## Abstract

In this study we present detailed aerodynamic and thermal field measurements downstream of an annular cascade of fully-cooled nozzle guide vanes (NGVs). The experiments were conducted in the Engine Component Aerothermal (ECAT) facility at the University of Oxford, at engine-matched conditions of Reynolds number and Mach number, and high turbulence intensity. The experimental data are unusually high-fidelity and allow for detailed comparison with modern computational fluids dynamics (CFD) methods. We compare the experimental data to simulations of fully-featured geometry (resolved internal geometry and film cooling holes). We analyze distributions of whirl angle, kinetic energy loss, and non-dimensional temperature at three axial planes downstream of the NGVs. The aerodynamic and thermal wakes are also characterized in terms of their spreading and decay rates. The analysis is deepened with detailed comparison to a previous data-set for a different design of heavily-cooled NGV. The analysis is a useful reference point for assessing the accuracy of the current state-of-the-art numerical methods used in the engine design process.

## Introduction

Preliminary turbine design is usually performed using mean-line calculations and through-flow codes, followed by design optimization using Reynolds-averaged Navier–Stokes (RANS) and unsteady RANS (URANS) codes. The very early correlations (e.g., Ainley and Mathieson [1]) on which the original mean-line and through-flow codes were built have since been updated to account for secondary flows and tip losses (e.g., Dunham and Came [2]) and for off-design operating points (e.g., Benner et al. [3]), but there is still limited data for transonic designs with significant cooling flow. This leads to inaccuracy in the design starting point, which would be improved by re-benchmarking these codes against data from more current engine components. Likewise, known errors in RANS and URANS methods can lead to inaccurate performance prediction. The most significant contributors to error are generally the limitations associated with artificially modeling the effects of turbulence rather than explicitly resolving it, with models relying on calibration against experimental datasets that are not particularly representative of most real engine environments. The original calibration datasets are often from low-speed experiments with lower-than-engine turbulence levels (see, for example: Launder and Spalding [4], Wilcox [5], Spalart and Allmaras [6], and Menter [7]). Obtaining turbulence *closure coefficients* from such tests might yield acceptable results in environments similar to the calibration environment, but at the sacrifice of good performance in engine-like environments. It is well-established that optimum closure coefficients depend on both the flow and the location within the flow (see, for example: Chambers et al. [8], Georgiadis and Yoder [9], and Costa Rocha et al. [10]). For high-accuracy simulations, application-specific model recalibration becomes a priority. This point is adequately demonstrated by considering recent comparisons of nozzle guide vane (NGV) flows at engine-realistic conditions (transonic and fully-cooled) with corresponding uncalibrated RANS and URANS. We review this literature now.

Yasa et al. [11] and Burdett and Povey [12] investigated kinetic energy loss and mixing rates downstream of fully cooled NGVs, and compared results to RANS simulations using the *k–ω* shear stress transport (SST) model. In Burdett and Povey, RANS simulations overpredicted profile loss by 22%, when compared to experiments [12]. This was largely attributed to over-estimation of loss local to the film cooling holes, which arose as a result of incorrect film momentum flux ratio due to the use of a point-source method, which is common in design practice. The recommendation was to use fully resolved cooling hole geometries to mitigate this problem. The simulations of Yasa et al. [11] overpredicted profile losses by approximately 40%. Here, it is noted that film cooling was not considered as part of the CFD model; a better match in loss predictions may have been achieved had this been included. In both studies the spreading and decay rates of the wake were investigated. On average, Burdett and Povey reported a 31% underprediction in wake width and a 91% overprediction of wake depth at midspan. The corresponding values for Yasa et al. were a 12% underprediction in wake width and 32% overprediction of wake depth. Overall, CFD overestimated loss and under-predicted mixing rates when compared to experiments. These differences are important for correct estimation of the unsteady aerothermal load on the rotor.

Trailing edge (TE) dynamics are complex (see review of Sieverding et al. [13]), with significant unsteady vortex shedding. The difference between RANS and URANS for mixing rates in the near and far-wake were investigated by Kopriva et al. [14] for a transonic linear cascade with TE cooling, comparing CFD to the experimental data of Kapteijn et al. [15]. Kopriva et al. [14] did not report integral loss, which we have estimated here using simple area-weighted integrals of the extracted wake profiles. In line with Yasa et al. [11] and Burdett and Povey [12], in the study of Kopriva et al. [14] RANS overpredicted overall loss by 27% on average, across 4 measurement planes, whilst under-predicting mixing rates of both the aerodynamic wake and the thermal wake. When compared to the experiment, the RANS aerodynamic and thermal wake depths were overpredicted by 60% and 119% respectively, with corresponding wake widths 18% and 38% narrower. URANS showed a similar 35% overprediction of overall loss but an improved match of mixing rates when compared to the experiment, both in terms of the peak intensity and the wake width of aerodynamic and thermal wakes. The improved accuracy of URANS was attributed to the modeling of unsteady vortex shedding, though the underlying mechanism was not explicitly addressed. The performance of RANS and URANS was also compared by Burdett and Povey [12], with URANS closing the gap on experimental data by on-average 12% for the peak aerodynamic loss and 14% for the wake width. In both studies [12,14], URANS led to an improved match with experiments in terms of mixing characteristics but had little effect on the overall loss.

Although URANS has been shown in the literature to be able to reproduce vortex shedding behavior for nozzle guide vanes with some success (see e.g., Leonard et al. [16]), and has been shown to improve the match of the wake profiles with experimental data [12,14], there is some debate as to the mechanism by which the improvement is achieved. Fadai-Ghotbi et al. [17] argue that for problems without external forcing, RANS and time-averaged URANS should converge to the same solution. We might therefore attribute the enhanced mixing that has been observed in URANS to numerical diffusion from the time‐space discretization. The point is that, whilst phenomenologically URANS appears to give closer match to experimental data than RANS for many examples of TE flows, the fundamental basis for the improvement is questionable. This has led some authors (e.g., Fadai-Ghotbi et al. [17] and Spalart [18]) to question whether URANS should be preferred over RANS for statistically-stationary problems—noting the attendant increase in computational cost—or whether effort would be better devoted to improving turbulence models for use in steady-RANS and hybrid solvers combining RANS with Large Eddy Simulations. Whatever position one takes on this unresolved question, there is little disagreement about the importance of high-fidelity experimental work for validation of particular methods, particularly the question of turbulence modeling.

In this study we consider the aerodynamic and thermal field development for a fully-cooled transonic NGV from a modern aero-engine. The parts were tested at engine-representative Mach and Reynolds number. In addition to a comprehensive analysis of the aerodynamic wake in terms of loss and whirl angle, the axial evolution of the thermal field downstream of the vanes is studied experimentally and via RANS simulations of the fully‐resolved NGVs. The aim is to explore the strengths and limitations of RANS solvers for this environment, and to quantify and understand the differences between the CFD and experimental data. Comparisons against high-fidelity data of this type are rare, and provide an opportunity to test commonly used methods against a relevant experimental benchmark. We compare results to the earlier work of Burdett and Povey [12], noting that the engine-parts of the current study are from an earlier generation engine of the same family.

## Experimental Methods

The experiments in this study were conducted in the Engine Component AeroThermal (ECAT) facility, described by Kirollos et al. [19]. The ECAT facility is a transonic annular cascade configured for testing high-pressure NGVs at engine-matched conditions of Mach number, Reynolds number, and coolant-to-mainstream pressure ratio. The vanes used in this study were from a modern turbo-fan engine. In this section we describe the layout and instrumentation of the facility working section.

### Test Section Layout and Instrumentation.

The layout of the working section is illustrated in Fig. 1. The inlet measurement plane, NGV, and three exit measurement planes are marked. The flow exhausts through a parallel annular duct to a silencer at atmospheric pressure.

At plane 1, both total temperature and total pressure were measured using rakes (8 radial measurement points) at each of four circumferential locations (32 measurements for each). Hub and case coolant plena feed conditions were measured using 4 total temperature and 4 total pressure probes per plenum.

The NGV exit static pressure was measured with 32 static pressure tappings on the hub and casing platform overhangs of the NGV, approximately 0.3 axial chords downstream of the midspan TE. The tappings were evenly distributed over four vane platforms (8 tappings per platform). The axial location of the static pressure tappings is indicated in Fig. 1.

Exit measurement planes 2, 3 and 4 are located 0.25 *C _{x}*, 0.50

*C*, and 0.75

_{x}*C*downstream of the midspan TE. They were surveyed with a five-hole aerodynamic probe (total pressure; pitch; yaw; static pressure) and a thermocouple probe (dual-sensor; K-type; shielded and vented; 25 μm wire diameter). The near-wall regions, defined as within 10% of the hub and case walls, were traversed with a Pitot probe. All probes were pre-yawed by 68 deg and pre-pitched by between −5 deg and +10 deg, depending on the location.

_{x}The frequency response of the five-hole probe was measured to be approximately 50 Hz using an exploding balloon impulse response experiment. The thermocouple time-constant was estimated to be around 10 milliseconds using Paniagua et al. [20]. The traverse speed was 6 deg s^{−1} for the five-hole probe and Pitot measurements, and 3 deg s^{−1} for the thermocouple measurements, giving effective spatial resolutions in the circumferential direction (five time constants) of 0.4 mm and 0.5 mm at midspan, respectively. The radial traverse resolution was 1.6 mm (3.0% of span) for the five-hole probe, and 0.8 mm (1.5% of span) for the Pitot probe. For the thermocouple probe, the radial step size was approximately 3.2 mm (6.0% of span), except within 15% of span of the hub or casing walls, where an increment of 1.6 mm was used.

