure turbine blade. Three-dimensional simulations have been performed with two levels of perturbations added at the inflow of the domain. This paper discusses the numerical set-up and presents the results obtained from these direct numerical simulations.

## Flow Configuration and Methods

### Numerical Approach.

**u**= (

*u*,

*v*,

*w*) represents the instantaneous streamwise, wall-normal and spanwise velocity components,

*p*is the pressure, and Re is the Reynolds number. Nek5000 is based on the spectral element method (SEM) by Patera [19], which has the advantage of combining the geometric flexibility of the finite element method with the high accuracy of spectral methods. Following the $PN\u2212PN\u22122$ [20] formulation, we perform the spatial discretization in each element where velocity is represented by high-order Lagrange interpolants through the Gauss-Lobatto-Legendre (GLL) quadrature points. In contrast, the pressure is represented on the staggered Gauss–Legendre (GL) quadrature points. The equations are advanced in time using a third-order conditionally stable backward differentiation and extrapolation scheme (BDF3/EXT3), employing an implicit treatment of the viscous term and explicit treatment of the non-linear term. For removing aliasing errors, we apply the over-integration.

### Mesh Design and Boundary Condition.

Figure 1 is a schematic of the computational domain that is following the experiments by Lengani and Simoni [21]. The test section was constituted of a seven-blade large scale planar cascade, representative of low-pressure turbine blade profiles. In the simulations, the flow past only one blade is computed, with periodic boundary conditions in the cross-flow direction to account for the cascade. Chord length *c* is selected as the characteristic length scale and mean inflow speed *U*_{in} as the characteristic velocity. Based on these reference scales, the simulation Reynolds number is Re = 40,000. One blade pitch which is *g* = 0.685*c* separates the top and bottom computational boundaries.

The streamwise extent of the domain is 1.8*c*, while the spanwise size is 0.685*c*. The spanwise extent is approximately four times the integral length scale at the leading edge and 11 times the maximum boundary-layer thickness for the case of low turbulence intensity. The inflow plane is at *x*/*c* = −0.5, at which a mean velocity (*U*_{in} cos(*α*),*U*_{in} sin(*α*), 0) is prescribed using Dirichlet boundary condition with super-imposed Fourier modes, where the inflow angle is *α* = 40 deg. At the outflow *x*/*c* = 1.3, we apply stress-free outflow boundary conditions. In the spanwise direction, we enforce periodic boundary conditions. The no-slip boundary is applied on the surface of the blade. In Table 1, we report the main geometrical parameters of the cascade.

Chord length | c = 120 mm |

Aspect ratio | h/c = 2.5 |

Pitch-to-chord ratio | g/c = 0.685 |

Inlet metal angle | $\alpha m1=43.95$ |

Outlet metal angle | $\alpha m2=\u221265.8$ |

Trailing edge thickness to throat ratio | t/o = 0.046 |

Zweifel number | Zw = 1.052 |

Chord length | c = 120 mm |

Aspect ratio | h/c = 2.5 |

Pitch-to-chord ratio | g/c = 0.685 |

Inlet metal angle | $\alpha m1=43.95$ |

Outlet metal angle | $\alpha m2=\u221265.8$ |

Trailing edge thickness to throat ratio | t/o = 0.046 |

Zweifel number | Zw = 1.052 |

In order to properly resolve the turbulent flow, Schlatter et al. [22] suggest the mesh criteria of Δ*x*^{+} < 10, Δ*y*^{+} < 1 (at the wall), and Δ*z*^{+} < 5, where *x*, *y*, and *z* are the streamwise (tangential), wall-normal and spanwise coordinates. By following this criterion, the mesh was build with characteristics stated in Table 2.

Extent | Streamwise resolution | Wall-normal resolution | Spanwise resolution |
---|---|---|---|

$xc<0.02$ | Δx^{+} = 0.75 − 1.65 | ||

$0.02<xc<0.9$ | Δx^{+} = 1.65 − 17 | $\Delta ywall+=0.7$ | Δz^{+} = 6 |

$xc>0.9$ | Δx^{+} = 0.3 − 4.2 |

Extent | Streamwise resolution | Wall-normal resolution | Spanwise resolution |
---|---|---|---|

