Performance of Rough-Ribbed Low-Pressure Turbine Blades Under Varying Loading and Operating Conditions

Low-pressure turbines (LPTs), which power the fan, form one of the heaviest components of modern high bypass ratio turbofan engines. Due to the necessity of reducing the weight and maintenance cost of LPT, this component has received considerable research attention over the years. The ultra high-lift (UHL) blade designs employed by modern engines are at an increased risk of performance deterioration resulting from the tendency of flow to separate from the suction surface under strong local adverse pressure gradients (APGs). Researchers [1,2] have shown that the freestream turbulence (FST), incoming wakes, and surface roughness contribute positively toward reducing the separation bubble-related losses in the transitional regime. However, the ensuing boundary layer losses due to bypass transition increase the profile loss in the turbulent regime. Recent research [3–5] has started to explore the possibilities of employing passive flow control strategies, using riblets, to reduce the profile loss of turbine and compressor blades.

Riblets are ordered surface patterns that are widely investigated as a drag-reducing passive flow control device for channel flows and zero pressure gradient turbulent boundary layers (ZPGTBLs). The mechanism of drag reduction involving riblets is well documented in the literature [6,7]. Walsh and Lindemann [8] and Bechert et al. [6] thoroughly investigated the performance of riblets of different shapes and sizes for ZPGTBLs. Their studies showed that scallop and V-shaped riblets with spacing and height in the range $15<s+,h+<25$ are the two shapes that are capable of reducing the skin-friction drag varying from 5% to 10%. Of the very few studies exploring the performance of riblets under APGs, the experimental works of Debisschop and Nieuwstadt [9] and Sundaram et al. [10] are notable. They showed that the riblets have the potential to exhibit superior drag reduction performance (ranging from 13% to 16%) under APGs. The manufacturing challenges for riblets for turbomachinery applications are explored by Klocke et al. [11] and Klumpp et al. [12]. It was found that scallop-shaped riblets for compressor and turbine blades can be produced through an incremental rolling process and the durability of these structures can be improved through strain hardening and micro-structure deformation.

In the context of LPT blade flows, Dellacasagrande et al. [3] recently conducted experimental measurements over blades with riblets. They observed a drag reduction of 8% using V-shaped riblets for a viscous spacing of 20 units. Our recent studies [5] revealed that scallop-shaped riblets with $s+=17$ result in a skin-friction reduction of 7.3% under high-lift (HL) loading at cruise Re. Building on the potential of massive scale-resolving simulations to discern the riblet performance for LPT applications [13], the goal of current research is to explore the robustness of the proposed “rough-ribbed” LPT blade [5] under various loadings and Re. To this end, we perform several high-fidelity scale-resolving simulations on the configuration of a flow developing over a flat plate under the influence of FST and prescribed LPT loading. It has been shown before [14,15] that this framework successfully captures the key unsteady flow features over a realistic turbine blade surface. The present work’s novelty lies in applying roughness and flush-mounted riblets in different flow regimes under design and off-design conditions.

The outline of the paper is as follows. First, the computational framework is described providing details of the flow domain, boundary conditions, test cases, spatial and temporal resolutions, and in-house solver used to perform the simulations. A grid refinement study is presented to demonstrate the resolution adequacy of high Re simulations. In the next section, instantaneous flow features and time-averaged statistics are compared and contrasted among different test cases, identifying their implications on the suction surface profile loss. Subsequently, the overall performance of the “rough-ribbed” LPT blade is assessed based on the key parameters that are important for LPT blade flows. Finally, key findings are summarized along with concluding remarks and future recommendations.