Accurate measurement of flow angles using multi-hole aerodynamic probes can be challenging in flows with strong pressure gradients. To increase the accuracy of the pitch and whirl angle measurements, flow-angles were corrected using a method similar to that of Vinnemeier et al. [21]. The recovery factor for the downstream total temperature probe was taken to be 0.975, a value typical for a shielded bare-bead thermocouple at transonic conditions (see e.g., Paniagua et al. [20] and Glawe et al. [22]). Experimental measurement uncertainty is covered in Appendix A.

### Experimental Operating Conditions.

Aerodynamic and thermal traverse measurements were performed at conditions of Mach number, Reynolds number and coolant-to-mainstream mass flow rate ratio matched to the engine conditions at cruise. These conditions are summarized in Table 1. Aerodynamic measurements were performed with ambient flow temperature. Measurements of the downstream thermal field were performed at a mainstream-to-coolant temperature ratio of 1.20 (using a 2 MW heater).

### Experimental Inlet Profiles.

Inlet measurements (plane 1; see Fig. 1) of total pressure and total temperature were conducted using both fixed rakes and an automated two-axis traverse system, which has been described by Amend and Povey [23]. The fixed pressure probes were beveled-inlet Pitot probes, and the traverse probe was of Kiel-head design. The thermocouple probes on the rake were 125 μm diameter K-type butt-welded wire. An Omega 125 μm diameter K-type thermocouple with improved error was used for the traverse probe. Circumferentially averaged profiles are shown in Fig. 2. Data were real-time-normalized against average instantaneous values from the fixed rakes (32 measurement points for each). We refer to these normalized measurements as *p*_{01,norm} and *T*_{01,norm}.

First, consider the normalized total pressure in Fig. 2(a). The profiles measured by both rake and traverse system have a peak at approximately mid-span, with a variation of approximately ±0.1% from the mean: i.e., the radial trend is very flat. The two profiles agree extremely well with maximum variation between the two trends of approximately 0.04%, which is within the combined bias error of the transducers of 0.06% (see Appendix A).

The normalized total temperature is shown in Fig. 2(b). The profiles are gently peaked toward midspan between 10% and 90% span with a variation of ±0.2% from the mean in this range. Toward the casing, a thermal boundary layer, which was measured with the traverse only, develops due to convective heat transfer from the hot flow to the relatively cold test section walls. The peak deficit here was roughly 5% (16 K) of the nominal value (*T*_{01} = 340 K). In this region the probe could be traversed to the casing wall (retractable probe). The hub thermal boundary conditions are essentially identical to the casing. Because of this, and because the measured profile was symmetric about midspan in the region measured, the hub thermal deficit was assumed to be the same magnitude as that measured at the casing. Measurements from the fixed rakes and traverse system agree to within 0.3%, which is well within the bias error of the measurement (0.49%; see Appendix A).

Turbulence was characterized at plane 1 by traversing the inlet with a pair of split-fiber probes [23]. Profiles of turbulence intensity, *I*, and normalized turbulence length scale, ℓ/*C*_{x}, are shown in Fig. 3. Turbulence intensity was calculated from turbulence kinetic energy and velocity magnitude. Finite probe size meant that the 0–5% span region could not be surveyed. A *k*–*ɛ* CFD simulation of the turbulence grid was used to inform extrapolation of the experimental profiles to the walls.

First consider the radial profiles of turbulence intensity, which are shown in Fig. 3(a). The experimental traverse profile (solid line) is relatively flat (average of 13%) with a slight bias for being higher in the lower half of the span (average of 14%) than the upper half of the span (average of 12%). The turbulence intensity rises sharply near the casing (95–100% span) but this is thought to be caused by a local interaction between mainstream and traverse probe access port, resulting in disturbed cavity flow. As this is local to the probe and at a single circumferential location (vane wakes were surveyed at a different circumferential location) we use a CFD-informed extrapolation in the 95–100% span region. The CFD-predicted profile of turbulence intensity (dotted line) has a similar general trend to experiments, but with a slightly higher average value (16%) and two moderate peaks (maximum of 21%) at 10% and 87% span. At the hub and casing, CFD predicts turbulence intensities of 15% and 13%, respectively. Low gradient of turbulence intensity in the approach to the wall is a common feature of *k–ɛ* CFD predictions because the inner boundary layer is not resolved. Extrapolations of the experimental data at the hub and case are shown as dashed lines.

Consider now the profiles of normalized turbulence length scale, ℓ/*C*_{x}, which are shown in Fig. 3(b). For the traverse data, ℓ was calculated from the product of the time-averaged velocity magnitude and the integral time scale (average of three principal components). Time scales were determined using the method of Nix et al. [24], integrating the normalized autocorrelation function up to the first zero crossing. CFD time scales were calculated from turbulence kinetic energy, *k*, and turbulence dissipation rate, $\u03f5$, using the well-known relation $\u2113\u221dk1.5/\u03f5$. Both the traverse and CFD profiles of turbulence length scale are relatively flat in the 20–80% span range, with values of approximately 1.10 *C _{x}* and 1.26

*C*, respectively. The traverse data has two moderate peaks (maximum value 1.40

_{x}*C*) at 10% and 89% span. These are not seen in the CFD and are thought to originate from flow separation at a step contraction of the duct, at the interface between the feed manifold and the test section. In the approach to the wall both the experimental and CFD data show reducing length scales, as we expect. It was not possible to perform measurements at the hub wall due to a probe size restriction, and the casing measurements were disturbed by cavity flow interaction. For this reason, in the near-wall region a patching method was used for this experimental data, informed by CFD and the proportionality between length scale and wall distance in the log-layer [5]. Patched data is shown by the dashed line.

_{x}Overall, the agreement between traverse data and CFD is good, especially as RANS frequently fails to predict the length scale parameter correctly.

## Numerical Methods

To complement the experimental measurements, back-to-back CFD simulations were performed in ANSYS CFX (version 22.2). The computational model was based on the engine geometry with fully-resolved internal features. Steady simulations were performed at the nominal rig operating point. We now describe the computational domain, grid, boundary conditions and solver settings.

### Computational Domain.

The computational domain includes two vanes (designed in pairs, with slightly different cooling schemes for *leading* and *trailing* vanes) and both external and internal flows. This is shown in Fig. 4. The internal features include all vane internals, fully resolved cooling holes and the coolant feed plena. The upstream domain extent is the inlet cassette measurement plane (plane 1; see Fig. 1), and the downstream extent is approximately three axial chords downstream of the NGV TE (parallel duct). The domain exit is approximately co-incident with the exit of the annular duct downstream of the vanes in the experiment.

### Computational Grid.

A hybrid grid was generated with Boxermesh (version 3.8.3). The mesh consists primarily of unstructured hexahedra, with variable grid refinement near the walls and in the wake. A 23‐layer inflation region was used near the walls, giving average *y*^{+} values of 0.7, 1.3, 0.6, and 0.8 on the pressure side, suction side, endwalls, and internals of the vane, respectively. Local mesh refinement around small features was down to 0.04 mm, giving approximately 35 cells across a cooling hole diameter. The overall mesh had approximately 312 million cells. Detail views of the TE/wake refinement, film cooling holes, and internal rib turbulators are shown in Fig. 5. Sensitivity to grid refinement was evaluated by refining the mesh globally. This is discussed in Appendix B. Results are analyzed and discussed using data from the most refined grid.

### Boundary Conditions.

The mainstream inlet boundary conditions of total pressure, total temperature, turbulence intensity, and length scale were set to the extrapolated traverse profiles at plane 1 (see Figs. 2 and 3). The hub and case coolant feeds were mass flow rate boundary conditions; 5% turbulence intensity and a viscosity ratio of 10 were specified. A radial equilibrium condition was used for the outlet static pressure, with the average outlet pressure adjusted iteratively to match the experimental vane pressure ratio (inlet total to platform static). A conformal periodic rotational interface was used for the circumferential boundaries of the domain. All remaining boundaries were modeled as adiabatic no-slip walls.

### Solver Settings.

Simulations were steady-state using the *k–ω* SST turbulence model, with a turbulent Prandtl number of 0.50, which is optimized for free shear thermal mixing layers (see e.g., Chambers et al. [8]). The *high-resolution* discretization scheme, which is formally *second-order-accurate*, was employed for both the momentum equations and the turbulence closure method. The convergence criteria were that residuals had entered a limit cycle, with monitors of inlet pressure and outlet entropy oscillating around a constant average value. At convergence, domain imbalances were oscillatory, with a maximum amplitude of 0.02% for mass and enthalpy.

## Performance Metrics

In this section, we summarize the performance metrics for our analysis of the aerodynamic and thermal field downstream of the NGVs.

### Whirl Angle Definition.

*α*, by

*V*

_{x}, and $V\theta $ are, respectively, the axial, and circumferential components of velocity in a cylindrical coordinate frame coincident with the machine axis.

### Kinetic Energy Loss Coefficients.

As measures of the aerodynamic efficiency of the NGV, we define four kinetic energy loss coefficients: a *local* KE loss coefficient, a *circumferentially-averaged* KE loss coefficient, a *plane‐averaged* KE loss coefficient, and a *mixed-out* KE loss coefficient.

*local*kinetic energy (KE) loss coefficient,

*ζ*

_{1}(

*r*,

*θ*), is defined by

*p*

_{01}is a

*reference*inlet total pressure,

*p*

_{02}is the downstream total pressure (measured at plane 2, 3 or 4, in our experiments), and

*p*

_{2}is the corresponding static pressure. The exponent

*χ*is defined by

*χ*= (

*γ*− 1)/

*γ*, where

*γ*the ratio of specific heats of air, which was assumed constant and equal to 1.40. Here the reference inlet total pressure (

*p*

_{01}) is calculated in real-time from the arithmetic mean of the fixed-rake total pressures (

*p*

_{01,rake}) using a fixed, linear scaling factor

*k*

_{p}, which is derived from the rake data and the more detailed area survey data. That is:

*p*

_{01}=

*k*

_{p}

*p*

_{01,rake}, where

*p*

_{01,rake}is the instantaneous average of the pressure measurements from fixed rakes. The local KE loss coefficient represents the ratio of the local kinetic energy deficit and the energy associated with an isentropic expansion from the mainstream total pressure,

*p*

_{01}, to the downstream local static pressure,

*p*

_{2}. We use it as a measure of the local aerodynamic inefficiency of the expansion.