$xc<0.02$ | Δx^{+} = 0.75 − 1.65 | ||

$0.02<xc<0.9$ | Δx^{+} = 1.65 − 17 | $\Delta ywall+=0.7$ | Δz^{+} = 6 |

$xc>0.9$ | Δx^{+} = 0.3 − 4.2 |

The spectral element mesh is structured and orthogonal near the turbine blade surface. The final mesh is composed of a total of ∼226 million points. Figure 1 illustrates the final mesh, with only spectral elements plotted. The grid resolution at the blade surface varies with the streamwise location, with the criterion from Table 2. Moreover, 35 points are positioned below *y*^{+} = 10 region in the direction away from the blade surface. It can be noticed that there is a slight deviation from the suggested criterion in the streamwise location, with values of Δ*x*^{+} = 17. Streamwise spacing reaches this value for two elements in the mid-chord, where we are confident that we do not have a turbulent boundary layer, so this limit becomes flexible. The small deviation is also present in the spanwise direction, where for the slightly higher value of Δ*z*^{+}, without losing the accuracy, we save computational time, by reducing the total number of elements.

All these values are based on the friction velocity from the simulation with the highest turbulence intensity. Scaling was provided in the viscous units, where *l** = *ν*/*u*_{τ} is the viscous length, *ν* is the fluid kinematic viscosity, and $u\tau =\tau w/\rho $, with *ρ* being the fluid density. Furthermore, we replicate this grid in the spanwise direction for the three-dimensional simulation. Extra care must be taken into account for the periodic boundary condition, i.e., top and bottom side because the streamwise position of the points must coincide as can be seen in Fig. 1. The final resolution utilizes a polynomial of ninth order.

### Free-Stream Turbulence.

Isotropic, homogeneous free-stream turbulence is prescribed at the inlet to introduce disturbances to the flow-field. The free-stream turbulence is prescribed as a superposition of Fourier modes with a random phase shift. We prescribe the maximum and minimum amplitudes of the wavenumber vector. The wavenumber space is divided into a set of concentric spherical shells. Points are chosen randomly on each shell, where the location of each point represents the three-dimensional components of the wavenumber vector. The amplitude of the free-stream modes on each spherical shell is scaled using the von Kármán spectrum. More about this method for generating free-stream turbulence for simulations of flat-plate boundary layers can be found in Refs. [6,13,23], for wind turbine simulations in Ref. [24] and wing simulation in Ref. [25].

Two levels of the free-stream turbulence intensity were simulated Tu = {0.19, 5.2} $%$ (further in the text they are referred to as low FST and high FST cases, respectively) with the integral length scale *L* = 0.167*c*, measured 0.083*c* upstream of the leading edge. Figure 2 shows the downstream decay of turbulence intensity for both cases, upstream of the blade. Note that values for turbulence intensity are scaled with respect to the maximum of turbulence intensity. We can use the following form Tu = *A*(*x* + *B*)^{C}, to express the evolution of the turbulence intensity. The best fit for simulation is achieved when *C* = −0.95, which is in agreement with the decay rate for isotropic turbulence. Deviation from the power decay at the end of the domain is happening because of the influence of the nearby leading edge and related potential pressure field, which are making free-stream turbulence less homogeneous. To assess the isotropy of the imposed free-stream turbulence, we can compare the decay rate of the different velocity fluctuations *u*, *v*, and *w*. Figure 3 shows the evolution of the averaged fluctuation intensity, *u*_{rms}, *v*_{rms}, and *w*_{rms} over the flow domain upstream of the blade. Within the main portion of the streamwise extent, the maximal variance of the RMS values is below 5$%$.

## Results

### Loading Distributions.

Figure 4(a) shows a comparison with the experiments of the pressure distributions around the blade surface for both high and low free-stream turbulence intensity levels. In the low FST case, the boundary layer starts to transition from the laminar to the turbulent state via a separation bubble as it is indicated by the changing pressure gradient close to the trailing edge. From this plot, one can identify the peak on the suction surface, after which the slope decreases as the flow is experiencing an adverse pressure gradient. Eventually, the adverse pressure gradient is too strong, and separation occurs. With increasing turbulence intensity level, the detectable laminar separation bubble disappears almost completely.

The higher free-stream turbulence intensity conditions promote a slight increase in skin friction, which causes a delay of the boundary-layer separation. In Fig. 4(b), the skin friction curve is plotted for both cases and shows that we have somewhat higher values of skin friction in high FST case than in the low FST case. We can also observe the extent of negative skin friction coefficient near the trailing edge in the low FST case, which confirms the occurrence of boundary layer separation. Interestingly, the boundary layer fails to reattach at the blade trailing edge, consistently with the experimental results reported in Ref. [21]. On the pressure side of the blade, we can also see that for the low FST case we have a greater extent of separation length, indicated by the negative skin friction between $20%$ and $50%$ of the chord.

