Effects of Localized Non-Gaussian Roughness on High-Pressure Turbine Aerothermal Performance: Convective Heat Transfer, Skin Friction, and the Reynolds’ Analogy

Surface roughness arising through in-service wear and/or the manufacturing process plays a critical role in determining high-pressure turbine (HPT) aerothermal performance [1,2]. While it is well known that roughness tends to augment heat transfer at the expense of an even greater augmentation in drag [3–5], detailed accounts of the fundamental flow parameters that govern convective heat transfer (wall heat flux, Nusselt, and/or Stanton numbers) and boundary layer losses (wall shear stress, skin friction coefficient, momentum thickness Reynolds number) on rough vanes are still lacking, especially at engine-relevant conditions. Obtaining these data experimentally can be extremely challenging, and although there are some notable exceptions to this trend [6,7], the most detailed studies to date have focused on smooth vanes, e.g., see work by Thole et al. [8].

At the same time, high-fidelity simulations of flow past HPT vanes with realistic levels of roughness have only very recently become possible [9]. This is because of the tremendous computational resources required not only to resolve the full spectrum of hydrodynamic and thermodynamic scales in the flow field but also resolve the topographical scales of the surface roughness. The very first compressible direct numerical simulation (DNS) of flow past an HPT vane with realistic levels of roughness was conducted in Refs. [10,11], which built upon a precursory series of compressible large-eddy simulations performed by Ref. [12]. These past studies quantified the aerothermal performance of the (uncooled) VKI LS89 nozzle guide vane [13] covered with numerically generated multiscale surface roughness [14] at an exit Reynolds number of 590,000 and an exit Mach number of 0.92, relative to a smooth vane at matched conditions. While Refs. [10–12] highlighted the profound impact that surface roughness can have upon HPT aerothermal performance, these studies mainly focused on near-Gaussian roughness (i.e., roughness with a symmetric height distribution) with a uniform roughness height on the suction and pressure surfaces.

However, as shown in measurements by Refs. [15,16], roughness is not homogeneously distributed over the surface of in-service vanes. In the context of first-stage HPT vanes, the highest roughness levels tend to occur in the leading edge region and along the pressure surface due to direct exposure to combustor exhaust gases, whereas the lowest roughness levels occur along the sheltered portion of the suction surface. Furthermore, it is well known that most turbine blade surface roughness is non-Gaussian [17], i.e., the height distribution is nonsymmetric and has nonzero skewness and kurtosis not equal to 3. As a result, systematic studies of how localized, non-Gaussian surface roughness—as it truly occurs on in-service vanes—affects HPT aerothermal performance are of significant practical interest.

To this end, a first-of-its-kind series of compressible DNSs with up to $14.6×109$ grid points per simulation have been conducted, with the specific goal of performing a direct computational assessment of convective heat transfer, skin friction, and the Reynolds” analogy on realistically rough HPT vanes at engine-relevant conditions. Particular attention is directed toward understanding how systematically varying the axial extent of leading edge roughness affects the instantaneous (and averaged) quantities that govern HPT aerothermal performance, i.e., the skin friction coefficient and the Nusselt and Stanton numbers, and how these quantities are augmented (or reduced) compared to smooth-vane levels. The impact of varying the inflow turbulence intensity and integral length scale for a fixed roughness topography is also investigated, and, where appropriate, the Reynolds number scaling of the HPT data is compared against existing correlations for turbulent heat transfer and drag.

The blade generation procedure adopted here is an extension of the method devised by Refs. [10–12]. However, in those past studies, the main focus was on nonlocalized (i.e., uniformly distributed) roughness with a near-Gaussian distribution, as opposed to the localized non-Gaussian roughness under investigation here.

HPT vanes covered with localized non-Gaussian roughness were generated by taking the following steps. First, a surface generation algorithm [18] was used to create a doubly periodic planar roughness distribution, $h(x′,z′)$⁠, with zero mean height, specified root-mean-squared (RMS) roughness height, $Rq≡A−1∬Ah2(x′,z′)dA$⁠, and non-Gaussian statistical properties, i.e., nonzero skewness $S≡A−1Rq−3∬Ah3(x′,z′)dA≠0$ and kurtosis not equal to 3, $K≡A−1Rq−4∬Ah4(x′,z′)dA≠3$⁠, where $A$ is the planform area of the heightmap. Here, $x′$ and $z′$⁠, respectively, denote the streamwise and spanwise directions of the planar roughness distribution. The statistical properties of $h(x′,z′)$ were inferred from a set of roughness measurements provided by GE Aerospace. The key amplitude parameters of the planar roughness distribution are $(Rq/c,S,K)=(4.2×10−3,0.42,3.75)$⁠, where $c$ is the axial blade chord. For a typical axial blade chord of, say, $c=25mm$⁠, the physical RMS roughness height is therefore $Rq≈100μm$⁠, which is typical of the most severely degraded portions of vanes, particularly in the leading edge region [17].