*circumferential-average*KE loss coefficient (see discussion of Burdett and Povey [12]),

*ζ*

_{3}, as

*p*

_{01}and

*p*

_{02}have the same meaning as in the preceding equation. Overbars represent mass-flux averaging in the case of

*p*

_{02}, and area averaging in the case of

*p*

_{2}. The area and mass-flux averages were evaluated across two vane pitches in the circumferential direction, with the circumferential limits arranged to track the same wakes at each axial location downstream of the vanes. When reporting midspan data, averages were taken between 45% and 55% span.

*plane-averaged*KE loss coefficient is defined according to the definition of Lawaczeck [25]:

*p*

_{0c}is the coolant total pressure. The overbars represent mass-flux averaging for

*p*

_{02}, and area averaging for

*p*

_{2}, both over the full span.

*T*

_{01}is the total temperature at plane 1, and

*T*

_{0c}is the mass-weighted average total temperature of the hub and case coolant feeds. This plane-averaged KE loss coefficient represents the kinetic energy deficit of the bulk flow, relative to the mass flow rate-weighted kinetic energy of hypothetical isentropic expansions of both coolant and mainstream flow to the area-average downstream static pressure, $p2\xaf$. We use it as a single scalar measure of the inefficiency of the entire expansion, accounting for excess pressure in the coolant stream. For cold testing,

*T*

_{01}=

*T*

_{0c}such that the temperature terms cancel out, and the definition of plane-averaged KE loss coefficient used by Burdett and Povey [12] is recovered.

*mixed-out*KE loss coefficient, using the Dzung mixing model [26] (see also discussion in [12]), by

*p*

_{02}and

*p*

_{2}(indicated with tilde) are performed according to the Dzung-method [26] in which a mixed-out state is calculated from the local flow field according to one-dimensional conservation laws for mass, momentum in the meridional plane, angular momentum around the machine axis, and energy. The equations for this are summarized by Pianko and Wazelt [27]. The mixed-out KE loss coefficient considers the loss to a hypothetically mixed state, and so accounts for loss that would arise from the complete mixing out of existing secondary flow. This allows comparison between flows which are mixed to different degrees, for example, a comparison between planes 2, 3 and 4 in the experiment, or between experiment and CFD at a particular plane.

*secondary kinetic energy*coefficient (SKE), which is defined as the difference between mixed-out and plane-averaged KE loss coefficients, i.e.,

A reduction in SKE indicates the mixing out of a non-uniform flow, with a value of zero corresponding to the fully-mixed state. In real flows, over finite length scales, SKE may decay to a stationary, but non-zero value. This occurs if the real flow does not mix-out to the same equilibrium conditions as the mixing-model used to estimate *ζ*_{5} (see comparison of mixing models by Burdett and Povey [28]).

### Flow-Effectiveness: The Non-Dimensional Flow Temperature.

*flow effectiveness*, Θ, by

*T*

_{01}is a

*reference*inlet total temperature,

*T*

_{02}is the downstream total temperature to be measured, and

*T*

_{0c}is the mass-weighted-average total temperature of hub and case coolant feeds. In similarity to the reference total pressure, the reference total temperature is calculated in real-time from the arithmetic mean of the fixed-rake total temperatures (

*T*

_{01,rake}) using a small, fixed, linear scaling factor

*k*

_{T}, which is derived from the rake data and the more detailed area survey data. That is

*T*

_{01}=

*k*

_{T}

*T*

_{01,rake}, where

*T*

_{01,rake}is the instantaneous average of the temperature measurements from fixed rakes. A physical interpretation of the flow effectiveness, Θ, is that the related variable

*λ*= 1 − Θ is equal to the cold gas mass fraction.

## Experimental and Numerical Results

In this section we compare and analyze the experimental and numerical results. Estimates of the experimental measurement uncertainty are summarized in Appendix A.

### Nozzle Guide Vane Exit Circumferential Static Pressure Profiles.

Experimentally-measured and CFD-predicted circumferential distributions of NGV platform overhang pressure are presented in Fig. 6. The purpose is to validate the operating pressure ratio in CFD. Static pressures are normalized by the upstream total pressure. Flow over two passages is shown, with the circumferential coordinate, *θ*_{NGV}, normalized by the vane pitch, *s*. The vane TE positions correspond to *θ*_{NGV}/*s* = 0, 1.

Overall the CFD prediction is in reasonable agreement with the experimental data, with the radial equilibrium component well-matched, but with an exaggerated circumferential potential field disturbance in the CFD when compared to the experiment. Minima in the pressure distribution occur at approximately *θ*_{NGV}/*s* = −0.6 and *θ*_{NGV}/*s* = 0.4, at the point of peak-suction on the uncovered suction side. Maxima occur at approximately *θ*_{NGV}/*s* = −0.1 and *θ*_{NGV}/*s* = 0.9, corresponding to the TE region where the PS and SS flows meet. These can be explained with reference to the CFD-predicted potential field at midspan, which is shown in Fig. 7. The axially-projected TE locations (at *θ*_{NGV}/*s* = 0.0 and 1.0) are marked TE1 and TE2. Regions around these lines are associated with high-pressure disturbances. These modify the whirl angle as the flow passes through them. We consider this effect in the next section.

### Whirl Angle Profiles.

Experimentally-measured and CFD-predicted whirl angle (*α*) distributions at downstream planes 2, 3, and 4 are shown in Fig. 8. The data are datumed to the mean midspan metal angle of the vane, *α*_{TE}, which is in the typical range for a modern transonic turbine with approximately 50% reaction; i.e., we present *α* − *α*_{TE} in Fig. 8. The experimental traverse data and CFD analysis consider the same two wakes at each plane. That is, the data analysis region moves around the annulus following the wakes. We differentiate between the NGV-fixed-reference-frame circumferential coordinate, *θ*_{NGV}, and the wake-following-reference-frame circumferential coordinate *θ*_{wake}. The wake-following circumferential coordinate translates around the annulus at the same rate as the mass-mean flow. This is shown diagrammatically in the lower half of Fig. 8. The wakes, located around *θ*_{wake}/*s* = 0.0 and 1.0, are marked W1 and W2, respectively. In the upper half of Fig. 8, a contour line of normalized local KE loss corresponding to *ζ*_{1}/*ζ*_{ref} = 0.5 is shown, where *ζ*_{ref} is defined as the experimentally-measured mixed-out KE loss coefficient at plane 4. Axially projected locations of the NGV TEs are marked TE1, TE2 and TE3. The regions axially downstream of the TE are associated with a high-pressure perturbation in the vane potential field (see Fig. 7).

First, consider the experimental whirl angle distribution at plane 2. The radial distribution of flow-angles is primarily dictated by spanwise metal-angle distribution, which is approximately 3 deg higher in the near-casing region than the near-hub region. Comparing the CFD to the experiment, we see very good agreement between the distributions, with a maximum local difference of approximately 5 deg.

Downstream of the TE, *local* whirl angle modulation can be introduced by the vane potential field. As the flow passes through the vane potential field, it is deflected toward the axial direction in a positive axial pressure gradient (i.e., toward lower pressure) and away from the axial direction in a negative axial pressure gradient. The direction of deflection is indicated with arrows in Fig. 7. The effect explains the slightly lower whirl angles on the immediate right of the high-pressure regions axially downstream of the vane TEs (aligned with TE1, TE2 etc.) and slightly higher whirl angles axially downstream of the vane SS (midway between vane TEs). The result is a slightly sinusoidal distribution of whirl angle with circumferential displacement.

Now consider the experimental whirl angle at plane 3. Here the flow is partially mixed-out, with lower local variation. As we move downstream in the wake-following reference frame, the vane-fixed potential field moves toward the left. This is indicated using arrows in Fig. 8. The high-pressure perturbations now intersect plane 3 at approximately *θ*_{wake}/*s* = 0.1 and 1.1. Corresponding regions of low whirl appear at approximately around *θ*_{wake}/*s* = 0.4 and 1.4. At plane 3 the high whirl location (approximately half a vane pitch out of phase with the low whirl region) coincides with the wakes (*θ*_{wake}/*s* = −0.1; 0.9), resulting in an interesting interaction with the potential field: in the upper half of the span, the peak whirl angle is higher than at plane 2. This local *decrease* in uniformity relative to plane 2 appears to occur because the relatively low momentum fluid in the wake is more strongly affected by the potential field, which acts to increase whirl angle. A similar—but reversed—effect can be seen at plane 4, where the whirl angle in the lower half of the wake is lower than at plane 3 (increasing non-uniformity) due to the low momentum wake fluid being overturned toward the axial direction by the positive axial pressure gradient in this circumferential location at this plane. The effect is particularly pronounced for the CFD results.