The loss coefficient *c*_{pt} is compared with the experiments and reported for the low and high FST cases in Fig. 5. The largest values of *c*_{pt} are found in the wake of the blades. For the low FST case, the wake total pressure defect is significantly larger than for the high FST case, due to the non-reattaching state of the boundary layer for this latter condition. Interestingly, outside of the wake region, we can also see that the value of *c*_{pt} is different from zero for the high FST case. This shows us that the influence of the high free-stream turbulence has a great contribution to losses, also outside from the boundary layer region. Table 3 reports the total pressure loss coefficients measured for the different free-stream turbulence intensities. The values of the losses are scaled using as reference the highest one, which corresponds to the numerical simulation for low FST. Losses are sensibly smaller in the high FST case as a consequence of the separation suppression. Even though we can see that predicted losses for both cases in the DNS simulation are slightly higher than the ones found in the experiments, we can still report the excellent agreement.

### Mean Flow and Statistics in the Boundary Layer.

Depending on the free-stream turbulence intensity, the dynamics through which boundary layer transitions from laminar to turbulent state differ. An appropriate way of giving further insight into this dynamics is through close inspection of the time-mean boundary layer velocity profiles. Figure 6 shows the comparison with the experiments [21] of the time- and spanwise-averaged velocity profiles at different streamwise locations on the suction side. These curves demonstrate the progressive change from laminar to “turbulent” velocity profiles. In this figure, the horizontal axis corresponds to the normalized streamwise velocity by its maximum, and the vertical axis shows the wall-normal position normalized by the displacement thickness. The PIV spatial resolution within the boundary layer is sufficiently high to be compared with the DNS just for the low FST case.

In Fig. 6(a), the profiles of mean velocity for the low FST case illustrate a laminar boundary layer at *x*/*L* = 0.744 that has undergone separation in the proximity of *x*/*L* = 0.797, as confirmed by the almost null velocity gradient at the wall characterizing the pink curve. Further downstream, the revers flow region becomes more substantial, and reattachment does not occur before the blade trailing edge. In Fig. 6(b), the profiles of the mean velocity for the high FST case show that the flow is attached and laminar up to 90$%$ of the chord. In the following sections of the paper, we show that the presence of streaks in the near-wall region is keeping flow attached to the wall. The elevated level of free-stream turbulence, giving rise to the higher magnitude of streaks, have a strong stabilizing effect on the boundary layer by completely removing the inflectional part of the velocity profile at the wall.

We calculated the second-order moment *u*_{rms} at different distances from the leading edge along the wall-normal direction. Figure 7 depicts the distributions of normalized *u*_{rms} by the free-stream velocity (*U*_{∞}) versus normalized *y* by the displacement thickness (*δ**). The color scheme shows the downstream location.

For the low FST case, RMS fluctuations of streamwise velocity show that during the first half of the bubble, the growth rate of velocity perturbation is small with the appearance of the double peak. The outer peak in the shear layer grows rapidly in the second half of the bubble, because of amplification of disturbances due to K-H instability, leading to the formation of large scale eddies and breakdown. Interestingly, all the profiles exhibit their maximum at the inflection point of the mean flow, further supporting the dominant role due to the inviscid instability in the transition process for this low FST case. The peak in the high FST case has a higher magnitude and is closer to the wall than it is in the low FST case. In the high FST case, the growth of streaky structures is responsible for the amplification of fluctuations into the boundary layer.

The variation of max (*u*_{rms}) with the downstream direction is shown in Fig. 8 for both FST cases. We can see that the maximum value of *u*_{rms} increases with the downstream location, but the slope is different for the two cases.

To properly characterize the boundary layer state along the rear part of the blade suction side, Fig. 9 shows the boundary layer shape factor (*H*_{12} = *δ**/*θ*), where *δ** is displacement thickness and *θ* is momentum thickness. The shape factor provides distinct values for either laminar and turbulent flow. Therefore, it is a good measure for the onset of transition and the general structure of the boundary layer. For the lower FST case, the shape factor is significantly higher. The lower shape factor in the high FST case indicates that high turbulence intensity level helps to keep the separation bubble thinner by promoting more mixing in the shear layer over the bubble and by inducing earlier transition and reattachment. In the low FST case, the shape factor has higher values at the end of the blade (with values still higher than 3), indicating that the boundary layer is not reattached.

In the high FST case, the shape factor is characterized by values of around 2.5 in the former part of the domain, which is typical of a laminar attached condition of the boundary layer. The shape factor reduces moving toward the trailing edge, with values that drop quickly toward 1.5, which is typical of a turbulent state.

