High-order accurate direct numerical simulation of ﬂow over a MTU-T161 low pressure turbine blade

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Introduction
Commercial air transportation moved over 4.3 billion passengers in 2018 and produced upwards of 800 million tonnes of CO 2 and other Greenhouse Gases (GHG) [1,2] .The European Union and the United States have embarked on multi-billion dollar research projects to reduce the environmental impacts of aviation.For example, the renewed European Union Clean Sky 2 project [3] has the goal of developing and validating technology to enable a 75% reduction in CO 2 emissions and a 90% reduction in NO X emissions by 2050 compared with 20 0 0 levels.
Jet engine turbines extract energy from hot exhaust gasses that exit the combustion chamber of the engine, and use it to drive the compressor, fan, and other auxiliary systems.Turbine stages typically comprise approximately one-third of the total engine weight [4] , with half of this contribution coming from the turbine blades.In order to reduce weight and CO 2 emissions, modern turbines are designed to use as few a blades as possible.However, this results in individual blades being subjected to higher-loading, which can lead to fully-separated flow over the aft-portion of each blade, and the introduction of complex, unsteady, three-dimensional flow phenomena.Taken together, these can have a detrimental effect on overall aerodynamic efficiency [5] , which translates almost directly to an increase in specific fuel consumption [6] , and GHG emissions.Therefore, there exists a trade-off between reducing the number of blades -and thus total turbine weight -while maintaining an appropriate level of aerodynamic performance.
At present, turbines are designed and optimised using a combination of RANS simulations [5,7] , and wind tunnel testing [8-10] .However, RANS simulations have limited accuracy, espe- cially for unsteady flows, and experimental wind tunnel testing can be costly, time consuming, and only provides limited data.Consequently, there is an emerging interest in use of scale-resolving simulations, including Large Eddy Simulations (LES) [11][12][13] or Direct Numerical Simulations (DNS) [14,15] .Such simulations could generate insight and data to underpin development of improved RANS models for turbine design via e.g.direct analysis of the physics and/or machine learning based approaches.Additionally, they could underpin a virtual turbine wind tunnel capability, that is cheaper, quicker, and more data-rich than experiments, for characterising blade shapes and exploring the effect of modifications.
The current study applies PyFR [16,17] , a Python based Computational Fluid Dynamics (CFD) solver, to fifth-order accurate petascale DNS of compressible flow over a three-dimensional MTU-T161 Low-Pressure Turbine (LPT) blade with diverging end walls at Re = 20 0 , 0 0 0 on an unstructured mesh with over 11 billion degrees-offreedom per equation.This case is representative of general trends in modern turbine design, with fewer highly-loaded blades per stage.Section 2 details the adopted methodology and simulation setup, including the governing equations, domain, mesh, boundary conditions, and data extraction protocol.Section 3 details the results, including comparisons with available experimental data and a more detailed analysis of various flow features, including separation/transition on both the suction and pressure sides of the blade, end wall vortices, and wake evolution at various span-wise locations.Finally, Section 4 presents conclusions and directions for future work.

Test case
In this study we undertake DNS of air flow over a highly loaded MTU-T161 turbine blade, bounded by diverging end walls.The configuration has been subject to an extensive experimental campaign, undertaken by MTU Aero Engines AG at a range of Reynolds and Mach numbers [9,10,18] .Here we consider a setup where the outflow isentropic Reynolds number based on the turbine blade chord is 20 0,0 0 0, and the outflow isentropic Mach number is 0.601.

Governing equations
It was assumed that air behaves as a viscous ideal gas, governed by the compressible Navier-Stokes equations with constant viscosity, which can be written as where u = ρ, ρv x , ρv y , ρv z , E T is the solution, with ρ the fluid density, v = { v x , v y , v z } T the fluid velocity, and E the total energy of the fluid per unit volume, and where where p is the pressure, which for an ideal gas is given by Fig. 7. Distribution of p t,l along a line with constant x = 1 .206 and z = 0 in the wake obtained from PyFR and from the experiments with 2% and 4% turbulence intensity.The numerical results are in agreement with the experimental data, capturing both the width of the wake and its shape.However, the predicted wake is slightly offset in the negative y direction relative to the experimental data, and the peak p t,l is slightly over-predicted.
with γ = c p /c v , where c p and c v are specific heat capacities at constant pressure and volume, respectively, and where with μ the dynamic viscosity, = μc p /P r, with P r the Prandtl number, and T is the temperature, which for an ideal gas is given by In the current study values of γ = 1 .4 , P r = 0 .71 , and μ = 2 .7672 × 10 −6 were employed.