Radial profiles of circumferentially-averaged (mass-flux-weighted) whirl angles are shown in Fig. 9. The general trend is a decrease in whirl angle in the direction from casing to hub with excursions from this trend in the near-wall secondary-flow regions. Between 25% and 75% span the CFD-predicted profiles are in excellent agreement with the experimental data, with maximum differences of only 0.5 deg between the two data. Above 75% and below 25% span, in the near-wall regions there is greater disagreement between CFD and experiment, with CFD predicting higher turning than measured by up to 1.6 deg. As a result, mass-flux-weighted whirl angle was slightly greater in CFD (evaluated over the area common with experiments), with values (relative to experiments) of +0.65, +0.20, and +0.02 deg at planes 2, 3, and 4, respectively. Discrepancies of the mass-flux-weighted whirl angle are similar in magnitude to the bias uncertainty of the experimental data, which we estimate to be around ±0.8 deg (95% confidence interval; see Appendix A)

An increase in absolute mass-flux-averaged whirl angle between planes 2 and 3 is seen for both the experimental data (+0.51 deg) and numerical results (+0.16 deg). This is thought to occur due to a slight reduction in axial velocity caused by a slight increase in passage area in the downstream direction (see Fig. 1), with approximate conservation of angular momentum of the bulk flow.

A notable feature of both experimental and numerical results is the double-peak whirl angle pattern between 55% and 90% span. Here, the flow angles are distorted by a clockwise-rotating (when viewed from downstream) vortex centered at approximately 80% span, which raises whirl angle around 85% span and reduces whirl around 75% span. The minimum at approximately 75% span gives rise to a weak second peak at approximately 60% span. The vortex develops near the leading-edge SS, seemingly as a result of an interaction between the casing endwall cooling flow and the vane leading-edge potential field. Noting the location of origin and direction of rotation, it is interesting that the vortex is not identifiable as either a conventional *passage vortex* or a conventional *counter-vortex* according to the generally-accepted terminology. This highlights the complexity of secondary flows for heavily cooled vanes. Similar modulations of the whirl angle, visible around 10%, 30%, and 55% span in the CFD data, correspond to smaller vortex structures in the flow.

### Local Kinetic Energy Loss.

Experimentally-measured and CFD-predicted distributions of local KE loss coefficient at downstream planes 2, 3, and 4 are shown in Fig. 10. The wakes are marked W1 and W2. Both experimental and CFD-predicted values of local KE loss coefficient, *ζ*_{1}, have been normalized by the experimentally-measured *mixed-out* KE loss coefficient at plane 4. We refer to this value as the reference KE loss coefficient, *ζ*_{ref}.

First, consider the normalized experimental local KE loss coefficient at plane 2. The wakes have the shape of the positively compound leaned TE, with relatively uniform peak KE loss coefficient across the span. Local peaks at approximately 12% and 79% span correspond to hub and casing secondary flows, respectively. The casing endwall BL is marked by a region of low momentum fluid which is thicker toward the SS corner of the vane due to the cross-passage migration effect. A similar pattern can be observed at the hub, but with—on average—greater BL thickness than at the case. This likely results from a combination of higher speed flow near the hub wall, and local separation caused by diffusion on the lee side of a high-curvature blend in the aft hub platform just upstream of plane 2 (see Fig. 1). The CFD predictions are in good agreement with the experiment, with similar overall distributions of loss, but with evidence of undermixing in several regions: narrower wakes with higher peak values; thinner BL regions near the endwalls. This results from undermixing in the CFD method, a seemingly common problem when simulating transonic NGV flows using turbulence models (see, e.g., [12,14]).

Now consider the experimental KE loss at plane 3 and plane 4. As expected, the wakes and secondary flow features mix out as the flow progresses downstream. The wakes also become stretched-out in the circumferential direction due to radial variation in the progression of the wake. That is, radial variation in the ratio of angular displacement to axial displacement, which depends on the local whirl angle and radius as *dθ*_{NGV}/*dx* ≈ tan(*α*)/*r*. The stretching-out of the CFD-predicted and experimentally-measured wakes is similar, but with greater local variation in the shape for CFD than experiment in the near-casing region (70% to 100% span) for planes 3 and 4. This is explained by the greater local angle variation in the CFD than experiment. The CFD wake also becomes slightly more asymmetric in appearance at planes 3, and 4 (i.e., less swept-back near the casing). This indicates greater angular displacement in CFD than in the experiment, which is consistent with the overprediction of whirl angles in this region (see Fig. 9).

Radial profiles of circumferentially-averaged kinetic energy loss coefficient, *ζ*_{3}, are shown in Fig. 11 for planes 2, 3 and 4 for both experiment and CFD. The values are normalized by *ζ*_{ref}.

First consider the *experimental* data at plane 2. The experimental profiles of KE loss are relatively flat between 20–60% span, with local peaks around 15% and 75% span, which are associated with the secondary flow loss cores. Near the walls, KE loss increases steeply on account of the endwall boundary layers. In the corresponding CFD prediction for plane 2, the radial distribution of KE loss—including the secondary flow peaks—is captured reasonably well. In the *profile loss region*, which we define to be the central 80% of the span, the profile KE loss is slightly lower in the CFD simulation than the experiment. The CFD-predicted boundary layers are also thinner than for the experiment.

Moving axially downstream, to planes 3 and 4, we observe boundary layer growth in both experiments and CFD. We also note an increase of KE loss in the 20–60% span range. This is associated with the conversion of secondary kinetic energy (SKE) into realized losses (i.e., the generation of loss through mixing-out of non-uniform flow). We consider mixing characteristics quantitatively in the next section by examining the change with streamwise distance of wake widths, peak loss and integrated loss.

### Circumferential Kinetic Energy Loss Distributions at Midspan: Wake Width, Peak Loss, and Integrated Loss.

In this section we consider the wake development with streamwise distance in more detail. Midspan distributions of normalized KE loss at planes 2, 3, and 4 are shown in Fig. 12 for both experimental measurements and CFD predictions. The wakes are marked W1 and W2. We analyze these distributions in terms of wake width, peak loss, and integrated loss. We determine wake width and peak loss from the radially mass-flux-weighted average profile of *ζ*_{1} between 45% and 55% span, where the width is the full width at half maximum. Similarly, integrated midspan loss is calculated using Eq. (3), where area and mass averages are evaluated between 45–55% span. Reported values are the average of both W1 and W2. An important result of Burdett and Povey [12] is that apparently disparate trends of these parameters when presented as a function of downstream *axial* distance from a nominal TE plane collapse extremely well when presented as a function of *real streamwise distance* from a local TE position. For this reason, we present trends of these parameters as a function of streamwise distance. In the calculation of streamwise distance constant whirl angle is assumed between the downstream measurement planes. The resulting trends are shown in Fig. 13. For the current experiment, midspan data at planes 2–4 have equivalent streamwise distances, *d*, of approximately 0.99 *C _{x}*, 1.99

*C*, and 3.00

_{x}*C*.

_{x}First consider the experimental and CFD data in Fig. 12 at downstream plane 2. The most striking feature is the undermixed CFD wake in comparison to the experiment. Here the peak KE loss coefficient exceeds that of the experiment by 30% for W1 and 43% for W2. The CFD wake is correspondingly narrower: we analyze the relationship between peak loss and wake width in the following paragraphs. In the experimental data, W1 and W2 are broadly similar, but *peak* loss for W2 is 13% lower than for W1. This is thought to be an artifact of spanwise loss redistribution resulting in a radial variation in the wake depth. An example of this can be seen at approximately 50% span for W2 in both the plane 2 and plane 3 data of Fig. 10, where there is a *pinch* in the wake. This can occur by local streamline divergence on the vane surface, causing boundary layer redistribution, which is a common effect particularly for the SS flow. Similar effects can be seen in all three planes of the CFD-predicted distributions of Fig. 10, with distinct radial variation along the wake centerline. Another interesting feature of the experimental data is the non-zero loss in the freestream (*θ*/*θ*_{wake} ≈ 0.5 and 1.5). Ames and Plesniak [29] showed that these so-called *background losses* occur at elevated inlet turbulence intensities, but their origin is not fully understood. Looking now at the CFD-predicted data in Fig. 12, W1 and W2 have *peak* loss within 4% of each other. This lends credibility to the earlier theory that the 13% difference in the experimental peak intensities is an artifact from loss redistribution, likely due to hardware variations. The CFD data do not show background losses.

Data at plane 3 and plane 4 show the general trend of wakes broadening with peak loss coefficient values reducing, as the wakes mix out. The rate of mixing is higher in the experiment than in CFD. We now quantitatively analyze these mixing rate trends.

The trends of wake width, peak loss, and integrated loss (Eq. (3)) with streamwise distance, *d*, are shown in Fig. 13. The CFD and experimental trends from the current study are compared to experimental and CFD-predicted data from Burdett and Povey [12]. This is an interesting reference point because it is a second example of a heavily cooled NGV from a modern aero-engine. The study [12] was conducted two years before the current experimental campaign, using a different working section, with largely independent instrumentation, and a separate processing team. This is relevant because it shows the independence of the two datasets. Although both engines had identical vane count, and were from the same family, an important difference between [12] and the current study is that [12] used a centered-ejection TE design (see Burdett and Povey [30]) whereas the current study used a SS-overhang TE design. As we will see, the centered-ejection design has higher overall loss, higher wake thickness, and lower peak loss: the reasons for this are complex and are influenced by the size of the TE *base region* and the unsteady shedding in this region, which is itself influenced by the TE coolant ejection design and momentum flux [30]. For both studies, we consider the restacked-wake-average between 45% and 55% span in which before averaging, each radial band of data is shifted circumferentially so that the peaks are aligned. The reason for comparing restacked data is that there can be significant lean of the mid-span wake, which in a direct averaging approach would lead to artificial mixing of misaligned peaks.

In the study of Burdett and Povey [12] the non-dimensional TE thickness (TE thickness normalized by the axial chord) was 1.27 times thicker than the current study. To facilitate closer comparison of the *streamwise trends* on a like-for-like basis with the data of the current study, in Fig. 13 we also present *rescaled* trends of wake width and peak intensity from the study Burdett and Povey [12]. Wake width is divided by 1.27 and (to preserve the product of normalized width and normalized peak loss) normalized peak loss is multiplied by 1.27. This is a crude rescaling method but allows first-order correction for the difference in geometries.