Figure 1 presents the schematic of the computational domain with the boundary conditions. A uniform inflow superimposed with FST is specified at the inlet. The FST is simulated by employing the synthetic eddy method proposed by Jarrin et al. [16]. The exit is set to a constant pressure condition with streamwise grid stretching employed beyond $x/S0=1.2$ to avoid any contamination of the flow field from acoustic reflections that might emanate from the outflow boundary [17]. Adopting the numerical framework of Rao et al. [18], the bottom boundary upstream of the elliptic leading edge (⁠$x/S0<0$⁠) is set as a free-slip wall and a no-slip condition is imposed beyond $x/S0≥0$⁠. The distributed hemispherical roughnesses and riblets are represented using the boundary data immersion method (BDIM), which inherently enforces the no-slip condition via a set of meta equations [19]. Inset plots in Fig. 1 show the magnified views of a roughness element and the riblet geometry. The top wall of the computational domain is contoured to employ the streamwise varying pressure gradient typical of low-pressure turbine blades, which is gradually flattened beyond $x/S0=1$ thereby creating ZPG conditions. To prevent flow separation from the top wall, a free-slip condition is imposed. In this study, the shape of the contoured top wall is varied to test the riblet efficacy under three progressively increasing loading conditions. Periodicity is enforced in the spanwise direction. The chosen spanwise extent of $15%S0$ is sufficient to accommodate all the dominant modes developing in the transitional and turbulent boundary layers as this value exceeds the recommended $8%S0$ of Wissink and Rodi [20] and $12%S0$ of Lardeau et al. [21].

Three different loadings ranging from a diffusion factor of 16–42% [22] are investigated in this study as shown in Table 1. The low-lift (LL), high-lift, and ultra high-lift loadings correspond to the designs F, C, and A from Ref. [22], respectively.

Figure 2 depicts the top wall contouring for different loadings along with the layout and cross sections of roughness and riblet geometries investigated. In Fig. 2(a), notice that the throat cross section (⁠$x≈42%S0$⁠) progressively reduces with an increase in the loading from LL to UHL, thereby increasing the diffusion factor from 16% to 42% in the APG region. Figure 2(b) shows the top view of the flat surface with distributed hemispherical roughness and riblets on the $x$–$z$ plane. In line with the recommendations of Vera et al. [23], the roughness elements are optimally placed downstream of the peak suction and upstream of the separation point to effectively suppress the separation bubble. Following the strategy of our recent research [5,24], the riblets are employed downstream of the surface roughness to reduce the turbulent wetted area in the APG region. Figure 2(c) presents the streamwise cross-sectional view of this layout in the $x$–$y$ plane. This view specifically shows the height and relative positioning of the roughness and riblet configuration. To avoid any losses due to the shear layer generated from an abrupt surface geometry change [25], the riblets are flush-mounted (where the riblet tip is positioned in line with the upstream flat surface level) and a cosine function [26] is employed to ramp down the riblets to facilitate a smooth transition from the flat surface to the riblet surface. A ramp length of $6%S0$ is used for the present study. Figure 2(d) shows the cross sections of the scallop-shaped riblets employed for this investigation. First, Rib1 is tested for both the Reynolds numbers and subsequently the riblet dimensions are scaled down to optimum values (Rib2) based on our previous investigation [5] to account for the effects of an increase in Reynolds number. In this research, the scallop-shaped riblets are selected for their superior drag-reducing efficacy [6,25,27]. The absence of sharp tips and corners renders the scallop-shaped riblets suitable for LPT applications and hence is a realistic choice for the riblet manufacturers [11,12]. Furthermore, on a computational perspective, the absence of sharp features makes achieving grid independence results relatively easier.

Table 2 defines the key physical dimensions of the roughness and riblets. Based on the flat surface test case without any roughness and riblets under high-lift loading [25], a roughness height of half the local boundary layer thickness at Re1 is employed for the boundary layer tripping. The roughness elements are distributed equidistantly such that spanwise periodicity is satisfied. Based on the research of Strand and Goldstein [28], a ramp length of 6% of the suction surface length is adopted in this study. The physical dimensions of riblet spacing and height corresponding to Rib1 and Rib2 are $0.0057S0$ and $0.0038S0$⁠, respectively. These values can alternatively be deduced from Fig. 2(d). For Rib1, $0.0057S0$ along with the span and streamwise averaged friction velocity (⁠$uτ,av$⁠) from the turbulent regime of HSRe1 test case sets the riblet dimensions in viscous units. For Rib2, $0.0038S0$ in conjunction with $uτ,av$ from the USRe2 case dictates the riblet dimensions in wall units.