### Instantaneous Flow Field.

Figure 10 visualizes the response of the boundary layer on the suction side for both cases in the blade to blade plane. Contours of instantaneous vortical structures identified with *λ*_{2} [26] are shown (colored with streamwise velocity). *λ*_{2} criterion indicates areas where rotation dominates strain, which gives us more information about the vortical structures in the flow. With *λ*_{2} criterion, we can also see structures that resemble Kelvin–Helmholtz (K–H) rolls. We observe that rolls appear around 80$%$ of the chord, and they get convected downstream. We can also notice the appearance of the laminar separation bubble. In the high FST case, we have quite distinct streamwise streaky structures (Fig. 10(b)). Streaks reach an amplitude of roughly 10$%$ of free-stream velocity for the high FST case before the breakdown to turbulence which is slightly less than what experiments by Ref. [21] found and they reported the value of $20%$. Turbulent spots appear at around 90$%$ of the chord. The turbulent boundary layer that is formed close to the trailing edge is attached to the blade surface.

To better judge, if the boundary layer undergoes a transition, contours of spanwise vorticity are shown in Figs. 11 and 12 for low FST case and high FST case, respectively. In the low FST case, we can observe that the laminar separation bubble does not reattach to the blade surface. We observe shedding of strong coherent (clockwise-rotating) vortices that started to penetrate the boundary layer at the very end of the blade, but it does not reattach and only start transition close to the end of the blade, as suggested by the drop in *H*_{12}. If we focus on this small region where small vortices are concentrated in areas with strong spanwise coherent rollers, we can see that the spanwise-oriented structures are visible before the flow breaks down to smaller scales. These structures are resulting from a saturation of the 2-D disturbances associated with the primary (Kelvin–Helmholtz) [27] instability of the 2-D mean flow profile. In the high FST case, we observe that the boundary layer stays attached until the end of the blade, where large-scale vortices have penetrated the boundary layer. Shedding of strong coherent (clockwise-rotating) vortices lead to a transition to turbulent state at the trailing edge. Results here shown are qualitatively in agreement to those reported in Ref. [4].

To better discuss structures that may penetrate the laminar boundary layer for both FST cases, Figs. 13 and 14 show wall-normal vorticity contours in the wall parallel plane. For the low FST case, we cannot observe the appearance of streamwise streaky structures and wall-normal vorticity is almost zero elsewhere, thus indicating that free-stream structures do not penetrate the boundary layer, and the K–H rolls develop in an almost 2D pattern. Conversely, in the high FST case, elongated perturbation (in the streamwise direction) could be recognized. These are the trace of low and high-speed streak structures, that experienced instability (sinuous and varicose type according to [28]) preceding turbulent breakdown, and the formation of turbulent spots close to the blade trailing edge, which is consistent with the low FST case results reported in Ref. [29].

Contours of streamwise vorticity in the cross-stream plane at the different streamwise locations are presented in Fig. 15 for the low FST case and in Fig. 16 for the high FST case, to gain more insight into the nature of the vortical structures and their evolution. Spanwise structures are not present just downstream of the separation position, nor into the shear layer above the bubble, neither (or only limited) into the dead air region close to the wall (see the plane at *x*/*c* = 0.82). They start generating only far downstream.

Then, they are lifted into the shear layer. Three-dimensionality appears to play a role in the vicinity of the trailing edge, and couple contour-rotating vortices extend toward the wall at a couple of different spanwise positions. Further downstream, these vortices start to fill out the boundary layer in the spanwise direction.

The picture changes when the turbulence intensity of free-stream increases, as previously seen in Fig. 10. Transition moves upstream compared to the case with low FST, and the mechanism by which transition is happening is different. Figure 16 shows contours of streamwise vorticity in different streamwise locations. The vortical disturbances are present in the free-stream, with significantly higher levels of vorticity. The free-stream vortical disturbances penetrate the boundary-layer and manifest themselves as streaky structures (as seen in the previous plane), that appears as counter-rotating vortices in this cross-stream plane as also observed in Ref. [30]. These counter-rotating vortices facilitate the exchange of momentum by transporting the low-momentum fluid away from the wall and pushing the high-momentum fluid toward the wall. As the streaks grow downstream, they undergo wavy motions which precede the breakdown into regions of turbulent spots. The spots grow in size by the end of the trailing edge and start to merge to form a turbulent boundary layer.