Consider first the evolution of wake width with streamwise distance, which is shown in Fig. 13(a). The wake width, *w*, has been normalized by the vane pitch, *s*. In the current experimental data, the wake successively grows from approximately 0.15 *s* at plane 2, to 0.26 *s* and 0.34 *s* at planes 3, and 4 respectively. The trend is approximately linear with streamwise distance. The slightly decreasing gradient of wake width with streamwise distance is consistent with the self-similarity solution for the far-wake, where the halfwidth is proportional to the square root of the streamwise distance (see e.g., Wilcox [5]). Now consider the experimental data of Burdett and Povey [12]. In this data the wake widths were higher (0.29 *s*, 0.35 *s*, and 0.45 *s*, at planes 2, 3, and 4, respectively) than the current experiment by—on average—a factor of 1.45. The greater wake width for the experimental results of Burdett and Povey [12] can be explained in part by the 27% greater non-dimensional TE thickness, considering that we might—simplistically—expect the wake width at the TE to be proportional to the sum of SS and PS momentum thickness and the TE metal thickness. Indeed, the rescaled trends of experimental wake width, which are shown with cross markers in Fig. 13(a), indicate that the experimental trends of the current study and those of Burdett and Povey [12] almost collapse once the differences in TE thickness are adjusted for. At *d*/*C*_{x} ≈ 1, rescaling closes the gap to 0.08 *s*, whereas for *d*/*C*_{x} ≈ 2, and *d*/*C*_{x} ≈ 3, the difference is reduced to 0.02 *s*, and 0.01 *s*, respectively. The significantly greater discrepancy at the first plane might be expected, considering the effects of centered-ejection on wake spreading characteristics [30].

Now consider the CFD results for both studies. Predictions of wake spreading rates have a number of implications for engine design, for example in terms of establishing the unsteady aerothermal load on the downstream rotor. The trends of CFD-predicted wake width with streamwise distance are broadly similar to the experimental data but with two distinct differences.

Firstly, there are a number of distinct plateaus in the rising trend, which are explained by interaction of the potential field and the wake, which periodically increases or decreases the wake whirl angle, leading to modulation in the wake width as measured on an axial plane. This behavior cannot easily be captured in the experimental data, due to the comparatively large spacing between the measurement planes. Secondly, an offset of the CFD-predicted wake width from the experimental data, with the wake width being underpredicted on average across three experimental planes by 34% in the current study, and by 23% in the study of Burdett and Povey [12]. This effect arises because of undermixing in the CFD solution. To take account of the larger TE thickness in [12], rescaled CFD trends from [12] are shown in Fig. 13(a) as dashed lines. This closes the gap with the current study by approximately half the initial difference. Overall, the experimental and CFD trends in these—essentially—independently constructed studies show remarkably similar characteristics.

Now consider the trends in the peak value of KE loss coefficient. These are shown in Fig. 13(b). For the current study, both CFD and experimental wake depth follow a trend of quasi-exponential decay with similar proportional streamwise decay *rates*. The peak CFD loss coefficient was consistently higher than the experiment (undermixing), with an average value (across three experimental evaluation planes) 45% higher. The experimental results of Burdett and Povey [12] have similar trends, though normalized peak intensity was on average only 66% of that in the current data. For the rescaled trend (i.e., multiplied by the non-dimensional TE thickness ratio), this figure is improved to 84%.

For the CFD data of [12], the trend was, on average, 77% higher than the corresponding experimental trend. Higher peak loss is—of course—concomitant with lower wake width. For the current study and the study of Burdett and Povey [12] we might crudely consider that the 34% and 23% deficits, respectively, in wake width, might—for the same overall loss—be accompanied by a corresponding excess in wake depth of 52% and 30% (inverse percentages). The former value (current study) is in good agreement with the difference in wake width between CFD and experiment. That is, we expect CFD to predict integrated loss relatively accurately, even though the flow is undermixed. The latter value (Burdett and Povey [12]) accounts for only half of the excess wake width predicted by CFD. The remainder is accounted for by an overestimate in integrated loss. In this case the CFD is both undermixed and overestimates integrated loss. We now consider this in more detail.

Trends of integrated midspan loss with streamwise distance are shown in Fig. 13(c). For the current study, experimental loss increases slightly with axial distance, from 0.60 *ζ*_{ref} at plane 2 to 0.68 *ζ*_{ref} at plane 4, along a shallow linear trend. The CFD trend is consistently lower, with an average difference of 14% across three measurement planes. Based on the 34% under-prediction of wake width and the 45% over-prediction of wake depth, an underprediction of integrated loss of 4% would have been expected. The additional 10% deficit can be explained by non-zero freestream loss, which is significant for the experimental data only (see circumferential profiles in Fig. 12). As we have mentioned, these so-called background losses have been observed to occur at high levels of inlet turbulence [29]. Now consider the CFD trend of integrated loss with streamwise distance. CFD-predicted midspan KE loss increases approximately linearly, reaching a maximum around *d*/*C*_{x} = 2, after which it begins to reduce again. A reduction indicates redistribution of loss to other radial positions. This occurs due to the tendency of low momentum fluid in the wake to migrate radially-inward. This highlights the limitations of using a narrow radial band as a proxy for profile. The experimental data of Burdett and Povey [12] are relatively similar in both magnitude and form to the current study. This is not unexpected as the engine parts were taken from engines of the same family (but different generations). So far as the CFD comparisons of Burdett and Povey [12] are concerned, the trend is similar to the experimental data (approximately constant), but the integrated loss is overestimated by on average 23%. Presumably, this is related to the excess loss, which arises from modeling film cooling holes using point-sources as opposed to resolving them, as was done in the current study.

### Mixed-Out and Plane-Averaged Kinetic Energy Loss Coefficients.

In this section we consider the mixed-out (*ζ*_{5}) and plane-averaged (*ζ*_{4}) KE loss coefficients for the entire flow. These are presented in Fig. 14 as a function of the normalized axial coordinate for both the experimental data (three planes) and CFD (evaluated at axial intervals of 0.4 mm). The data are normalized by the experimentally-measured mixed-out loss coefficient at plane 4, *ζ*_{ref}. We also decompose the plane-averaged loss coefficient into a profile-region loss coefficient, *ζ*_{p} and an endwall-region loss coefficient, *ζ*_{e}. A common approach is to define endwall loss as the difference between plane-averaged loss, and profile loss (see e.g., Coull [31]). A profile-region KE loss coefficient was evaluated by calculating the mass-flux-weighted total pressure over the central 80% span and substituting this value for *p*_{02} in Eq. (4). Because the choice of span range for calculating profile loss is somewhat arbitrary, it has been verified that the sensitivity of profile KE loss to the span range is acceptable. For averaging windows that include 40% to 90% of the *total span* but arranged symmetric about the midspan, the profile loss varied with averaging window size by less than ±3% at any one axial plane.

First, consider the CFD-predicted normalized mixed-out KE loss coefficient (solid line). At the TE plane, normalized mixed-out KE loss coefficient takes a value of 0.90. The trend with axial distance is that it decreases to a minimum of 0.80 at approximately *x*/*C*_{x} = 0.2, before rising approximately linearly to 0.88 at *x*/*C*_{x} = 0.75. In principle, we would expect the mixed-out loss coefficient to remain constant by definition. The initial *decrease* occurs because of continued flow turning at the hub and casing, where the airfoils extend up to 0.2 *C*_{x} (recall the TE datum, *x*/*C*_{x} = 0.0, is defined at midspan). In the context of the Dzung mixing framework, the additional flow turning reduces mixed-out loss, because it increases the downstream KE. A quasi-linear *increase* of mixed-out loss—starting around 0.20 *C*_{x}—is the result of KE dissipation through friction at the endwalls. Over the axial range of experimental data, the CFD-predicted mixed-out KE loss trend is similar to the corresponding experimental trend (triangular markers), but with CFD underpredicting mixed-out loss by 12% on average across three measurement planes. The fact that the trends are similar indicates that the *rate* of endwall loss production is accurately captured by CFD, this being the only mechanism for increasing mixed-out KE loss for a constant-area horizontal duct.

Now consider the CFD-predicted normalized plane-averaged loss coefficient (dashed line). At the TE (*x*/*C*_{x} = 0.0) plane-averaged KE loss increases steeply due to mixing of the PS, SS, and TE coolant streams, which are all initially at different velocities. The plane-averaged loss approaches the CFD-predicted mixed-out KE loss trend (solid line) quasi-asymptotically, as secondary kinetic energy (SKE) is converted into realized losses. Around *x*/*C*_{x} = 0.9, the lines representing CFD-predicted mixed-out and plane-averaged KE loss intersect such that *ζ*_{4} > *ζ*_{5}. This is surprising because it implies that SKE is negative, which is *conceptually* non-physical. Mathematically, however, SKE—calculated using Eq. (6)—can adopt small negative values, because mixed-out loss depends on the *assumed* mixed-out velocity profile (see e.g., comparison of free and forced vortex models in [28]). The fact that SKE can only be guaranteed everywhere-positive when the real flow mixes out to the same conditions as assumed by the mixing model points to the limitations of mixing models, and the arbitrariness of the definition of SKE.