Solver and numerical method
Simulations were undertaken on 5760 Nvidia K20X GPUs of the Titan supercomputer at Oak Ridge National Laboratory using PyFR [16,17] , which is based on the high-order Flux Reconstruction approach of [19] .Discontinuous Galerkin flux correction functions were employed, a Rusanov Riemann solver was used to calculate the inter-element inviscid fluxes, the Local Discontinuous Galerkin (LDG) approach was used to calculate the inter-element viscous fluxes, an explicit RK45 scheme with adaptive time-stepping via a PI controller was employed to advance the solution in time, volume and surface flux anti-aliasing was applied to stabilise the solution, and all runs were performed using double precision arithmetic.Specifically, a patched version of PyFR v1.5.0 was employed.The patch is provided as Electronic Supplementary Material.Key functionality of PyFR v1.5.0 is summarised in Table 1 .

Domain and mesh
The computational domain is shown in Fig. 1 .The blade has a chord of 1.0, an axial chord of 0.8663, a pitch of 0.9557, and the domain has a span of 2.287 in the uniform span section.The origin of the domain is located at mid-span on the leading edge of the turbine blade.The inflow plane is located 0.92 upstream of the leading edge of the turbine blade.The outflow plane is located 1.86 downstream of the leading edge of the turbine blade.
Meshing was undertaken using Gmsh [20] .Regions of the mesh are shown in Fig. 2 .A structured region of second-ordercurved hexahedra, 23 elements deep, was located adjacent to the surface of the turbine blade and adjacent to the end walls.The remainder of the domain was meshed with second-ordercurved unstructured hexahedra.Elements were sized to achieve DNS resolution throughout the domain.In total the mesh consisted of 90,760,192 elements, resulting in 11,345,024,0 0 0 degrees-offreedom per equation when fourth-order polynomials were used to represent the solution within each element.

Boundary conditions 2.5.1. Inflow plane
At the inflow plane total temperature T t,i , total pressure p t,i , and inflow direction n i were prescribed.Specifically, T t,i was chosen such that c p T t,i = 3 .5 uniformly, , (7) where p i = 0 .9106 , u i = 0 .4515 , h z = 1 .14391 and η = 5 .07514 , a bl = 0 . 33245 ,n bl = 1 . 6 6 67 ,δ bl = 0 .2070 were chosen to fit the resulting inflow velocity profile to available experimental data, and laminar, in the sense no turbulent fluctuations are injected.such the empirical correction δ p o was applied in order to achieve the desired static pressure at the outflow plane.

Turbine blade surface and end walls
At the turbine blade surface and at the end walls an no-slip adiabatic boundary condition was applied.

Top and bottom
Periodic boundary conditions were applied to connect the top and bottom of the domain.

Startup process
The simulation was initiated at t = 0 and advanced in time through to t = 230 .0 via a series of partial-span-periodic and then full-domain runs, with second-order and then third-order polynomials used to represent the solution within each element of the mesh, and p o = 0 .7910 uniformly at the outflow plane.This was done to wash out large-scale initial transients.
The simulation was then restarted from the solution at t = 230 .0 , but with fourth-order polynomials used to represent the solution within each element of the mesh, and hence fifth-order accuracy in space, and p o = 0 .7910 + δ p o uniformly at the outflow plane, where δ p o = 0 .002285 .This was run until t = 240 .0 in order to achieve the experimentally observed static pressure at the outflow plane, and to wash out further transients, and then run until t = 258 .5 , during which time data was extracted for analysis.This latter period from t = 240 .0 to t = 258 .5 is henceforth referred to as the Data Extraction Period, and corresponds to approximately 11 flow passes over the turbine blade.