The weighting function, $W$⁠, is plotted against axial position normalized by the axial chord, $x/c$⁠, in Fig. 1, where $x/c<0$ corresponds to the vane suction surface and $x/c>0$ corresponds to the vane pressure surface. Looking from left-to-right across Fig. 1, it can be observed that all the control points remain fixed with respect to $x/c$⁠, with the exception of control point IV (white circles), which takes on one of four possible values along the suction surface, i.e., $x/c={0.28,0.40,0.50,0.60}$⁠. In terms of amplitude, the weighting function increases from control point I at the leading edge (LE) $(x/c=0.0)$⁠, to a maximum at control point III $(x/c≈0.15)$⁠, before decreasing to a minimum at control point IV, beyond which the weighting function remains constant until control point V, which is located at the vane trailing edge $(x/c=1.0)$⁠. In contrast, the weighting function has a constant amplitude between control points I and VI, which covers most of the pressure surface. However, the weighting function alters the local roughness height, skewness and kurtosis remain unaffected, since standardized moments of a probability density function are scale invariant by definition (see Table 1).

A zoomed-in view of the non-Gaussian surface roughness in the leading edge of each HPT vane under investigation here is provided in Fig. 2. While the relative changes in the roughness topography may at first appear too small to significantly affect HPT aerothermal performance, the results of this investigation suggest otherwise.

Transonic flow past rough HPT vanes was simulated using the HiPSTAR flow solver [19], which solves the compressible Navier–Stokes equations nondimensionalized by a reference length (⁠$c=$ axial blade chord), velocity (⁠$Uin=$ inflow velocity), density (⁠$ρin=$ inlet density), and temperature (⁠$Tin=$ inflow temperature). The governing equations for mass, momentum, and total energy solved by HiPSTAR can be written in the tensor form as follows:

HiPSTAR solves Eqs. (2)–(4) on structured overlapping grids using the fourth-order accurate compact finite differences for spatial discretization and an ultra-low storage frequency-optimized explicit fourth-order accurate Runge–Kutta method for time integration [19]. The computational domain is shown in Fig. 3(a) and is composed of three overlapping blocks: (i) a background Cartesian H-type grid (block 1); (ii) an outer curvilinear O-type grid (block 2), and (iii) an inner curvilinear O-type grid (block 3). At the overlapping block boundaries, continuity conditions are imposed as variables and are interpolated with a fourth-order Lagrangian method [20]. A zoomed-in view of the inner and outer O-meshes is shown in Fig. 3(b), which shows the typical mesh resolution around the roughness profile. Note that only every fourth in-plane point is shown.

To represent the geometry of each rough HPT vane, a point cloud of the three-dimensional blade coordinates (Eq. (1)) was immersed within the inner O-grid of the computational domain (Fig. 3), and a compressible variant of the boundary data immersion method (BDIM) [21] was used to enforce no-slip boundary conditions on velocity $(ui=0)$ and an isothermal boundary condition on temperature (⁠$T=Tw$⁠, where $Tw$ is blade surface temperature).

To drive the flow through the passage, an outlet static pressure was prescribed and an inlet static pressure and temperature were imposed as Riemann conditions using standard isentropic flow relations. To mimic disturbances from an upstream combustor, synthetic eddies with a specified inlet turbulent intensity, $Tu/Uin$⁠, and an (isotropic) integral length scale, $L/c$⁠, were fed into the domain inlet using a compressible version of the digital filter technique [22]. A nonreflective zonal characteristic boundary condition was enforced at the outlet [23]. Periodic boundaries were imposed in the pitchwise $(y)$ and spanwise $(z)$ directions to simulate a linear HPT cascade without end-walls.

For groups 1 and 2, the background mesh (block 1) has dimensions of $(Nx,Ny,Nz)=(1470,716,576)$ and the outer O-mesh (block 2) has dimensions of $(Nx,Ny,Nz)=(8165,237,576)$⁠, where $Nx$⁠, $Ny$⁠, and $Nz$ are the number of grid points along the axial $(x)$⁠, pitchwise $(y)$⁠, and spanwise $(z)$ directions, respectively. The inner O-mesh (block 3) has dimensions of $(Nx,Ny,Nz)=(35595,230,1200)$⁠, except for cases X28_0820 and X28_2020 in group 2, where $(Nx,Ny,Nz)=(35595,230,2400)$ to account for the wider computational domain. For all cases presented here, the viscous-scaled grid resolutions in the blade tangential direction $(s)$⁠, blade normal direction $(n)$⁠, and spanwise direction $(z)$ are such that $Δs+<5.0$⁠, $Δn+<1.5$⁠, and $Δz+<8.0$⁠, where superscript $+$ denotes nondimensional by the local (mean) viscous length scale. The viscous-scaled grid resolution adopted here is therefore commensurate with that in past work related to the present study [10,11]. The accuracy and reliability of the HiPSTAR flow solver and BDIM algorithm in simulating flow over rough (and smooth) HPTs have been validated extensively in the previous works related to the present study [10–12].

For each rough HPT case listed in Table 2, a series of extreme-scale compressible DNSs were performed on the Summit supercomputer based in the Oak Ridge National Laboratory, USA. Where possible, all seven cases were executed concurrently as a single job using 1152 Summit nodes, meaning that 6912 Nvidia Tesla V100 GPUs were required for each production-scale calculation. In terms of computational resources, each production scale simulation on Summit required approximately 168 h of wall-time to reach statistical convergence, meaning that $∼194,000$ node-hours (or $∼1116,000$ GPU-hours) were consumed in total. Some calculations were also performed on Summit’s successor, Frontier, using a comparable number of AMD Instinct M1250x Accelerators. For a typical job with $∼5.0×106$ points per GPU, Frontier delivered a speed-up of three times compared to Summit. Herein, all statistical quantities were collected for a minimum of four nondimensional time units (based on the axial inlet velocity and axial blade chord) following an initial transient.