Now consider the experimentally-measured normalized plane-averaged KE loss coefficient (circular markers). In close similarity to the CFD trend, the experimental plane-averaged KE loss also approaches the corresponding (experimental) mixed-out KE loss asymptotically. The CFD-predicted plane-averaged KE loss is 12% lower than the corresponding experimental trend, on average across three measurement planes. This is interesting because the difference is very similar to that for the CFD-predicted mixed-out KE loss, which is lower than the experimentally-measured mixed-out KE loss by on average 12%. As a result, the SKE coefficients (dash-dot-dotted line for CFD; cross markers for experiments) for experiments and CFD are exceptionally well matched, even though the absolute mixed-out and plane-averaged KE loss are not.

To further explore the numerical under-prediction of KE loss, we decompose plane-averaged KE loss into profile-region and endwall-region KE loss coefficients. By definition, profile-region KE loss and endwall-region KE loss coefficients are summative to the plane-averaged KE loss coefficient.

CFD-predicted profile-region loss (dash-dotted line) rises sharply with axial distance before leveling off at a value equal to 0.66 times *ζ*_{ref}, at approximately *x*/*C*_{x} = 0.55. Experimentally-measured profile-region loss (diamond markers) is consistently greater than the CFD prediction, with values of 0.69, 0.76, and 0.77 *ζ*_{ref}, at planes 2, 3, and 4, respectively. The average underprediction of profile-region loss by CFD was 12%, which accounts almost precisely for the deficit of mixed-out and plane-averaged loss.

Trends of normalized endwall-region KE loss (square markers for experiments; dotted line for CFD) are very similar to each other, increasing approximately linearly with axial distance. This is expected because circumferentially-averaged skin friction is relatively constant downstream of the vanes (see dissipation coefficient correlations of Denton [32]). A minor deviation from the linear trend is evident for the experimental data at *x*/*C*_{x} = 0.25, where accurate experimental characterization of the endwall flow is particularly challenging due to extremely thin boundary layers.

The trends of mixed-out loss, plane-averaged loss, SKE, and profile loss from the study of Burdett and Povey [12] are plotted alongside those from the current study in Fig. 15. Results from [12] were also normalized by the experimental mixed-out KE loss coefficient at a location equivalent to plane 4.

First consider the comparison of normalized mixed-out KE loss. We recall that downstream of the vanes, endwall-region losses are the only driver for an increase of mixed-out loss with axial distance. The experimental data of [12] have a slightly steeper gradient of mixed-out KE loss with axial distance than the current study, indicating a higher rate of KE loss production in the endwall boundary layer. This might be caused by a small endwall flow separation bubble in the experimental data of [12], with accordingly higher losses.

The trend of CFD-predicted plane-averaged KE loss from [12] has a trend which runs in parallel to—but below—the mixed-out KE loss from [12]. It is surprising, perhaps, that the plane-averaged loss does not approach the mixed-out loss asymptotically. The corresponding CFD-predicted SKE coefficient (dash-dot-dotted line) decays to a constant value of approximately 0.11 *ζ*_{ref} (around 0.75 *C*_{x}). As discussed earlier, constant non-zero SKE indicates that the real flow does not mix-out to the same equilibrium state as the mixing model. The relatively high value (0.11 *ζ*_{ref}) suggests a significant difference for this particular flow. Whilst the trends indicate important truths about the mixing rate, we note the difficulty of finely ranking different NGV designs with absolute values of SKE coefficient.

Finally, we consider the profile-region loss coefficient. Experimentally-measured normalized profile loss in both studies increases monotonically with axial distance, accounting on average for 56% of mixed-out loss in [12] and 78% in the current study. One possible explanation for the significantly lower *proportion* of profile loss in [12] is the greater rate of boundary layer growth, which increases endwall losses thus inflating the normalizing reference KE loss, *ζ*_{ref}. Indeed, the *absolute* value of the reference KE loss coefficient, *ζ*_{ref}, was approximately 1.45 times greater in the study of Burdett and Povey [12]. Therefore, the average *absolute* profile KE loss coefficient in [12] exceeds that of the current study by only 4%, as a fraction of the current profile loss coefficient. A slightly greater *absolute* profile loss coefficient for the centered-ejection design [12] might be expected due to the non-dimensional TE thickness being 1.27 times greater than the current study. We now consider CFD predictions of normalized profile-region KE loss coefficient. Compared to experiments, CFD under-predicted profile loss by 12% in the current study, whereas the CFD of [12] over-predicted profile loss by 18%. Burdett and Povey [12] hypothesize that their over-prediction of loss is associated with the point-source method used for modeling film cooling. This hypothesis is compatible with the current CFD study giving lower-than-experimental losses with fully-resolved film-cooling holes. The 12% underprediction of KE loss in the current study (±3% bias uncertainty; see Appendix A) might be explained in part by surface roughness effects, which were not modeled in CFD, or turbulence modeling. Surface roughness is known to increase skin friction and KE loss for NGV flows, as demonstrated in the studies of Erickson et al. [33] and Michaud et al. [34].

### Downstream Thermal Field.

A unique feature of the current experiments was that both the downstream loss distribution and the associated thermal field were measured in detail. This data is important for establishing rotor-frame boundary conditions, which dictate the thermal and aerodynamic forcing of the parts, and unsteady temperature segregation effect within the rotor row.

Non-dimensional temperature distributions, or flow effectiveness maps, Θ, for experiments and CFD at planes 2, 3, and 4 are shown in Fig. 16. For completeness we note that the thermal traverse was taken downstream of a different vane pair than the pressure traverse data in previous sections, but for nominally identical parts. Aerodynamic data (KE loss, flow angles etc.) was also collected for this vane pair, and to compare locations of the thermal wake and the momentum wake we superimpose contour line of KE loss corresponding to *ζ*_{1}/*ζ*_{ref} = 0.5 on the data in Fig. 16. The same data were not used in the pressure analysis sections because it did not extend all the way to the endwalls. We recall that a physical interpretation of the effectiveness, Θ, is that its complement *λ* = 1 − Θ is equal to the cold gas mass fraction.

As expected, the experimental effectiveness distributions at plane 2–4 are similar in general form to the KE loss distributions but with generally broader wakes. This is expected because—up to the TE at least—the *thermal mixing layer* is (in general) thicker than the aerodynamic boundary layer *for a film cooled part* because of both the mass-displacement effect (coolant entering the domain) and because of direct mixing of the coolant and mainstream (coolant jet mixing). Although both of these effects contribute to total pressure loss, the primary mechanism for total pressure loss—up to the TE—is friction with the wetted surface: i.e., the classical aerodynamic boundary layer. This classical aerodynamic boundary layer has an attendant thermal boundary layer for the case of cooled parts, this latter being slightly thicker (for air; Prandtl number of 0.71). Importantly, it is the direct mixing of coolant—however—which is responsible for most of the thermal mixing layer temperature deficit. As discussed, this occurs at greater scale than both the aerodynamic and thermal boundary layers.

For the case of film cooled parts with TE coolant ejection the situation becomes complex. The TE can be seen as a direct and significant source of both pressure loss (base pressure drag) and temperature deficit (slot injection of coolant) followed by enhanced mixing in the vortex street downstream of the TE, which enhances the mixing of both the thermal mixing layer (now a thermal wake) and the aerodynamic wake (now enhanced by the base pressure drag). In this region, there are, of course additional thermal diffusivity terms, but these might be expected to be small in comparison to the effective momentum diffusivity terms (large scale vortex structures). That is, in this region, we might expect the thermal wake and pressure wake to mix at similar rates dominated by large scale vortex structures.

To confirm that the experiment was quasi-steady during the test period, we compare the expected (energy conservation based on hot and cold mass flow rates) mass-averaged effectiveness to that measured (based on full-plane mass flow-averages) at the three planes. The theoretical value of flow effectiveness was 0.868 for all three planes, compared to values of 0.871, 0.865, and 0.868 at planes 2, 3, and 4, respectively. The discrepancies are within 0.17 K (or ±0.3%) of the 55 K hot-to-cold stream temperature difference, which is well within the measurement uncertainty of K-type thermocouples (see uncertainty propagation in Appendix A). This points to excellent conservation of energy, indicating effectively-steady flow. For CFD, the mass-average effectiveness was exactly 0.872 at all three planes; this was very slightly higher than for experiments because the coolant-to-mainstream mass flow ratio was not directly constrained (mainstream inlet was specified using a total pressure boundary condition).

Radial profiles of circumferentially-averaged (mass-flux-weighed) effectiveness are shown in Fig. 17 for experimental measurements and CFD predictions. Both experimental and CFD profiles have a relatively constant value between 20% and 70% span, with lower values toward the endwalls because of the endwall cooling flows. In the hub region (0% to 20% span) the experimental and CFD results are in reasonable agreement, but peaks and troughs in the CFD profile are not seen in the experimental data. These arise because of undermixed near wall cooling flow and undermixed secondary flows with entrained cold gas in the CFD solutions. This manifests as a distinct cold core at approximately 12% span at plane 2 in the CFD data of Fig. 16, where the near-wall flow is significantly undermixed. In the casing region between 70% and 100% span, there is greater difference between the experiment and CFD: the experimental distributions are well-mixed in the near-wall region, with the region of substantial temperature deficit extending to 70% span, suggestive of casing secondary flows spreading the cold endwall flow away from the casing wall (see Fig. 16 experimental data at plane 2). In contrast the CFD predictions are substantially undermixed, both for the near wall flow, suggesting less entrainment of the cooling flow into secondary flows, and the wake flow than expected, leading to distinct low temperature core at 75% span (see Fig. 16 CFD data at plane 2, and the corresponding Fig. 10 KE loss core).

For both the experimental and CFD data, as we progress from plane 2 to plane 4 the effect is for the profiles to become slightly more radially uniform, as gradients in the flow mix out (see near-wall regions). The effect is slight because the majority of the gradients and associated mixing are in the circumferential direction.