Table 3 lists the high-fidelity simulations performed in the current investigation. For each of the three loadings, the simulation of a tripped boundary layer developing over a smooth surface (in the APG region) is carried out. The boundary layer is tripped using three equally spaced hemispherical roughness elements with a peak-to-peak spacing of $0.05S0$ ensuring periodicity in the spanwise direction. Simulations are carried out at $Re1=83,000$ with roughness tripping alone and serve as baseline cases to subsequently compare the riblet efficacy at respective loadings. These three test cases also demonstrate the ability of the roughness to suppress the separation bubble typically encountered in the APG region with increased loading. For each of the baseline cases, simulations are carried out with two-dimensional scallop-shaped riblets with a viscous spacing (⁠$s+$⁠) and height (⁠$s+$⁠) of 22 units (referred to as Rib1 in this paper), respectively. These dimensions are chosen based on our earlier study where the effect of riblet spacing and height has been tested under high-lift loading conditions [5]. In addition to the six simulations described so far, the effect of off-design conditions on the ultra high-lift configuration is examined by increasing the Reynolds number to $Re2=166,000$⁠. Three additional cases are simulated on the UHL configuration which include the baseline case (with roughness tripping alone) and simulations with riblet dimensions of $s+=h+=22$ and $s+=h+=17$ referred to as Rib1 and Rib2, respectively.

As stated in Table 1, for Re1, a grid comprising of $515×125×1040$ is used to discretize the computational domain in the streamwise, wall-normal and spanwise directions upstream of the riblets (⁠$x/S0<0.52$⁠). Beyond $x/S0>0.52$⁠, around $484×157×1040$ points are used to resolve flow over the riblets. The grid points across the interface $x=0.52S0$ are point matched and since $Ny$ upstream and downstream of this location varies, the computational domain is split into different blocks at this location with a six-point overlap of grid points across the block interface. The spanwise resolution required per riblet element for the riblet simulations is established based on a thorough grid refinement study in our recent studies [24,25]. It was observed that grid-independent results for the first- and second-order statistics can be achieved beyond 30 points per riblet element resolution along the span. For all the riblet cases at low Reynolds number, Re1, each riblet element is resolved using 40 points in the spanwise direction. As listed in Table 1, the grid is further refined in the streamwise and wall-normal directions for cases at higher Reynolds number, Re2, to ensure adequate resolution of fine-scale turbulence. For simulations involving Rib2 at Re2, the spanwise direction is further refined to ensure a minimum of 30 points/riblet resolution. A near-wall grid spacing of $Δyw+=uτΔy/ν<0.9$ and $Δyw+<0.7$ is achieved for all the Re1 and Re2 simulations, respectively. The streamwise grid resolution within the pressure gradient region varies between $1.0<Δx+<10$ for both the Re1 and Re2 simulations. The streamwise grid spacing is finest over the riblet ramp and the hemispherical roughness elements where adequate resolution needs to be ensured. A minimum spanwise resolution of $Δz+=0.6$ is ensured for all the test cases. For instance, Fig. 3 presents the streamwise variation of grid resolution in viscous units for the USRe1 test case.

All the Re1 simulations are performed at a time-step of 1.7 $×$$10−5$$S0/Uin$ (corresponding to Courant–Friedrichs–Lewy number, $CFL≈1$⁠) and further reducing the time-step has a negligible effect on the first- and second-order statistics. Due to the grid refinement in the streamwise and wall-normal directions for the Re2 simulations, the time-step is reduced to 1.2 $×$$10−5$$S0/Uin$ to ensure numerical stability. For all the test cases, the initial transients are flushed out for three through flows (defined in terms of the suction surface length and inflow velocity) and the flow statistics are then collected for a period of five through flows. It has been ensured that the primitive variables, Reynolds stresses, and turbulent kinetic energy (TKE) have all converged and that further time averaging has a minimal effect on the flow statistics.