Traditionally, roughness effects have been quantified in fully developed turbulent flows in canonical configurations, e.g., circular pipes [24], plane channels [25], and flat-plate boundary layers [26], often with the goal of mapping skin friction measurements to an equivalent value of Nikuradse’s sandgrain roughness, $ks$ [27]. However, $ks$ is only strictly valid for turbulent flows in the fully rough regime, i.e., a rough-wall flow where the skin friction coefficient is Reynolds number independent. Whether the fully rough regime is reached in the context of the present investigation is debatable, since detailed accounts of how skin friction drag varies as a function of the Reynolds number is typically limited to smooth HPT vanes [8,13,28]. Furthermore, despite having units of length, it is important to note that $ks$ cannot be reliably determined from topographical data alone (e.g., from a digital scan of an ablated turbine blade [2]). This is because $ks$ is a hydraulic length scale, i.e., a hydrodynamic property of the flow, not a topographical property of the surface, that must be determined on a surface-specific basis. In fact, as noted in a recent review article [29], determining $ks$ remains the major bottleneck for accurate predictions of roughness effects in fully rough flows. Whether this bottleneck applies to the present study raises the following question: do any of the rough HPT vanes under investigation here attain fully rough conditions?

To answer this question, the fractional contribution of pressure drag, $Fp$⁠, and viscous drag, $Fν$⁠, to the total drag, $Ftot=Fp+Fν$⁠, can be computed. To reach the fully rough regime, it is generally accepted that the tangential component of the (inviscid) pressure force acting on the roughness elements overwhelms its viscous counterpart, i.e., $Fp≫Fν$ [30]. A previous experimental investigation of surface roughness effects on the fully developed turbulent pipe flow [31] suggests that the onset of the fully rough regime occurs when pressure drag accounts for at least 80% of the total drag force, i.e., $Fp/Ftot≥0.80$⁠. Whether this threshold is exceeded in the context of the present study remains unclear, since very little is known regarding the relative magnitude of viscous and pressure forces acting against rough HPT vanes—with one recent exception being the work presented in Ref. [11]. Yet, knowledge of the drag partition over rough HPT vanes is of critical importance, especially when one considers that any $ks$-based modeling approach, e.g., surface-specific wall functions [32] or roughness-induced transition models [33], implicitly assume that fully rough conditions are attained.

The fractional contribution of pressure drag to the total drag, $Fp/Ftot$⁠, is plotted against the axial position for groups 1 and 2 in Fig. 4. Here, $−1<x/c<0$ corresponds to the pressure surface, $0<x/c<1$ corresponds to the suction surface, and the horizontal dashed line depicts the threshold of $Fp/Ftot=0.8$⁠, which is required to attain fully rough conditions according to Ref. [31]. At first glance, the data in Fig. 4 bear a striking resemblance to the shape of the weighting function used to control the roughness amplitude around the vane (see Fig. 1). To be specific, group 1 data (Fig. 4(a)) show that the fractional contribution of pressure drag correlates with regions of high-amplitude roughness, i.e., in the forward half of the suction surface $(0≲x/c≲0.5)$ and along the majority of the pressure surface $(−0.9≲x/c≲0.0)$⁠. A simple explanation for this behavior is that the larger amplitude asperities have more surface normals with an axial component on which pressure drag can act (see integrand of Eq. (7)). This also explains why increasing the axial extent of larger amplitude roughness results in larger values of $Fp/Ftot$ along a greater proportion of the vane suction surface. Quantitatively, the fractional contribution of pressure drag to the total drag reaches a peak value of $Fp/Ftot≈0.6$ at $x/c≈0.25$ and $x/c≈−0.85$ on the suction and pressure surfaces, respectively, meaning that viscous drag still accounts for up to two fifths of the total drag, since $Fν/Ftot≡1.0−Fp/Ftot$⁠. On the other hand, in the aft portion of the suction surface, where the RMS roughness amplitude is four times smaller than elsewhere on the vane (see Fig. 1 and Table 1),– pressure drag accounts for just one tenth of the total drag force, i.e., $Fp/Ftot≈0.1$⁠. Looking at Fig. 4(b), the variation of $Fp/Ftot$ with respect to $x/c$ is almost identical for all cases in Group 2, suggesting that pressure drag is relatively insensitive to changes in inflow conditions, although this is not necessarily the case since the magnitude of a force cannot be inferred from a ratio of forces. Nevertheless, beyond an axial distance on the suction surface of, say, $x/c≳0.6$⁠, the viscous drag force for all cases in groups 1 and 2 is approximately nine times greater than the pressure force, i.e., $Fν≫Fp$⁠, which is the opposite of what is required to reach fully rough conditions.

Overall, the data in Fig. 4 imply that fully rough conditions are not achieved for any of the HPT vanes under investigation here. The fractional contribution of pressure drag to the total drag fails to exceed the threshold $Fp/Ftot=0.8$ of Ref. [31] at any point on the pressure or suction surface. As a result, the bottleneck of determining an equivalent value of Nikuradse’s sandgrain roughness [29] does not apply here, since it is not possible to ascribe a physically meaningful value (or values) of $ks$⁠. The lack of fully rough conditions is most apparent in the aft portion of the suction surface, where viscous drag accounts for up to 90% of the total drag. As will be shown later, the flow in this region perceives the blade as an aerodynamically (and thermodynamically) smooth surface. Consequently, the Reynolds number scaling of the local skin friction and heat transfer coefficients agree well with the existing empirical correlations for (turbulent) flow over smooth surfaces.