Solution snapshots
During the Data Extraction Period, solution snapshots containing instantaneous distributions of ρ, ρv , and E throughout the entire domain were obtained every 0.25 time units.These acted as restart checkpoint files, and allowed inspection of the full threedimensional flow field.Snapshots containing instantaneous distributions of ρ, ρv , and E throughout the entire domain taken at t = 240 .0 and t = 258 .5 are available in a publicly accessible AWS S3 bucket with ARN arn:aws:s3:::pyfr-mtu-t161-dns-data .

Time-Averaging
During the Data Extraction Period various quantities were timeaveraged throughout the entire domain, including: where i, j, k, l each run through x, y, z, the numbers in parenthesis indicate the term count.Distributions of these time-averaged quantities are available in a publicly accessible AWS S3 bucket with ARN arn:aws:s3:::pyfr-mtu-t161-dns-data .

Point probes
During the Data Extraction Period, values of ρ, ρv , and E were obtained at 660 point locations every 30 time-steps.This data allowed spectra to be obtained.The time series of ρ, ρv , and E at each of the 660 point locations are available in a publicly accessible AWS S3 bucket with ARN arn:aws:s3:::pyfr-mtu-t161-dns-data .

Boundary layer resolution
Fig. 3 plots variation of y + = y ρ τ /μ along the suction side of the turbine blade surface, where y is the normal distance from the blade surface to the first solution point, ρ is the local time-averaged density, and τ is the local time-averaged wall shear stress.y + < 1 at all locations.Hence, it is concluded that the mesh resolution is sufficient for DNS of the boundary layer [21] .

Free-Stream resolution
An estimate of the Kolmogorov length scale η can be obtained at each point in the computational domain, as can the characteristic solution point spacing δ.Moving from the laminar flow region on the suction side of the turbine blade into the turbulent wake, the ratio δ/η is seen to grow monotonically, peaking Fig. 14.Plots of time-averaged wall-parallel velocity v as a function of wall-normal distance s ⊥ along lines P i /S j for i = 6 (a), i = 9 (e), i = 10 (i) and i = 12 (m), for various j.Plots of Favre averaged xy Reynolds stress σ xy as a function of wall-normal distance s ⊥ along lines P i /S j for i = 6 (b), i = 9 (f), i = 10 (j) and i = 12 (n), for various j.Plots of turbulent kinetic energy E tk as a function of wall-normal distance s ⊥ along lines P i /S j for i = 6 (c), i = 9 (g), i = 10 (k) and i = 12 (o), for various j.Plots of velocity magnitude power spectral density | v | psd as a function of Strouhal number St at points P i /S j for i = 6 (d), i = 9 (h), i = 10 (l) and i = 12 (p), for various j.Note that line P i /S j resides in S j , intersects the turbine blade surface normally in the x -y plane, and passes through the z-wise projection of point P i onto slice S j .Also note that point P i /S j is the z-wise projection of point P i onto slice S j .at δ/η ≈ 5 and then decreasing along line L2 as per the plot in Fig. 4 , asymptoting towards δ/η ≈ 2 in the far field.Based on previous results [22] , δ/η ≈ 5 is sufficient for DNS resolution of first and second-order statistics.Hence, it is concluded that the mesh resolution is sufficient for DNS of the wake.

Blade surface
To validate the accuracy of our numerical simulations we compare our results against the experimental data of MTU Aero Engines.The distribution of time-averaged isentropic Mach number M over the blade at the mid-span location is shown in Fig. 5 .The DNS results achieve very good agreement with the experimental data at all measurement locations on both the pressure and suction sides of the blade.Furthermore, a comparison between the Line Integral Convolution (LIC) of time-averaged wall shear from the DNS results and experimental oil streak lines is shown in Fig. 6 .On the suction side we observe that the DNS results accurately predict the turbulent end-wall effects and their development downstream on the suction side of the blade.Also, the location and curved shape of the mid-span separation line, which is demarcated by a horizontal edge in the oil streak lines ahead of the mid-chord, is also predicted accurately.Finally, we note that the behaviour on the pressure side of the blade is also qualitatively consistent with the experimental data.