*λ*

_{2}criterion. We can see that the structures experience an initial decay followed by the growth in magnitude further downstream. It is interesting to note the appearance of longitudinal vortical structures. We can notice that in the high FST case (Fig. 12), there are more longitudinal structures than in the low FST case (Fig. 11), which was also observed in Ref. [7]. In work by Wu and Durbin [3], these structures were observed for the first time, where the authors investigated the interaction of an incoming wake with a stationary turbine blade. They argue that these structures do not arise due to Görtler instability, despite that they emerge on a concave surface. They associate these structures to the straining of the passing wake. Although in our simulations, the source of perturbation is due to homogeneous isotropic turbulence imposed at the inlet of the domain, we still observe these type of structures. To check if we have this type of instability, we can calculate Görtler number on the pressure side using

*U*

_{∞}is freestream velocity,

*θ*is the momentum thickness,

*ν*is the kinematic viscosity, and

*R*is the radius of curvature. The critical value of Görtler instability is known to be

*G*

_{θ}= 3 [31]. In our simulations, the local values of

*G*

_{θ}are always below the critical one. Therefore, we may exclude Görtler instability as a reason for the appearance of these structures, which is in agreement with previous studies by Refs. [3,32].

## Conclusions

We have performed DNS of the free-stream turbulence-induced boundary layer transition on a low-pressure turbine blade. All the simulations were performed with the spectral-element code Nek5000. Two levels of the free-stream turbulence intensity were simulated (Tu $=0.19%$ and $5.2%$). We observed that in the case of low free-stream turbulence on the suction side, the Kelvin–Helmholz instability dominated the transition process and full-span vortices were shed from the separation bubble. The inspection of the three mutually orthogonal planes makes evident that structures start affecting the boundary layer transition process into the dead air region just close to the blade trailing edge. Transition on the suction side proceeded more rapidly in the high FST case. In this case, streaky structures reduce the inflectional state of the time-mean velocity profile, retarding or even suppressing separation. Instantaneous flow visualizations in the wall-normal and cross-stream plane showed the population of streaky structures. These are responsible for momentum transfer removing separation, and the consequent breakdown events generating turbulent spots causing the boundary-layer transition just close to the blade trailing edge. On the pressure side, we have identified the appearance of the longitudinal vortical structures, where we could observe that an increase of FST gives rise to more longitudinal structures. They seem not to be caused by Görtler instability and further works are needed to clarify their origin and impact on the flow over the pressure side of the blades. The transition mechanisms observed using numerical simulations are in accordance with the experimental results. Moreover, the losses and the wall-normal velocity profiles for the low FST case showed a good agreement.

## Acknowledgment

The European Research Council provided financial support for this work under grant agreement 694452-TRANSEP-ERC-2015-AdG. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the PDC Center for High-Performance Computing at the Royal Institute of Technology (KTH) and the National Supercomputer Centre at Linköping University.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper. Data provided by a third party listed in Acknowledgment. No data, models, or codes were generated or used for this paper.

## Nomenclature

*c*=chord length

*g*=blade pitch

*h*=blade height

*o*=cascade throat

*p*=static pressure

*t*=trailing edge thickness

**u**=velocity vector

*L*=integral length scale

*R*=radius of curvature

*T*=time

*c*_{f}=skin friction coefficient $=\tau w/(0.5\rho U\u221e2)$

*c*_{p}=pressure coefficient $=(pt1\u2212p)/(pt1\u2212p2\xaf)$

*c*_{pt}=total pressure coefficient $=(pt1\xaf\u2212pt2\xaf)/(pt1\xaf\u2212p2\xaf)$

*c*_{x}=axial chord length

*p*_{t}=total pressure

*u*_{τ}=friction velocity

*G*_{θ}=Görtler number

*H*_{12}=shape factor

*U*_{in}=inflow velocity

- $l*$ =
viscous length

*x*,*y*,*z*=streamwise, wall-normal, spanwise coordinate

*Zw*=Zweifel number = 2

*g*/*c*_{x}cos^{2}*α*_{2}(tan*α*_{1}− tan*α*_{2})- Re =
Reynolds number

- Tu =
turbulence intensity

*α*=absolute flow angle

*α*_{m}=blade metal angle

*δ** =displacement thickness

*ν*=kinematic viscosity

*ω*=total pressure loss coefficient

*ρ*=fluid density

*τ*_{w}=wall shear stress

*θ*=momentum thickness

### Superscripts and Subscripts

- + =
viscous units

*rms*=root-mean-square

- ∞ =
Undisturbed, free-stream

- 1 =
cascade inlet

- 2 =
cascade outlet

- $a\xaf$ =
mass-averaged quantities

## References

*re*

_{θ}= 2500 Studied Through Simulation and Experiment