Attention is now turned toward evaluating HPT aerothermal performance. This is achieved by quantifying the sensitivity of the skin friction coefficient and wall heat flux with respect to changes in the axial extent of leading edge roughness (group 1) and changes in inlet conditions (group 2).

Snapshots of the instantaneous skin friction coefficient are shown on the suction and pressure surfaces in Fig. 5. Isovolumes of the $Q$-criterion [34] are also shown over the right-hand half of each vane. Looking at the data, it is clear that the suction-side (SS) $Cf$ distributions exhibit a greater sensitivity with respect to changes in either leading edge roughness (Figs. 5(a) and 5(b)) or inlet conditions (Figs. 5(c) and 5(d)) compared to their pressure-side (PS) counterparts. Comparing the suction-side data for both groups, it is also clear that $Cf$ is more sensitive to changes in the axial extent of leading edge roughness (Fig. 5(a)) than variations of inlet turbulence (Fig. 5(c)), at least for the seven cases considered here. For instance, although the $Cf$ distributions and vortical structures for cases X28_0808 and X40_0808 show that the boundary layer breaks down into turbulence in the final $∼15%$ of the suction surface (Fig. 5(a)), the remaining group 1 cases show a different behavior. In particular, case X60_0808 exhibits a streaky $Cf$ distribution and fine-scale vortical motions across its span and over most of the suction surface—flow characteristics that are synonymous with wall turbulence [35]. Overall, the data in Fig. 5 show that even a small change in the axial extend of leading edge roughness can trigger a large change in the suction-side boundary layer state, underlining the critical role that localized surface roughness plays in determining HPT aerothermal performance.

Next, the mean skin friction coefficient, $C¯f$⁠, was obtained by integrating the instantaneous $Cf$ values (Eq. (8)) with respect to time and the spanwise direction. The variation of $C¯f$ along the vane suction surface for groups 1 and 2 is plotted against the axial position in Fig. 6. Here, the gray and black lines represent the $C¯f$ values corresponding to the smooth and rough vanes vane, respectively. The light and dark contours represent the $Cf$ histogram for the smooth and rough vanes, respectively, and illustrate the frequency distribution of instantaneous $Cf$ values at a given $x/c$ location. Note that a vertical shift of 0.1 units has been applied to consecutive cases in groups 1 and 2 for brevity. Looking at the data in Fig. 6, it is clear that these statistical data support their instantaneous counterparts shown in Fig. 5 in the sense that: (i) Increasing the axial extent of leading edge roughness causes an abrupt upstream shift of the (mean) transition point, e.g., compare the $C¯f$ curve for case X60_0808 against that of X40_0808, X28_0808, or the reference smooth vane. (ii) Compared to the effect of increasing the axial extent of leading edge roughness, increasing the inlet turbulence intensity or integral length scale has a relatively weaker effect upon the (mean) transition point, i.e., cases X28_0808 and X28_0820 closely resemble the smooth-vane data, the upward kink in the $C¯f$ curves for cases X28_2008 and X28_2020 shows a more gradual upstream shift. Further understanding of the near-wall boundary layer state can be gained from comparing the shape of the $Cf$ histograms included in Fig. 6. Focusing on the data in the trailing edge region, say, $0.7≲x/c≲1.0$⁠, it is clear that the presence of intermittent flow phenomena across the span (and in time) produce a wider distribution of instantaneous $Cf$ values, compared to those that break down to turbulence closer to the vane leading edge. For instance, comparing the $Cf$ histogram for case X60_0808 with that for X50_0808 or X28_2020, it is obvious that the broadband distribution of $Cf$ values for the latter cases is driven by a rapidly varying boundary layer state due to the inception, growth, and coalescence of turbulent spots along the suction surface—the foot print of these structures are clearly visible on the suction surface of each vane (see Figs. 5(a) and 5(c)). Similar observations have been made in the context of smooth HPT vanes, e.g.. see Ref. [36].