Wake
Fig. 7 plots the time-averaged total pressure loss p t,l = ( p t,i − p t ) / ( p t,i − p o ) along a line with constant x = 1 .206 and z = 0 in the wake, where p t,i is the time-and y -averaged total pressure at the inflow plane at z = 0 , p o is the time-and y -averaged pressure at the outflow plane at z = 0 , and p t is the time-averaged total pressure.The numerical results are in agreement with the experimental data, capturing both the width of the wake and its shape.However, the predicted wake is slightly offset in the negative y direction relative to the experimental data, and the peak total pressure loss is slightly over-predicted.This latter discrepancy is consistent with the laminar inlet conditions used in the current study, and their effect on bypass transition and subsequent wake formation.
Fig. 8 plots contours of p t,l in the plane x = 1 .206 with z < 0 .The main features of the distribution are captured.However, the simulation results appear to exhibit a slightly larger and stronger end-wall vortex.This discrepancy is also consistent with the laminar inlet conditions used in the current study, which will cause over-thickening of the upstream boundary layer.Plots of Favre averaged xy Reynolds stress σ xy as a function of wall-normal distance s ⊥ along lines P i /S j for i = 14 (b), i = 16 (f), i = 18 (j) and i = 20 (n), for various j.
Plots of turbulent kinetic energy E tk as a function of wall-normal distance s ⊥ along lines P i /S j for i = 14 (c), i = 16 (g), i = 18 (k) and i = 20 (o), for various j.Plots of velocity magnitude power spectral density | v | psd as a function of Strouhal number St at points P i /S j for i = 14 (d), i = 16 (h), i = 18 (l) and i = 20 (p), for various j.Note that line P i /S j resides in S j , intersects the turbine blade surface normally in the x -y plane, and passes through the z-wise projection of point P i onto slice S j .Also note that point P i /S j is the z-wise projection of point P i onto slice S j .Fig. 18.LIC of time-averaged wall shear on one of the domain end-walls slack.

Suction-Side
Fig. 12 shows instantaneous snapshots of Q-Criterion isosurfaces, which define vortical structures, over the suction-side of the turbine blade.It can be seen that turbulent transition in the mid-span region occurs near the aft portion of the blade.However, for the end-wall region transition occurs near the leading edge of the blade and propagates inwards towards the mid-span.This demonstrates that turbulent transition near the ends of the blade is dominated by end-wall effects, whereas turbulent transition in the mid-span region is relatively uniform across the span.The image also makes visually apparent the amount, and fidelity, of data made available by a DNS compared with e.g.wind tunnel experiments.
Fig. 13 shows a snapshot of the magnitude of the density gradient, an LIC of the time averaged in-plane velocity, and turbulent kinetic energy on a plane at the mid-span location of the blade.Similar to Figs. 12, 13 a shows that turbulent transition on the suction side occurs near the trailing edge aft of a Kelvin-Helmholtz instability.By comparing the density gradients in Fig. 13 a and the time averaged velocity in Fig. 13 b it can seen that transition actually occurs partway along a small closed laminar separation bubble.Fig. 13 c shows that turbulent transition initiates in the shear layer between the outer edge of the laminar separation bubble and the mean flow.The turbulence intensity grows along this interface, rapidly amplifies in the aft portion of the separation bubble, and then propagates downstream as a turbulent wake.  i = 27 (f), i = 28 (i) and i = 29 (l), for various j.Note that line L i /S j is the z-wise projection of line L i onto slice S j , and s is measured from the end of the line with the smallest x -coordinate.Also note that the point P i /S j is the z-wise projection of point P i onto slice S j .
St at the same locations.In the mid-span region flow is attached ( Fig. 14 a) and laminar ( Fig. 14 c/d) on the upstream part of the blade surface.Moving downstream, mid-span flow then separates ( Fig. 14 e) and transition begins near the inflection point of the velocity profile ( Fig. 14 g) driven by a Kelvin-Helmholtz instability.The mid-span separation bubble then grows in size and strength ( Fig. 14 i), and coherent structures from the Kelvin-Helmholtz instability become apparent ( Fig. 14 l) with a peak around St = 10 .Finally, by the trailing edge of the blade, the mid-span flow reattaches ( Fig. 14 m) and a turbulent wake emerges ( Fig. 14 o/p).
In the end-wall region flow is attached ( Fig. 14 a) and turbulent (( Fig. 14 c/d) on the upstream part of the blade surface.Moving downstream in the end-wall region, wall-parallel flow remains persistently attached and turbulent at all stream-wise locations.Specifically, the turbulent kinetic energy starts off high ( Fig. 14 c) inside the region where end-wall vortices dominate, but then decreases moving downstream as these structures move away from the wall ( Fig. 14 k/o).