A high-order structured multi-block in-house solver—COMP-SQUARE—is employed to solve the 3D compressible Navier–Stokes equations in generalized curvilinear coordinates. The code is equipped with explicit finite difference (FD) schemes of up to fourth-order accuracy and compact schemes of up to sixth-order accuracy to compute the spatial derivatives as discussed in Ref. [29]. In the current set of simulations, all the spatial derivatives are computed using the fourth-order accurate centered FD scheme. The explicit four-stage fourth-order Runge–Kutta scheme is used for time marching. An implicit spatial filter of eighth-order accuracy with a tuning parameter of $αf=0.48$ is employed to retain both stability and high accuracy. As noted in the previous sub-section, the grid used in the current study is sufficiently fine to accurately resolve the near-wall turbulent structures (⁠$Δx+<10$⁠, $Δyw+<1$⁠, $Δz+<1$⁠), and hence all the computations reported are indeed equivalent to direct numerical simulations (DNS). COMP-SQUARE has been validated over a wide range of test cases and the readers are referred to Refs. [24,30] for more details. The solver scales on message passing interface (MPI) and multi-GPU (graphics processing unit) platforms and the simulations reported in this work are performed using NVIDIA Tesla A100 80GB GPU cards. For each Re1 simulation, the domain is decomposed into four blocks of approximately equal sizes with each block assigned to a GPU. Thus, four GPUs are employed for a simulation and the communication between them is established through MPI. For the Re2 simulations, the domain is split into eight blocks of approximately equal sizes to utilize eight GPUs for each simulation. Following Gaitonde and Visbal [31], a six-point overlap between the block interfaces is employed for preserving the numerical accuracy of the spatial discretization. All the Re1 test cases are run on a grid comprising of $≈$146 million grid points. The Re2 computations comprise of $≈$210 million grid points while the URib2Re2 case required $≈$236 million points to accommodate the increased number of riblet elements along the span. The wall-clock time required for one through flow of computation is estimated to be around 6.5 h for the Re1 cases and $≈$9.2 h for the Re2 cases.

The second-order accurate BDIM for compressible flows, proposed by Schlanderer et al. [19], is used to represent the hemispherical roughness elements and riblets. The BDIM offers the flexibility to use Cartesian grids and perform flow computations over complex geometries without having to generate intricate body-fitted meshes. The implementation of the BDIM in COMP-SQUARE has been validated on different canonical test cases and the readers are referred to Refs. [24,25] for more details.

A grid refinement study is performed for the high Reynolds number case Re2 = 166,000 on the URib1Re2 test case. Table 4 presents the details of three different grids—G1, G2, and G3—adopted for the study. Having established the required spanwise resolution already, the grid points are now progressively refined in the streamwise and wall-normal directions in the adverse pressure gradient region, $0.42<x/S0<1$⁠. Figures 4(a) and 4(b) show the streamwise variation of the time and span averaged momentum thickness and maximum turbulent kinetic energy. The latter is computed by first identifying the peak TKE values along both streamwise and spanwise directions. Subsequently, these values are span averaged to obtain the streamwise variation. From these results, it is evident that the grid independence is attained for both the first- and second-order statistics beyond G2, and hence grid G2 is chosen for performing the Re2 test cases.

To validate the numerical setup of the boundary layer developing under a streamwise varying pressure gradient in the absence of roughness and riblets, three simulations (not listed in Table 3) under the grid resolutions discussed earlier are performed under LL, HL, and UHL loadings. Figure 5 shows the surface velocity distribution for these cases. The experimental data from Ref. [15] for the HL loading and Ref. [22] for the LL loading are overlaid for comparison. In both cases, a favorable agreement is obtained, especially in the adverse pressure gradient regime of the flow. For the LL loading, the separation bubble is shorter in the experiments as the measurements were performed at a higher Reynolds number (Re = 268,000). The inset plot corresponding to each loading shows the separation bubble with recirculating streamlines. As expected, the increase in the size of the separation bubble with loading is evident from these insets.