### Mixing Rates of Thermal Field.

The circumferential distributions at midspan (average between 40% and 60% span) of the cold gas mass fraction, *λ*, at planes 2, 3, and 4 are shown for experiments and CFD in Fig. 18. The corresponding distributions of normalized KE loss coefficient at midspan from Fig. 12 have been superimposed for comparison. Here, the aerodynamic wake has been scaled such that average *experimental* peak values of KE loss and cold gas mass fraction across all three planes are equal. The wakes are marked W1 and W2. The distributions are similar in general form to those for kinetic energy loss coefficient distributions, but with broader wakes for both the experimental and CFD distributions. As with the kinetic energy loss coefficient distributions the wake broadens, and the peak reduces with streamwise distance. The CFD data offer an interesting insight into the relative locations of peak cold gas mass fraction and peak rescaled KE loss coefficient. On average, the aerodynamic wake trails the thermal wake by 0.02 *s* in the circumferential direction. This is consistent with the fact that streamlines of peak KE loss originate from the base pressure region of the SS overhang (far left of TE; see Fig. 5(a)), whereas streamlines of peak cold gas mass fraction originate from the TE coolant slot (center of TE). We now analyze the thermal wake distributions in terms of wake width, *w*, and peak cold-gas mass fraction, *λ*_{max}, as a function of streamwise distance, *d*. These trends are shown in Fig. 19, and compared to the corresponding trends for KE loss coefficient (data of Fig. 13). To compare the decay rate of both thermal and pressure fields, absolute values of the peak cold-gas mass fraction (*λ*_{max}) were normalized by the arithmetic average of this parameter for the *experimental* data across planes 2–4 $(\lambda maxexp\xaf)$: i.e., we present $\lambda max/\lambda maxexp\xaf$. In the same manner, the maximum KE loss coefficient values (*ζ*_{max}) were normalized by the experimental average of peak values across planes 2–4 $(\zeta maxexp\xaf)$.

Consider first the streamwise evolution of the thermal wake width. This is shown in Fig. 19(a), in which we also compare to the pressure wake width. Both wakes were restacked prior to radial averaging. The experimentally-measured thermal wake widths were 0.24 *s*, 0.36 *s*, and 0.43 *s* at planes 2, 3, and 4 respectively, on average 37% wider than the experimental pressure wake widths. This is in line with our earlier argument that the mass displacement resulting from coolant injection causes the thermal mixing layer on the vane surface to generally be thicker than the aerodynamic boundary layer, resulting in greater exposure to, and mixing with the turbulent freestream. The proportional growth rates of momentum and thermal wakes are similar, with factor increases in momentum/thermal widths of 1.69/1.52 from plane 2 to 3, and 1.33/1.18 between planes 3 to 4. So far as CFD results are concerned, both the initial width and the spreading rate of the thermal wake were lower than in experiments. At planes 2, 3 and 4 the CFD-predicted widths were 0.16 *s*, 0.23 *s* and 0.26 *s*, respectively: an average underprediction of 38%. As we have discussed in the context of Fig. 13, CFD under-predicted the aerodynamic wake width by 34% on average. The suggestion is that an improved turbulence model is likely to address undermixing of both the aerodynamic and thermal wakes.

The streamwise evolutions of rescaled peak cold gas mass fraction and rescaled peak KE loss coefficient are shown in Fig. 19(b). The normalized experimental trends of thermal and pressure wake intensity are remarkably similar. Likewise, there is excellent agreement between the two CFD data in the range of streamwise distance 1 < *d*/*C*_{x} < 5, but with differences from the experimental data of—on average—53% and 45% for the thermal and pressure wakes. The similar trends in the far-field, when the flow is already partially mixed out, presumably arise because both mixing of the pressure and thermal fields is dominated by turbulent transport rather than laminar thermal diffusivity (i.e. conduction). In the near-wake (*d*/*C*_{x} < 1), the normalized decay rate of the aerodynamic wake intensity is significantly greater than for the thermal wake. Presumably, the greater rescaled intensity of the aerodynamic near-wake is related to the viscous dissipation of kinetic energy. Viscous dissipation scales with the square of the strain rate, which explains why the effect is strongest in the near-wake. The viscous dissipation effect is twofold, because it represents both a significant source of KE *loss*, and a *redistribution* of total enthalpy away from the wake centerline (see discussion by Kopriva et al. [14]). This is likely compounded by a small spatial difference in origin of streamlines with peak KE loss and peak cold gas mass fraction (rooted at the SS overhang base pressure region, and the TE coolant slot, respectively), which causes the aerodynamic and thermal wakes to experience differing levels of shear stress.

A second interesting difference between the decay trends of the thermal and aerodynamic wakes is the absence of plateaus in the CFD trend of thermal wake intensity. This is surprising because the modulation of CFD-predicted thermal wake *width* (see Fig. 19(a)) is very similar to that of the aerodynamic wake. Following Thomas and Liu [35], the decay rate of peak KE loss is influenced by the *streamwise* pressure gradient, which varies due to the downstream potential field (see Fig. 7 and associated discussion). Burdett and Povey [12,28] attributed both the wake width and wake intensity modulation to streamwise pressure gradients. The absence of plateaus in the thermal wake intensity data of the present study suggests, however, that the modulation of aerodynamic wake width and wake intensity could be separate—but associated—effects, driven by the pressure gradient perpendicular, and parallel to the streamline, respectively. Firstly, the *streamline-normal* pressure gradient deflects the wake, with the modulation of wake width resulting from the projection of a wake with varying whirl angle onto an axial plane (affecting both pressure and thermal wake width). Secondly, the *streamwise* pressure gradient affects the decay of peak aerodynamic loss only (see Thomas and Liu [35]). We hope that the understanding of similarities and differences between the development of aerodynamic and thermal wakes that arises from this study will be useful in future turbine design.

## Conclusions

Aerodynamic and thermal traverse measurements downstream of a transonic annular cascade of heavily cooled NGVs have been presented. The experimental data were complemented by *k–ω* SST RANS simulations. In these simulations both the internal cooling passages and the film-cooling holes were fully resolved. Detailed pressure and temperature traverses were conducted at an upstream plane and three downstream planes, with clocking to track particular vane wakes. This allowed the streamwise development of whirl angle, kinetic energy loss, and thermal field to be studied. The data are for an NGV with a trailing-edge-cutback design (suction side overhang), but are compared to a similar dataset from the same facility for a centered-ejection trailing edge design [12]. This comparison is interesting because it allows us to compare common aspects of the datasets. We consider results for whirl angle, kinetic energy loss, and the thermal field in turn.

*Whirl angle*. We have presented area maps of whirl angle at three downstream planes for both experimentally measured data and CFD predictions. The general trend was an almost linear increase in whirl angle from hub to casing (dictated by the spanwise metal-angle distribution), with slight excursions on account of secondary flows. The distributions of whirl angle in CFD were generally well-matched to experiments, with CFD predicting marginally higher turning. Relative to experiments, mass-flux-weighted whirl angles were +0.65, +0.20, and +0.02 deg, at planes 2, 3, and 4, respectively, with a maximum *local* difference of 1.6 deg in the radial profiles. Area maps of whirl angle indicate a strong influence of the TE potential field, which periodically increases/decreases the whirl angle. This effect modulates the width of the wake, when evaluated on an axial-plane-projection.

*Kinetic energy loss coefficient*. Experimentally-measured and CFD-predicted area maps of KE loss at three downstream measurement planes were presented and analyzed. The NGV wakes showed the compound lean of the TE (as viewed on an axial cross-section). With increasing axial distance, the wakes deformed and mixed-out, whilst endwall boundary layers gradually grew in thickness. Mixing characteristics were investigated in terms of the streamwise development of wake width and the decay of peak KE loss at midspan. For experiments, aerodynamic wake width increased approximately linearly from 0.15 to 0.34 vane pitches between 0.25 *C _{x}* and 0.75

*C*, and wake depth (i.e., peak KE loss) decreased approximately inverse-linear over the same distance. The current experimental data for wake width and peak loss was found to correlate reasonably well with another study of NGV loss using cooled engine parts (Burdett and Povey [12]), with good collapse when accounting for differences in non-dimensional TE thickness. This suggests that trends of NGV wake spreading, and peak-loss decay rates are—to first order—applicable for a broad range of TE designs. Compared to experiments, CFD data in both the current study and the study of Burdett and Povey [12] were undermixed, with significantly narrower and more intense aerodynamic wakes. In the current study, wake width was underpredicted by 34%, and wake intensity was over-predicted by 45% (on average), with corresponding under- and over-predictions of 23%, and 77% in [12]. We also considered mixed-out and plane-averaged KE loss. CFD (

_{x}*k–ω*SST model) underpredicted the mixed-out and plane-averaged loss by 12%, on average, primarily due to an underprediction of profile-region KE loss rather than a difference in the endwall-region loss, which was well-matched between CFD and experiment. By comparison, the CFD of Burdett and Povey [12] over-predicted profile-region loss by 18%. An over-prediction of KE loss (as opposed to the under-prediction in the current study) is presumably related to the method of modeling film cooling: Burdett and Povey [12] used point sources, whereas the film cooling holes were fully resolved in the current study.

*Thermal field.* Detailed experimental area-traverse data of the thermal field at three downstream axial planes have been analyzed. The thermal wake was wider than the pressure wake by 37% on average across three planes. This is caused by the mass-displacement effect of injecting film coolant on the vane surface, such that the thermal mixing layer is on average much thicker than the attendant aerodynamic boundary layer. The effect is likely compounded by greater resulting exposure to freestream turbulence, which then enhances mixing. Whilst the thermal wake was wider in *absolute* terms, *proportional* spreading and decay rates were very similar to those of the aerodynamic wake. Compared to experimental data, the CFD-predicted thermal field was undermixed, having on average a 38% narrower and 53% deeper wake (versus 34% and 45%, respectively, for KE loss). The fact that the aerodynamic and thermal fields behave similarly suggests that an improved turbulence model could address undermixing of both the aerodynamic and the thermal wakes.