Pressure-side
Fig. 15 shows instantaneous snapshots of Q-Criterion isosurfaces, which define vortical structures, over the pressure-side of the turbine blade.It can be seen that the flow is laminar at the leading edge on the pressure side of the blade.Slightly aft of the leading edge the flow undergoes turbulent transition, leading to the formation of a number of large coherent vortices.The location of this transition point correlates with a separation bubble shown in the shear LIC lines of Fig. 6 .While some end-wall effects are visible, they are less pronounced than those on the suction side of the blade.The image also makes visually apparent the amount, and fidelity, of data made available by a DNS compared with e.g.wind tunnel experiments.
Fig. 16 shows a snapshot of the magnitude of the density gradient, an LIC of the time averaged in-plane velocity, and turbulent kinetic energy on a plane at the mid-span location of the blade.It can be seen that there is also a Kelvin-Helmholtz instability on the pressure side of the blade that forms around a small closed separation bubble.This instability initiates the formation of a number of large scale vortex structures inside the separation bubble, which are then propagated downstream inducing a pronounced increase in turbulent kinetic energy.These structures then gradually dissipate as they move downstream on the pressure side, which is apparent from a gradual decrease in turbulent kinetic energy moving towards the trailing edge of the blade.c), leading to a rise in turbulent kinetic energy ( 17 g).Moving downstream the flow then reattaches to the blade at all span-wise locations ( 17 i), and remains attached over the remainder of the blade surface ( 17 m), while turbulent kinetic energy decreases monotonically.In general, the magnitude of the turbulent kinetic energy on the pressure side of the blade is significantly lower than that observed on the suction side.

End-wall
Fig. 18 shows an LIC of time-averaged wall shear stress on one of the domain end-walls.A bow-like structure upstream of the turbine blade is apparent where the LIC field lines converge along a curve.The area contained within this curve shows the extent of the end-wall vortex relative to the size of the turbine blade.In addition to the stagnation point directly at the leading edge of the blade, a secondary stagnation point is also apparent from convergence of the LIC field lines at a point situated upstream of the leading edge.This stagnation point forms due to impingement of the reversed flow in the end-wall vortex on the incoming freestream velocity field.Additional secondary flow structures are also visible, including a stream-wise oriented structure underneath the pressure side of the blade.Fig. 19 shows instantaneous snapshots of density gradient magnitude in various planes of constant x near one of the domain endwalls.Growth of the end-wall vortical structures in the streamwise direction is apparent, as is their unsteady time-dependent nature.
Finally, Fig. 20 shows the time history of static pressure p recorded at points P 12 /S 3 , P 12 /S 4 , P 12 /S 5 , which are the z-wise projections of point P 12 onto slices S 3 , S 4 and S 5 .The pressure at point P 12 /S 4 is lower than that recorded at points P 12 /S 3 and P 12 /S 5 .This is to be expected since as per Fig. 19 m/n/o/p, P 12 /S 4 is located within the end-wall vortex.Moreover, the pressure signal recorded at P 12 /S 4 exhibits well defined low-frequency oscillations, indicating that the end-wall vortex undergoes quasi-periodic motions with approximately unit period.This latter result suggests that steady RANS approaches would fundamentally struggle to achieve accurate predictions in the end-wall region.However, given the separation of time-scales between the turbulence and the quasiperiodic motions, unsteady RANS approaches with a suitable model may be effective.