It also represents the product of the thermal conductivity and the temperature gradient at the wall in the direction perpendicular to the blade surface. Snapshots of the instantaneous wall heat flux on the vane suction and pressure surfaces are shown in Fig. 7. Isovolumes of the $Q$-criterion are once again included. In general, the spatial distribution of instantaneous heat flux data resembles their instantaneous $Cf$ counterparts in Fig. 5, on both the pressure and suction surfaces. As expected, regions of high instantaneous heat flux correlate with regions of high instantaneous skin friction drag, and the presence of fine-scale vortical motions in the near-wall region enhance the rate at which heat is transferred from the hot fluid into the (relatively) cool blade. Next, the mean wall heat flux, $Q¯$⁠, was obtained by integrating the instantaneous $Q$ values (Eq. (9)) with respect to time and the spanwise direction. The $Q¯$-distribution along the vane suction surface is plotted against the axial position for groups 1 and 2 in Fig. 8. Note that a vertical shift of 0.5 units has been applied to consecutive cases in groups 1 and 2 for brevity. Again, the gray and black lines represent the $Q¯$ values corresponding to the smooth and rough vanes vane, respectively. The light and dark contours represent the $Q$-histogram for the smooth and rough vanes, respectively. Data points from a past study of a smooth HPT vane resolved using a body-fitted mesh [28] are also included (white squares) for comparison. Looking at the data shown in Fig. 8, these $Q¯$-curves support their instantaneous counterparts (Fig. 7) in the sense that: (i) increasing the axial extent of leading edge roughness can cause an abrupt change in the mean transition point, e.g., compare the $Q¯$-curves for cases X40_0808 and X50_0808 in the range of $0.5≲x/c≲1.0$ in Fig. 8; (ii) changing the inlet condition also has a strong effect upon the levels of wall heat flux, particularly in the trailing edge region; and (iii) the presence of transitional flow phenomena (e.g., see localized “hot spots” on the suction surface of cases X50_0808 and X28_2020 in Fig. 7) results in the widest band of realizations of wall heat flux in their respective $Q$-histograms.

The preceding subsection demonstrated that changes in the axial extent of leading edge roughness and inlet conditions can profoundly impact skin friction and wall heat flux (in both an instantaneous and averaged sense), particularly along the vane suction surface (see Figs. 5–8). However, what is not yet clear is how heat transfer and drag are augmented compared to smooth vane levels or to some other suitable reference case.

To quantify the overall augmentation of heat transfer and drag, the ratio of the mean Nusselt number, $Nu¯/Nu¯0$⁠, and the mean skin friction coefficient, $C¯f/C¯f,0$⁠, were integrated over the suction surface and the pressure surface to obtain “global mean” quantities, herein denoted using angular brackets, i.e., $⟨Nu⟩/⟨Nu⟩0$ and $⟨Cf⟩/⟨Cf⟩0$⁠, where subscript $0$ denotes a specified reference state. Group 1 reference state is taken as the smooth-vane case, whereas Group 2 reference state is taken as case X28_0808. Hence, any augmentation in heat transfer or drag in group 1 is due to the effect of varying the axial extent of leading edge surface roughness (for a fixed inlet condition), whereas changes in group 2 are due to the effect of varying the inlet conditions (for a fixed roughness topography). If $⟨Cf⟩/⟨Cf⟩0>⟨Nu⟩/⟨Nu⟩0$⁠, then the fractional increase in heat transfer comes at the expense of a greater fractional increase in drag compared to the reference state, whereas $⟨Cf⟩/⟨Cf⟩0<⟨Nu⟩/⟨Nu⟩0$ indicates the opposite. Otherwise, if $⟨Cf⟩/⟨Cf⟩0=⟨Nu⟩/⟨Nu⟩0$⁠, then the fractional increase in heat transfer and drag matches those of the reference state. Note the (instantaneous) Nusselt number is defined as $Nu(xw,zw,t)≡(c/ΔT)/∂T(x,z,t)∂n|w$⁠, which represents the ratio of convective to conductive heat transfer at a given point on the blade surface. Here, thermal conduction is driven by the difference between the (constant) total inlet temperature and the (isothermal) blade surface temperature, i.e., $ΔT≡(Tintot−Tw)>0$⁠, whereas thermal convection is solely determined by the instantaneous temperature gradient at the wall in the direction perpendicular to the blade surface.

The relationship between $⟨Nu⟩/⟨Nu⟩0$ and $⟨Cf⟩/⟨Cf⟩0$ for cases in groups 1 and 2 is shown in Fig. 9, a plot that is inspired from the past work of Bunker, e.g., see Fig. 3 in [37]. Looking at the data, it is clear that all of the suction-side data points (squares) fall below the diagonal dashed line, i.e., $⟨Cf⟩/⟨Cf⟩0=⟨Nu⟩/⟨Nu⟩0$⁠, meaning that the fractional increase in (global mean) heat transfer comes at the expense of an even greater fractional increase in (global mean) drag, i.e., $⟨C⟩f/⟨Cf⟩0>⟨Nu⟩/⟨Nu⟩0$⁠. Data points from Ref. [11] that quantified the aerothermal performance of HPT vanes covered with uniformly distributed near-Gaussian roughness and the same inlet condition as group 1 (see Table 2) is included in Fig. 9 (white squares) and show a similar trend. The same behavior has been reported in the past studies of fully developed channel-flow turbulence over irregular roughness [32] and streamwise-aligned riblets [38], and as result, the breakdown in Reynolds analogy due to roughness observed here in some sense is not surprising. On the pressure surface, the data points (circles) fall closer to the diagonal dashed line, indicating a relatively small fractional increase in heat transfer and drag compared to the specified reference state, i.e., the smooth vane for group 1 and case X28_2020 for group 2.