In this study we have compared the loss and thermal field development downstream of two sets of heavily cooled engine parts operating at near-engine conditions including high turbulence intensity. The experimental data are unusually high-fidelity and allow for detailed comparison with modern CFD methods. The *k*–*ω* SST model gave rise to undermixing in simulation results, but with reasonably good match with experiment for mass-averaged exit conditions. The data provide a robust benchmark for assessing current CFD design methods.

## Acknowledgment

The financial and technical support of Rolls Royce plc, the Engineering and Physical Sciences Research Council, and Innovate UK are gratefully acknowledged.

## 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.

## Nomenclature

*C*_{x}=axial chord length [m]

*d*=streamwise path length [m]

*I*=turbulence intensity [%]

*k*=turbulence kinetic energy [m

^{2}s^{−2}]*k*_{p}=pressure scaling constant [–]

*k*_{T}=temperature scaling constant [–]

- ℓ =
turbulence length scale [m]

- $m\u02d9$ =
mass flow rate [kg s

^{−1}]- M =
Mach number [–]

*p*=pressure (total or static) [Pa]

- Re =
Reynolds number [–]

*r*=radius [m]

*s*=NGV pitch [deg]

*T*=temperature [K]

*V*_{x}=axial velocity [m s

^{−1}]- $V\theta $ =
circumferential velocity [m s

^{−1}]*w*=full angular wake width at half maximum [deg]

*x*=axial coordinate [m]

*y*^{+}=normalized wall distance [–]

### Greek Symbols

*α*=whirl angle [deg]

- $\u03f5$ =
turbulence dissipation rate [m

^{2}s^{−3}]*ζ*_{1}=local kinetic energy loss coefficient [–]

*ζ*_{3}=circumferentially-averaged kinetic energy loss coefficient [–]

*ζ*_{4}=plane-averaged kinetic energy loss coefficient [–]

*ζ*_{5}=mixed-out kinetic energy loss coefficient [–]

*ζ*_{p}=profile kinetic energy loss coefficient [–]

*ζ*_{e}=endwall kinetic energy loss coefficient [–]

*ζ*_{ref}=reference kinetic energy loss coefficient [–]

*θ*=Circumferential location [deg]

- Θ =
Flow effectiveness (non-dimensional total temperature) [–]

*λ*=Cold gas mass fraction [–]

### Subscripts

## Appendix A: Measurement Uncertainty

Estimated experimental bias uncertainties for upstream normalized total pressure and temperature, downstream flow angle, KE loss, and flow effectiveness are summarized in Table 2. Values are given with 95% confidence interval (CI).

Whirl angle bias uncertainty was estimated from alignment uncertainties during calibration and testing. Uncertainties of pressure-based measurements were calculated from the manufacturer-quoted bias uncertainties of the transducers. A special effort was made to reduce uncertainty by performing accurate calibrations of all transducers in an appropriate range (1–2 bar absolute). Uncertainties of temperature-based measurements were calculated assuming a bias uncertainty of ±1.50 K (95% CI) for most thermocouples, and ±0.75 K (95% CI) for the upstream traverse thermocouple with improved error (see Nakos [36]).

## Appendix B: Grid Sensitivity Study

In this section we examine the sensitivity of the CFD solutions to the resolution of the computational grid. In the context of the experimental validation of simulation results, a grid sensitivity study allows us to differentiate between model and discretization errors.

#### Grid Refinement Strategy

Starting from a *coarse grid* comprised of 110 million (110 M) cells, the mesh was globally refined by successively decreasing the background cell size of the octree mesh. This was performed in two refinement steps, with corresponding relative cell edge lengths (i.e., normalized by that of the 110 M grid) of 0.72 (*intermediate grid*), and 0.62 (*fine grid*) with respective cell counts of 221 million (221 M), and 312 million (312 M). The internal cooling passages and film cooling holes of the NGVs account for approximately half of the cells. We examine the sensitivity of local flow properties, wake mixing characteristics, and integral KE loss to grid resolution.

#### Grid Sensitivity of Circumferential Profiles at Midspan

Circumferential midspan profiles (radially mass-flux-averaged between 40% and 60% span) of whirl angle, local KE loss, and cold gas mass fraction at plane 2 are shown in Fig. 20.

First consider the circumferential profiles of whirl angle shown in Fig. 20(a). Values of whirl angle, *α*, are datumed to the midspan metal angle, *α*_{TE}. The maximum local variation across the three grids is 0.22 deg. Relative to the *fine grid* (312 M), differences in mass-flux-averaged whirl angle were −0.058 deg (112 M), and −0.010 deg (221 M). These values are extremely small, and much lower than the estimated experimental measurement bias uncertainty for whirl angle (±0.8 deg).

Next, consider the circumferential profiles of local KE loss coefficient, which are shown in Fig. 20(b). The values of local KE loss, *ζ*_{1}, have been normalized by the reference KE loss coefficient, *ζ*_{ref}. Peak normalized local KE loss, averaged across wakes W1 and W2, was 4.602 for the *coarse grid*, 4.678 for the *intermediate grid*, and 4.697 for the *fine grid* (difference of 2.0% across all three grids). A somewhat surprising result is that—contrary to what we might expect based on arguments of numerical diffusion—the peak KE loss intensity of W2 does not increase with grid refinement, whereas W1 does. This behavior could be related to the mixed numerical order of accuracy, when using the *high resolution* advection scheme of the CFX solver. Roy has shown that oscillatory convergence can occur when truncation errors from blended first-order and second-order discretization interfere constructively or destructively, depending on the grid spacing [37].

Finally, consider the circumferential profiles of cold gas mass fraction, which are shown in Fig. 20(c). We recall that the cold gas mass fraction, *λ*, is related to effectiveness, Θ, by *λ* = 1 − Θ. Peak cold gas mass fraction (average of W1 and W2) was 0.455 for the *coarse grid*, 0.468 for the *intermediate grid*, and 0.476 for the *fine grid* (difference of 4.4% across all three grids). Although the general trend is for an increase in peak cold gas mass fraction as the grid is refined, in similarity to the aerodynamic wake, only the thermal wake depth for W1 increases monotonically as the grid is refined.

#### Grid Sensitivity of Streamwise Trends of Aerodynamic and Thermal Wake Properties

We now consider the grid sensitivity of the streamwise trends of aerodynamic and thermal wake properties. Width and peak heights for the aerodynamic wake and thermal wakes are shown as a function of normalized streamwise distance in Fig. 21. These trends show the sensitivity of *mixing characteristics* to grid refinement. The method for determining wake width and peak intensity is described by Burdett and Povey [12] (direct averaging approach; without restacking). Average values for wakes W1 and W2 are reported.

The streamwise evolution of aerodynamic wake width is shown in Fig. 21(a). Here, the wake width, *w*, has been normalized by the vane pitch, *s*. Over the experimental range of streamwise distances (0.99 < *d/C _{x}* < 3.00), the aerodynamic wake width is effectively identical for all three grids (maximum local variation of 0.013

*s*). Above 3.0

*C*, the trends diverge slightly. At

_{x}*d/C*= 5, the aerodynamic wake widths on the

_{x}*coarse grid*and

*intermediate grid*are 0.061

*s*and 0.007

*s*greater (respectively) than on the

*fine grid*. This is consistent with the expectation that grid refinement reduces artificial diffusion.

Let us now consider the grid sensitivity of the peak KE loss coefficient, which is shown in Fig. 21(b). Compared to the *fine grid*, peak intensities of normalized local KE loss are on average 7.3% lower on the *coarse grid*, and 2.1% greater on the *intermediate grid*. We reference the earlier argument of Roy [37] to explain the non-monotonic decrease of peak loss. Overall, the observations are consistent with the expectation that numerical diffusion will tend to increase wake width and decrease wake depth.

The corresponding trends of thermal wake width and peak cold gas mass fraction with grid refinement are shown in Figs. 21(c) and 21(d), respectively. Relative to the *fine grid*, thermal wake width was on average 9.0% greater on the *coarse grid*, and 1.2% lower on the *intermediate grid*. Peak cold gas mass fraction was on average 12.4% lower and 0.2% greater for the *coarse* and *intermediate grids*, respectively. Approximately inverse percentage error for wake width and peak intensity are expected, because the mass-flux-averaged cold gas mass fraction is a conserved quantity.

#### Grid Sensitivity of Integral Measures of Loss

Finally, we consider the development of mixed-out KE loss, plane-averaged KE loss, and SKE, which are shown as a function of normalized axial location in Fig. 22 for three grid sizes. These trends are indicative of the grid sensitivity of *integral loss*.

As the grid is refined, both mixed-out KE loss (shown as solid lines), and plane-averaged KE loss (shown as dashed lines) reduce monotonically. On the first refinement step (112 M to 221 M), mixed-out and plane-averaged loss are reduced by 2.8% and 3.3%, respectively (ratio of ensemble averages between 0.0 and 1.0 *C _{x}*), with corresponding reductions of 0.7% and 0.3% on the second refinement step (221 M to 312 M). Using Richardson extrapolation, this translates to a predicted error of approximately 0.8% on the

*fine grid*for plane-averaged KE loss. Because both mixed-out and plane-averaged loss change by a similar amount with grid refinement, SKE (dotted lines) is essentially unchanged. This suggests that integral mixing characteristics are less grid-sensitive than the absolute levels of KE loss.

## References

*k*-

*ω*SST (Shear Stress Transport) Turbulence Model Calibration: A Case Study on Small Scale Horizontal Axis Wind Turbine