Figures 6(a)–6(f) compare the coherent structures visualized using Q-iso-surfaces (⁠$Q=250$⁠) for different loadings at Re1 = 83,000 over a smooth surface and that over a riblet surface. The iso-surfaces depicting the flow features are colored using streamwise velocity. Also presented for each case are contours of spanwise component of vorticity at $z/S0=0$ plane and a streamwise slice of vorticity magnitude showing the near-wall vortex motions close to the smooth/riblet surface. The latter is shown around the mid-span region at a streamwise location of $x/S0=0.62$⁠, which is also immediately downstream of the riblet ramp for the riblet cases. Figure 6(a) shows the flow features developing for the LSRe1 test case. Hemispherical roughness elements employed downstream of the peak suction (⁠$x/S0=0.42$⁠) perturb the boundary layer causing early transition. For all the smooth wall cases, the BDIM is used to represent the flat surface from $0.52<x/S0<1.7$⁠. This is done just to ensure consistency, with the riblet cases, in the application of numerical methods across different zones of the computational setup. The BDIM-represented geometries are shown as black color in the background. The development of horseshoe vortices around the roughness elements is visible (also marked in Fig. 6(b)). The shear layers shed from the roughness peak evolve downstream giving rise to a train of hairpin vortices. The leeward interaction between the legs of horseshoe vortices and hairpin structures creates secondary instabilities [32], which results in the spanwise spreading of the vortices. The adverse pressure gradient further destabilizes the evolving boundary layer resulting in a turbulent breakdown and a cascade of hairpin vortices (see Fig. 6(a)). These hairpin vortices enhance skin friction by wall-normal momentum transport [33]. The streamwise slice of vorticity magnitude shows the vortical motions associated with the evolution of hairpin structures downstream of the roughness in the transition region. Figures 6(b) and 6(c) show the boundary layer evolution under high-lift and ultra high-lift loadings over a smooth surface. Although the flow features qualitatively remain the same as for the case of low-lift loading, the secondary vortices downstream of the roughness elements become increasingly complex for the HSRe1 case (see Fig. 6(b)) and the USRe1 case (see Fig. 6(c)). As marked in these figures, the vortical structures appear to develop noticeable spanwise interaction for these cases. The comparison of vorticity contours on the streamwise slices at $x/S0=0.62$ indicates that an increase in the loading (a) promotes early transition and enhances the vortical motions, and (b) progressively thickens the boundary layer lifting the vortical structures away from the wall under stronger APGs. The latter effect together with an increase in freestream velocity with loading causes the hairpin structures to convect at higher velocities as evident from the streamwise velocity contours in Fig. 6(c).

Figures 6(d)–6(f) present the development of a boundary layer over a riblet surface under low-lift, high-lift, and ultra high-lift loadings. The flush-mounted riblets represented using BDIM are shown as black color in the background. Development of part-span rollers over the riblet surface is evident from the Q-iso-surfaces. This observation also finds support from DNS of riblet-mounted open channel flows by Endrikat et al. [34] and Rouhi et al. [35]. The latter research attributes the formation of rollers to the generation of a mixing layer between slow-moving flow within the riblet valleys and fast-moving flow above the riblet tips. The part-span rollers become increasingly prominent with loading (marked for the URib1Re1 case). Such rollers likely enhance the vortical fluctuations thereby hastening the transition process compared to the corresponding smooth surface case. Similar to the smooth wall cases, the vorticity increases with loading for all the riblet cases (compare the streamwise slices of vorticity in Figs. 6(d)–6(f)). However, the chosen riblet spacing (⁠$s+=22$⁠) is effective in shielding the majority of the riblet surface from near-wall vortical motions. This is attributed to the fact that the riblet spacing chosen is smaller than the typical diameter of quasi-streamwise vortices for zero and adverse pressure gradient flows (⁠$d+$ = 30) as noted by Choi et al. [7] and Lee and Sung [36]. As a result, the vortices cannot penetrate the riblet valleys although the riblet crests are exposed to enhanced vorticity, thereby reducing the drag.