To understand how the augmentation of heat transfer and drag vary along the pressure and suction surfaces, the ratios of the mean skin friction coefficient, $C¯f/C¯f,0$⁠, and the mean Nusselt number, $Nu¯/Nu¯0$⁠, for groups 1 and 2 are plotted against the axial position in Fig. 10. Note that a vertical offset of 10 units has been applied to the former ratio for clarity. Looking from left-to-right across Fig. 10(a), it is clear that the fractional increases in heat transfer and drag are (approximately) in balance across the entire pressure side and for axial positions less than $x/c≲0.4$ on the suction side. It is also clear that changes in the axial extent of leading edge roughness can lead to very significant augmentations of heat transfer and drag in the aft portion of the suction surface. To be specific, at an axial position of $x/c≈0.90$⁠, the mean skin friction drag for cases X50_0808 and X60_0808 is augmented by up to a factor of 22 compared to smooth-vane levels, whereas $Nu¯$ is augmented by just a factor of six at the same location. Quantitatively, the mean Nusselt number increases from $Nu¯0≈88$ to $Nu¯≈527$ at $x/c≈0.90$ on the smooth vane and case X50_0808, respectively. The remaining group 1 cases exhibit a much weaker augmentation of heat transfer and drag—consistent with the previous analysis of the skin friction coefficient and wall heat flux, e.g., see Figs. 5(a)–5(c) and Figs. 7(a)–7(c). Looking from left-to-right across Fig. 10(b), it is clear that changes in inlet turbulence can also lead to appreciable augmentations of trailing edge heat transfer and drag, although the fractional increases are less significant compared to the group 1 data. Nevertheless, at an axial position of $x/c=0.90$⁠, skin friction drag and convective heat transfer are, respectively, augmented by a factor of 12 and 4 for case X28_2020 compared to the reference rough-vane state (case X28_0808). The integrated effect of these large, local augmentations of heat transfer and drag in the trailing-edge region explain why the global mean ratios, $⟨Nu⟩/⟨Nu⟩0$ and $⟨Cf⟩/⟨Cf⟩0$ fall below the diagonal dashed line in Fig. 9. Compared to the suction surface, the peak augmentation of heat transfer and drag on the pressure surface is at least one order of magnitude smaller for all cases in groups 1 and 2, which explains why the global mean ratios closely follow the diagonal dashed line in Fig. 9.

To illustrate the differences in boundary layer state responsible for the augmentation of drag and heat transfer, profiles of the mean blade tangential velocity and mean temperature are plotted against blade normal distance at an axial position of $x/c=0.90$ in Fig. 11. Here, velocity is normalized by its corresponding mean value at the boundary layer edge, i.e., $U¯t/U¯e$⁠, temperature is shown in “reduced” form, i.e., $(T¯−T¯w)/(T¯e−T¯w)$⁠, where $T¯e$ is the mean temperature at the edge of the thermal boundary layer, and blade normal distance is normalized by the local boundary layer thickness, i.e., $n/δe$⁠. Looking at group 1 data first (Fig. 11(⁠$a$⁠)), it is clear that the thermal and hydrodynamic boundary layer slopes at the wall are far steeper for cases X50_0808 and X60_0808 compared to reference smooth-vane profile, which explains why skin friction and heat transfer are augmented so dramatically for these two particular cases (see Fig. 10(a)). Similar behavior for cases X28_0820 and X28_0820 (relative to case X28_0808) can be observed in Fig. 11(b). Also included in Fig. 11 are DNS data points for the mean streamwise velocity profile of an incompressible zero-pressure gradient (ZPG) turbulent boundary layer (TBL) over a smooth flat plate [39] at a momentum thickness Reynolds number of $Reθ≈1400$⁠. Remarkably, cases X50_0808 and X60_0808 show good levels of agreement with the DNS data points at all blade normal positions, whereas X28_2020 shows the best overall agreement among group 2 cases. For the remaining cases in groups 1 and 2, the blade normal velocity profiles show a point of inflection at around $n/δe≈0.3$ (a feature that is absent in the corresponding temperature profiles), which is consistent with the sharp drop in $C¯f$ over the range $0.85≲x/c≲0.95$ (see Fig. 6) and indicative of a flow approaching a (mean) separation as a result of adverse pressure gradient forcing.

To further characterize the velocity and temperature profiles plotted in Fig. 11, some nondimensional boundary layer parameters are given in Table 3. The parameters include the following: the friction Reynolds number, $Reτ≡Uτδe/νw$⁠; the momentum thickness Reynolds number, $Reθ≡Ueθ/νw$⁠; the boundary layer shape factor, $H≡δ*/θ$⁠; the mean skin friction coefficient, $C¯f$⁠; the mean Stanton number, $C¯h≡Nu¯/(RePr)$⁠; and twice the mean Stanton number divided by the mean skin friction coefficient, i.e., the Reynolds analogy factor, $2C¯h/C¯f$⁠. Looking at Table 3, stark differences between the various cases can be observed. For instance, although the smooth-vane friction Reynolds number reaches a value of $Reτ≈60$⁠, this quantity is a factor of nine greater for case X60_0808. Likewise, although the smooth-vane shape factor attains a value of $H≈3.0$⁠, a value of $H≈1.4$ is attained for case X60_0808—the latter typical of turbulent flow and the former for a near-separated flow. In terms the Reynolds’ analogy, which, according to Bons [40] and others, is preserved when $2C¯h/C¯f≈1$⁠, the data in Table 3 indicate that this approximation holds, but only for three out of the seven cases investigated here (X50_0808, X60_0808, and X20_2020). Two common features among these three cases are their relatively high friction Reynolds numbers $(Reτ≳400)$ and relatively low shape factors $(H≈1.4)$⁠, indicating the flow has attained fully turbulent conditions at $x/c=0.90$⁠. A further common feature among these cases (and all other cases) is that the fractional contribution of pressure drag to the total drag is far smaller than the viscous drag force (see Fig. 4). As a result, one may conclude that the preservation of Reynolds’ analogy, i.e., $2C¯h/C¯f≈1$⁠, for cases X50_0808 and X60_0808 from group 1 and case X20_2020 from group 2 is due to the presence of fully turbulent flow and the absence of fully rough conditions. Note that the Reynolds’ analogy would likely not be preserved if fully rough conditions were reached. This is because although $C¯f$ would asymptote a surface-specific constant, $C¯h$ would continue to decrease with the increasing Reynolds number and, as a result, the Reynolds’ analogy factor $2C¯h/C¯f<1$⁠, e.g., see rough-wall channel-flow studies of Refs. [32,41]. Put in other words, the breakdown of the Reynolds’ analogy in a fully rough flow is due to an absence of viscous-dominated transport in the near-wall region—the opposite of what is observed here (see Fig. 4)