Wake
Fig. 21 shows plots of time-averaged line-normal velocity v ⊥ and turbulent kinetic energy E tk as a function of distance s along 20 lines in the wake of the turbine blade, as well as the velocity power spectral density | v | psd as a function of Strouhal number St at points on these lines.In the mid-span region a well defined compact wake structure is present directly behind the turbine blade ( Fig. 21 a), that weakens and broadens on moving downstream ( Fig. 21 d/g/j).In contrast, the end-wall regions exhibit a more complex structure.In particular, the wake within S 4 , which cuts through the end-wall vortex, is displaced upwards from the mid-span wake ( Fig. 21 a/b), and one can also observe increased high-frequency power at this location ( Fig. 21 c).

Conclusions
The current study applied PyFR, a Python based Computational Fluid Dynamics (CFD) solver, to fifth-order accurate petascale DNS of compressible flow over a three-dimensional MTU-T161 LPT blade with diverging end walls at a Reynolds number of 20 0,0 0 0 on an unstructured mesh with over 11 billion degrees-of-freedom per equation.Various flow metrics, including isentropic Mach number distribution at mid-span, surface shear, and wake pressure losses were compared with available experimental data and found to be in agreement.Subsequently, a more detailed analysis of various flow features was presented.These included the separation/transition processes on both the suction and pressure sides of the blade, end-wall vortices, and wake evolution at various spanwise locations.The results, which constitute one of the largest and highest-fidelity CFD simulations ever conducted, demonstrate the potential of high-order accurate GPU-accelerated CFD as a tool for delivering industrial DNS of LPT blades.Such DNS could generate insight and data to underpin development of improved RANS models for LPT design via e.g.direct analysis of the physics or machine learning based approaches.Additionally, they could underpin a virtual LPT wind tunnel capability, that is cheaper, quicker, and more data-rich than experiments, for characterising blade shapes and exploring the effect of modifications.Future work should investigate the addition of further physics to the simulation setup, including Sutherland's law for temperature dependent viscosity, and the specification of fully turbulent inlet boundary conditions via e.g. a digital filter approach [23,24] .Future work should also explore using PyFR to simulate higher Mach number transonic cases, where shocks and shock-turbulence interactions are present.This will require implementation of specialised shock-capturing approaches [25][26][27] .

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Slice through the computational domain with constant z = 0 (a) showing the turbine blade profile, the location of points P 4 through P 12 , P 14 through P 22 and P 26 through P 29 , and the location of lines L 2 and L 26 through L 29 .View of the computational domain in x − z plane (b) showing the expanding end-wall profile, and the location of slices S 1 through S 5 .

Fig. 2 .
Fig. 2. Slices through computational mesh with constant z = 0 in the vicinity of the suction side (a) and trailing edge (b) of the turbine blade.

Fig. 3 .
Fig. 3. Variation of y + along the suction side of the turbine blade.

Fig. 5 .
Fig. 5. Distribution of time-averaged isentropic Mach number M over the pressure and suction sides of the turbine blade surface at mid-span obtained from PyFR and from the experiment with 2% turbulence intensity.

Fig. 6 .
Fig. 6.View of time-averaged wall shear LIC from PyFR (a) and oil-streak lines from the experiment with 2% turbulence intensity (b) on unwrapped surfaces of the turbine blade.The pressure surface is located on the bottom of each image, the suction surface is located on the top of each image, and the leading edge of the blade is located approximately half-way up each image.

Fig. 8 .
Fig. 8. Distributions of time-averaged total pressure loss p t,l in the plane x = 1 .206 with z < 0 from PyFR (a) and from the experiment with 2% turbulence intensity (b).

Fig. 9 .
Fig. 9. Distributions of time-averaged stream-wise vorticity ¯ in the plane x = 1 .206 with z < 0 from PyFR (a) and from the experiment with 2% turbulence intensity (b).

Fig. 10 .
Fig. 10.Distributions of time-averaged flow angle from the x -axis ¯ in the plane x = 1 .206 with z < 0 from PyFR (a) and from the experiment with 2% turbulence intensity (b).