Figures 7(a)–7(c) compare the turbulent structures over smooth and riblet surfaces at Re2 = 166,000 for the ultra high-lift loading. It is immediately noticeable from the USRe2 test case that the turbulent structures are finer at Re2 compared to those at Re1. Moreover, the shear layer development around and downstream of the hemispherical roughness appears much more complicated with richer flow features. The spanwise interaction of horseshoe vortices is visible (see Fig. 7(a)) and the appearance of a counter-rotating streamwise vortex pair is more prominent at Re2 (see Fig. 7(b)). When comparing the USRe1 and USRe2 test cases, it is evident that the spanwise spread rate of vortical structures has markedly increased for Re2. This is because of the much faster onset of secondary instabilities at higher Reynolds numbers [32]. This suggests that the transition length at Re2 is shorter compared to that at Re1, as expected. Figure 7(b) presents the URib1Re2 test case, which employs the same riblet geometry that is used for the Re1 simulations. Contrary to what is expected, the streamwise slice of vorticity magnitude contours shows the residence of energetic vortices within the riblet grooves. The strength of vorticity also appears to be marginally higher than in the corresponding smooth surface case, USRe2. To minimize the penetration of vortical motions into the groove region of the riblets, another riblet surface Rib2 with $s+=h+=17$⁠, defined based on the $uτ,av$ of USRe2 case, is tested under ultra high-lift loading. This test case is presented as URib2Re2 in Fig. 7(c). When the riblet spacing is reduced (by a factor of $1/3$⁠), the vorticity magnitude has reduced compared to that of the URib1Re2 case. However, the vortical motions still penetrate the riblet grooves and the net turbulent wetted area for the URib2Re2 case still appears to be higher than for the USRe2 case. This suggests that at higher Reynolds numbers, for skin-friction reduction, optimization of the riblet ramp is equally important in addition to employing appropriately sized riblets. This aspect will be revisited in a subsequent section.

Figures 8(a)–8(d) compare the time and span averaged boundary layer integral parameters for different test cases investigated. Figures 8(a) and 8(b) show the streamwise variation of displacement thickness (⁠$δ*$⁠) over both the smooth and riblet surfaces under different loadings and Re. For all the cases, a sharp change at $x/S0≈0.43$ occurs which corresponds to the location of the hemispherical roughness elements. As the loading increases from low-lift to ultra high-lift, $δ*$ increases in the APG region signifying thickening of boundary layers. For all the riblet cases, in the cosine ramp region (⁠$0.52<x/S0<0.58$⁠), there occurs an initial sharp increase in $δ*$ followed by a gradual growth downstream of the ramp. This is marked in Fig. 8(a). This was observed in our previous investigation [5] and is attributed to the formation of small pockets of reverse flow in the riblet troughs close to the ramp. For the LRib1Re1 and HRib1Re1 cases, this has a considerable effect in the downstream evolution of the boundary layer. Notably, at the hypothetical trailing edge (TE) for both these cases, the displacement thicknesses are higher than their corresponding baseline cases. This implies that for the LRib1Re1 and HRib1Re1 cases, there is an added mass deficit due to the application of the Rib1 surface. The recirculation region for the ultra high-lift loading is shown as an inset plot in Fig. 8(b). Interestingly, for this loading, $δ*$ over riblets at the TE is lower than in the USRe1 case as is evident from the figure. Approximately, a 7.5% reduction in $δ*$ is achieved at the TE for the URib1Re1 configuration compared to its baseline. This is ascribed to the fact that under ultra high-lift conditions, the APG strength has a dominant impact on the state of the evolving boundary layer. As a result, a reduction of $δ*$ is realized at the TE over riblets despite the existence of a recirculation zone close to the riblet ramp. This demonstrates that the riblets are effective in reducing the mass deficit (and hence passage flow blockage [37]) due to the boundary layers under UHL conditions. For the URib1Re2 case, the sharp rise in the $δ*$ has significantly reduced compared to the URib1Re1 case indicating a reduction in the strength of reverse flow close to the ramp. This further elucidates that the deleterious effect of the ramp in causing flow blockage disappears with an increase in both loading and Reynolds number. The URib2Re2 test case depicts the variation of $δ*$ over a riblet surface of smaller height and spacing values. It is clear that when the riblet spacing and height are optimized, the state of the boundary layer improves further as the recirculation zone shrinks even more. Compared to their respective baseline cases, at the hypothetical TE, the $δ*$ has reduced by a factor of $≈$16% and $≈$33% for the URib1Re2 and URib2Re2 cases, respectively.