To understand the Reynolds number dependence of heat transfer and drag, HPT Moody diagrams can be constructed. Here, the Re-scaling is examined in the context of the Chilton-Colburn analogy, which states that the (mean) friction factor coincides with the product of the (mean) Stanton number and two-third power of the (molecular) Prandtl number, i.e., $C¯f/2=C¯hPr2/3$⁠. While HPT Moody diagrams have been constructed using measurements acquired at mid-span over smooth HPT vanes, e.g., see Figs. 7 and 8 in the study by Thole et al. [8], detailed information in the context of rough vanes remains rare.

The Chilton–Colburn form of the mean skin friction coefficient based on the local dynamic pressure at the boundary layer edge, $C¯f≡τw/(ρ¯eU¯e2/2)$⁠, and mean Stanton number, $C¯h$⁠, are plotted against the momentum thickness Reynolds number, $Reθ$⁠, on the suction surface in Fig. 12. Also included on this plot are the Coles–Fernholz correlation [42] for the mean skin friction coefficient of an incompressible smooth-wall ZPG TBL, $C¯f=2(κ−1log(Reθ)+C)−2$ (black dashed line), with von Kármán coefficient of $κ=0.387$ and offset of $C=4.1$⁠, as well as TBL data points on the range $670≤Reθ≤2000$ [39]. Looking again at Fig. 12, it is clear that all of the HPT data points (in both groups 1 and 2) collapse onto the Coles–Fernholz correlation and the DNS data beyond $Reθ≳1000$⁠. This behavior not only indicates that the (viscous) skin friction losses scale in a manner similar to that of a smooth-wall ZPG TBL but also indicates that the mean Stanton number can be approximated as $C¯h≈4(κ−1log(Reθ)+C)−2Pr−2/3$ provided that the momentum thickness Reynolds number exceeds $Reθ≳1000$⁠. The collapse of the HPT data on to the Coles–Fernholz correlation occurs at increasingly small $Reθ$ values for cases X50_0808 $(Reθ≈800)$ and X60_0808 $(Reθ≈500)$ since increasing the axial extent of leading edge roughness causes the transition point to shift upstream (see Figs. 5 and 7). Ultimately, earlier transition leads to a larger value of $Reθ$ at the vane trailing edge since more (turbulent) momentum loss is accumulated along a greater proportion of the suction surface, relative to the other cases. Increasing the axial extent of leading edge roughness also causes the $C¯f$ and $C¯h$ curves to shift upward (relative to the smooth-vane level) on the range $400≲Reθ≲700$⁠, which is consistent with the remarks of Denton [43], who predicted greater losses in the transition region at higher values of roughness. The $C¯f$ curves for cases X50_0808 and X60_0808 also highlight how an early transition produces a boundary layer that is more resistant to the suction surface adverse pressure gradient, whereas $C¯f$ curves for the other (nonturbulent) cases exhibit a rapid reduction within range $400≲Reθ≲700$⁠, before finally transitioning to turbulence at around $Reθ≈1,000$⁠. A similar behavior is evident in the HPT Moody diagrams constructed in a past experimental study by Thole et al. [8], although those authors preferred to examine the $C¯f/2=C¯hPr2/3$ scaling as a function of Reynolds number based on distance from the leading edge of a smooth vane, as opposed to $Reθ$ scaling presented here.

The impact of localized, non-Gaussian surface roughness on HPT aerothermal performance was quantified at an exit Reynolds number of 590,000 and exit Mach number of 0.92. This was achieved by undertaking a series of extreme-scale compressible DNSs for a range of HPT vanes covered with systematically varied surface roughness for a range of realistic turbulence inflow conditions using up to 14.6 billion grid points per simulation. The key results of this study are as follows:

• Drag partition between pressure and viscous forces. The present study shows that the fractional contribution of pressure drag to the total drag falls short of the fully rough regime. In the aft portion of the suction surface, the (inviscid) pressure force is nine times smaller than its viscous counterpart, and, as a result, the (turbulent) near-wall flow perceives the blade surface as an aerodynamically (and thermodynamically) smooth surface. As a result, roughness effects cannot be modeled using an equivalent value of Nikuradse’s sandgrain roughness, since $ks$ is only meaningful if pressure drag overwhelms viscous drag, i.e., the opposite of what is observed here.