Fig. 11 .
Fig. 11.Distributions of time-averaged flow angle from the z-axis ¯ in the plane x = 1 .206 with z < 0 from PyFR (a) and from the experiment with 2% turbulence intensity (b).
At the outflow plane a characteristic boundary condition was applied.As such a density ρ o = 0 .8430 , velocity v o = 0 .6565 { cos (−65 • ) , sin (−65 • ) , 0 } and static pressure p o = 0 .7910 + δ p o were prescribed, where δ p o = 0 .002285 .Note that due to the nature of the boundary condition p o is not strictly enforced, and as

Fig. 12 .
Fig. 12. Instantaneous snapshot of a Q-criteria iso-surface above the whole suction side of the turbine blade and in the wake (a) and above the region of the suction side where transition occurs (b).

Fig. 13 .
Fig. 13.Instantaneous snapshot of density gradient magnitude (a), LIC of the time averaged in-plane velocity (b) and turbulent kinetic energy E tk (c) near the trailing edge of the suction side of the turbine blade.

Fig. 15 .
Fig. 15.Instantaneous snapshot of a Q-criteria iso-surface above the whole pressure side of the turbine blade and in the wake (a) and above the region of the pressure side where transition occurs (b).

Fig. 9
plots contours of time-averaged stream-wise vorticity ¯ = ( ¯ x cos (25 • ) + ¯ z sin (25 • )) / || u i || , where ¯ x and ¯ z are the time-averaged x and z vorticity, respectively.Figs. 10 and 11 plot contours of timeaveraged flow angles ¯ = arcsin ( ū x / | ū | ) and ¯ = arcsin ( ū z / | ū | ) relative to the x -and z-axis, respectively, where ū x and ū z are the time-averaged x and z velocity, respectively, and | ū | is the time averaged velocity magnitude.Once again, the main features of the distributions are captured.

Fig. 16 .
Fig. 16.Instantaneous snapshot of density gradient magnitude (a), LIC of the time averaged in-plane velocity (b) and turbulent kinetic energy E tk (c) near the leading edge of the pressure side of the turbine blade.

Fig. 17 .
Fig.17.Plots of time-averaged wall-parallel velocity v as a function of wall-normal distance s ⊥ along lines P i /S j for i = 14 (a), i = 16 (e), i = 18 (i) and i = 20 (m), for various j.Plots of Favre averaged xy Reynolds stress σ xy as a function of wall-normal distance s ⊥ along lines P i /S j for i = 14 (b), i = 16 (f), i = 18 (j) and i = 20 (n), for various j.

Fig. 14 Fig. 20 .
Fig. 20.Time history of static pressure p at points P 12 /S 3 , P 12 /S 4 , P 12 /S 5 , which are the z-wise projections of point P 12 onto slices S 3 , S 4 and S 5 , respectively.

Fig. 21 .
Fig. 21.Plots of time-averaged line-normal velocity v ⊥ as a function of distance s along lines L i /S j for i = 26 (a), i = 27 (d), i = 28 (g) and i = 29 (j), for various j.Plots of turbulent kinetic energy E tk as a function of wall-distance s along the lines P i /S j for i = 26 (b), i = 27 (e), i = 28 (h) and i = 29 (k), for various j.Plots of velocity magnitude power spectral density | v | psd as a function of Strouhal number St at points P i /S j for i = 26 (c), i = 27 (f), i = 28 (i) and i = 29 (l), for various j.Note that line L i /S j is the

Fig. 17
shows plots of time-averaged wall-parallel velocity v , Favre averaged xy Reynolds stress σ xy , and turbulent kinetic energy E tk as a function of wall-normal distance s ⊥ , at a total of 20 locations on the pressure-side of the turbine blade, as well as the velocity power spectral density | v | psd as a function of Strouhal number St at the same locations.The flow is separated at all spanwise locations near the leading edge of the blade ( 17 a).Turbulent transition occurs near the inflection point of the separated velocity profiles at all but the outermost span-wise location ( 17