Figures 8(c) and 8(d) compare the momentum thickness variation for the test cases under different loadings and Re. For all the cases, a monotonically increasing variation is observed. The momentum thickness over the riblet surface Rib1 remains consistently lower relative to the respective baseline cases in the APG region. Furthermore, it is interesting to note that the thickness reduction over the riblet surface tends to increase with loading. Through a control volume analysis, Denton [38] demonstrated that the blade profile loss is proportional to the trailing edge momentum thickness, $θTE$⁠. The present flat plate configuration lacks a physical trailing edge. Hence, it is sensible to associate $θ$ at the hypothetical trailing edge $x/S0=1$ as a measure of loss. The highest reduction in $θTE$ caused by the riblets is achieved under ultra high-lift loading. This shows that the riblets are effective in reducing the mixing losses under adverse pressure gradients and the efficacy is enhanced with loading. The momentum thickness values for the USRe2 case remain slightly lower than that of the USRe1 values. Over the Rib1 surface, the momentum thickness of the boundary layer is lower compared to the respective base line cases under UHL. With an increase in the Reynolds number, a $θTE$ reduction of $≈18%$ and $≈25%$ is realized for the URib1Re2 and URib2Re2 cases, respectively, compared to the USRe2 case. A higher $θTE$ reduction with the Rib2 surface is due to its more optimized dimensions at Re2.

Figures 9(a) and 9(b) compare the $Cf$ variation over the riblet and smooth surfaces under different loadings and Re. In all the curves, $Cf$ displays a peak at $x/S0≈0.43$ corresponding to the location of roughness elements and at $x/S0≈0.52$ corresponding to the riblet leading edge. However, since a cosine ramp is provided at $x/S0≈0.52$⁠, we retain a smooth variation in $Cf$ avoiding any undesirable step changes. Following Jacobs and Durbin [39], the transition and turbulence onsets in general are identified as locations where $Cf$ reaches its minimum and maximum values, respectively. However, in the presence of APGs, the turbulent breakdown can be better identified by associating with the location of peak $kmax$⁠.

For all the curves, the transition onset occurs at $x/S0>0.6$⁠. Moreover, the $Cf$ values decrease from low-lift to ultra high-lift loading due to a reduction in the wall-normal velocity gradients. A relatively large dip in the value of $Cf$ over riblet surfaces during transition is due to the occurrence of reverse flow close to the riblet ramp region. Comparing the LSRe1, HSRe1, and USRe1 cases, it is evident that the transition onset location shifts upstream along with a shortening of the transition length when loading increases. When the boundary layer growth over smooth and riblet surfaces are compared, it is clear that an appreciable reduction in $Cf$ is obtained at Re1 under all loadings and the extent of drag reduction is the highest for ultra high-lift loading. These results corroborate with the experimental observations made by Debisschop and Nieuwstadt [9] and Sundaram et al. [10] who reported drag reductions of $≈$13% and $≈$16%, respectively, under strong adverse pressure gradients. A comparison between the USRe1 and USRe2 test cases show that the transition onset shifts upstream, accompanied by a reduction in transition length at Re2. For the USRe2 case, subsequent to the turbulent onset at $x/S0≈0.62$⁠, the $Cf$ values are significantly lower than those of the USRe1 case. This is again due to reduction in the wall-normal velocity gradients caused by a thicker boundary layer growth at higher Reynolds number. The URib1Re2 case depicts an enhanced skin-friction coefficient in the transitional regime compared to its baseline USRe2 case. This is due to the fact that the Rib1 surface, which is optimized for Re1 and high-lift loading, becomes a drag increasing configuration at Re2 under ultra high-lift loading. Although a marginal reduction of $Cf$ compared to the USRe2 case is observable in the range $0.75<x/S0<1.0$⁠, overall the URib1Re2 configuration acts as a drag increasing one. To reduce the increase in the $Cf$⁠, the test case URib2Re2 is investigated wherein the riblet spacing and height are re-sized to 17 viscous units based on Re2 and $uτ,av$ from the USRe2 case. Although a reduction in $Cf$ is attained in the transitional regime compared to the URib1Re2, the configuration still remains a drag increasing one throughout the riblet length compared to the baseline case. This contrasting behavior will be investigated in more detail in Fig. 10(e).