• Augmentation of heat transfer and drag. While the augmentation of convective heat transfer comes at the expense of an even greater augmentation in drag—in both the global and local sense—significant augmentations only occur on the suction surface. For one of the rough HPT vanes investigated here, the local skin friction coefficient is a factor of 22 greater than the reference smooth-vane level, whereas the local Nusselt number is just six times higher in comparison. The rise in skin friction drag and heat transfer occurs as a direct result of contrasting boundary layer states over the smooth and rough vanes, which are driven by differing transition mechanisms and vortical structures along the suction surface.

• Reynolds’ analogy and Reynolds number scaling. The present investigation demonstrates that the Reynolds’ analogy, i.e., $2C¯h/C¯f≈1$⁠, can hold over the HPT vanes under investigation here, but only in regions (i) where the flow is fully turbulent and (ii) where viscous drag dominates pressure drag. Here, conditions (i) and (ii) are satisfied for three cases that undergo the earliest transition to turbulence on the vane suction surface. The Re scaling of heat transfer and drag was examined in the context of the Chilton–Colburn analogy. The HPT Moody diagrams demonstrate that the $C¯f/2=C¯hPr2/3$ scaling holds, but only for regions of turbulent flow where the momentum thickness Reynolds number was sufficiently large, i.e., $Reθ≳1000$⁠.

We are grateful for the permission of GE Aerospace to publish results from this study. Support from the ARC is acknowledged. This research used data generated on the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$c$ =

axial blade chord (m)

$h$ =

roughness height (m)

$k$ =

thermal conductivity (⁠$W/m/K$⁠)

$n$ =

blade normal distance (m)

$p$ =

pressure (⁠$kg/m/s2$⁠)

$s$ =

blade tangential distance (m)

$t$ =

time (s)

$x$ =

axial distance (m)

$y$ =

pitchwise distance (m)

$z$ =

spanwise distance (m)

$n$ =

surface normal unit vector (–)

$A$ =

planform area (⁠$m2$⁠)

$C$ =

log-law offset (–)

$E$ =

total energy (J)

$Q$ =

wall heat flux (⁠$W/m2$⁠)

$R$ =

specific gas constant (⁠$J/K/mol$⁠)

$T$ =

temperature (K)

$L$ =

turbulence length scale (m)

$cp$ =

specific heat capacity (⁠$J/K/kg$⁠)

$ks$ =

equivalent sandgrain height (m)

$qj$ =

heat flux vector (⁠$W/m2$⁠)

$t0$ =

initial time (s)

$ui$ =

velocity (⁠$m/s$⁠)

$Fp$ =

pressure drag force (⁠$kgm/s2$⁠)

$Fν$ =

viscous drag force (⁠$kgm/s2$⁠)

$Ftot$ =

total drag force (⁠$kgm/s2$⁠)

$Lz$ =

spanwise domain width (m)

$Nx$ =

number of axial grid points (–)

$Ny$ =

number of pitchwise grid points (–)

$Nz$ =

number of spanwise grid points (–)

$Rq$ =

root-mean-square roughness height (m)

$Tin$ =

inlet temperature (K)

$Tint$ =

sampling period (s)

$Uin$ =

inlet velocity (⁠$m/s$⁠)

$U¯τ$ =

mean friction velocity (⁠$m/s$⁠)

$txys$ =

smooth-vane tangential unit vector (–)

Tu =

turbulence intensity (⁠$m/s$⁠)

$γ$ =

heat capacity ratio

$δe$ =

boundary layer thickness (m)

$δij$ =

Kronecker delta (–)

$δ*$ =

displacement thickness (m)

$ΔT$ =

temperature difference (K)

$θ$ =

momentum thickness (m)

$κ$ =

von Kármán coefficient

$μ$ =

dynamic viscosity (⁠$kg/m/s$⁠)

$ν$ =

kinematic viscosity (⁠$m2/s$⁠)

$ρ$ =

density (⁠$kg/m3$⁠)

$τij$ =

viscous stress tensor (⁠$kg/m/s2$⁠)

$τw$ =

wall shear stress (⁠$kg/m/s2$⁠)

M =

inlet Mach number, $M≡Uin/γRTin$

$H$ =

shape factor, $δ*/θ$

$K$ =

Kurtosis

$S$ =

skewness

$Cf$ =

instantaneous skin friction coefficient, $2τw/(ρinUin2)$

$C¯f$ =

mean skin friction coefficient, $2τw/(ρ¯eU¯e2)$

$C¯h$ =

mean Stanton number, $Nu¯/(RePr)$

$Reτ$ =

friction Reynolds number, $U¯τδe/νw$

$Reθ$ =

momentum thickness Reynolds number, $U¯eθ/νw$

Nu =

instantaneous Nusselt number, $(c/ΔT)/∂T(x,z,t)∂n|w$

Pr =

Prandtl number, $cpμin/k$

Re =

inlet Reynolds number, $ρinUinc/μin$

$Nu¯$ =

mean Nusselt number, $(c/ΔT)/∂T¯∂n|w$

+ =

viscous-scaled quantity

$0$ =

reference quantity

$e$ =

boundary layer edge quantity

in =

inlet quantity

$r$ =

rough-vane quantity

$s$ =

smooth-vane quantity

$t$ =

tangential component

tot =

total quantity

$w$ =

wall quantity

$(⋅)¯$ =

time-then-spanwise average

$⟨(⋅)⟩$ =

time-then-surface-area average