Figures 9(c) and 9(d) present the maximum turbulent kinetic energy variation over smooth and riblet surfaces for various loadings and Re values. For all the cases, an abrupt rise in $kmax$ is observable immediately downstream of the roughness elements at $x/S0=0.43$⁠. This is due to the onset of pre-transitional fluctuations generated within the boundary layer from roughness tripping. Associated with these fluctuations a peak, P3, is prominent in the $kmax$ curves. For the Re1 cases, downstream of this peak, the disturbance growth continues its steep evolution reaching a second peak, P2, although the growth rate has marginally reduced compared to the initial one. The formation of P2 is likely associated with the emergence of secondary instabilities associated with the coherent hairpin vortices. Subsequently, the typical peak (P1) associated with the onset of turbulence develops downstream of which the $kmax$ gradually decreases due to diffusion in the wall-normal direction. Overall, it is observable that the $kmax$ values increase with an increase in loading. For all the riblet cases, the peak value of $kmax$ is lower than in the corresponding smooth surface cases. However, as observable from the upstream shift in the location of the $kmax$ peak, it is evident that the riblets favor early transition under all the loadings at Re1. This concurs with the experimental findings of Ladd et al. [40] under zero pressure gradient. Kozul et al. [4] from their DNS of compressor blade flows also noted that the riblets hasten the transition process. At Re2, for all the cases, the peak, P2, disappears and the growth rate and level of pre-transitional fluctuations are noticeably higher than in the Re1 cases. This results in the occurrence of a much larger peak, P3, at Re2. This is attributable to the fact that a larger roughness height, k, accelerates the non-linear breakdown process. It is to be noted that the roughness height, k, is not scaled down in accordance with the Reynolds number increase (see Table 2) for the Re2 cases. Accordingly, the transition mechanism is expected to be different at Re2 and Re1. For the USRe2 case, as a result of an increase in the Reynolds number, the $kmax$ peak, P1, increases and shifts upstream compared to the USRe1 case. The tendency of riblets to trigger early transition diminishes at Re2 and the Reynolds number appears to be the dominant parameter that dictates the transition onset over both smooth and riblet surfaces. For the drag increasing riblet configuration, URib1Re2, the peak P1 reaches larger values than in the baseline USRe2 case. However, $kmax$ drops below the corresponding USRe2 values beyond $x/S0≈0.63$⁠. For the URib2Re2 case, despite the $Cf$ increase observed earlier, the $kmax$ values consistently stay lower than in the baseline USRe2 case.

Figures 10(a) and 10(b) compare the variation of riblet spacing and height in viscous units for all the cases. At Re1, the viscous dimensions of the riblets stay in the range $15<s+,h+<25$⁠, which is known to be in the drag-reducing regime [8]. This is consistent with the skin-friction reduction observed for all the riblet cases at Re1. For the URib1Re2 case, compared to the baseline USRe2 case, the transitional $Cf$ is higher in the region $0.52<x/S0<0.75$ as noted in Fig. 9(b). In this streamwise extent, it is evident from Fig. 10(b) that the $s+,h+>30$ for Rib1. Previous studies [5,8,41] have demonstrated that these riblet dimensions fall in the drag increasing regime. It is worth noting that at the transition onset location of $x/S0≈0.57$⁠, $s+$ and $h+$ values are as high as 43. This means that the quasi-streamwise vortices having diameters of $d+≈30$ [36] can easily penetrate the riblet grooves, thereby enhancing the vortical activity in these regions. This justifies the drag increasing behavior of URib1Re2. However, the Rib2 surface that is designed based on the $uτ,av$ of the USRe2 case exhibits a significant streamwise variation of $s+$ and $h+$⁠. The viscous dimensions of the riblets vary from $≈$33 at $x/S0=0.52$ to $≈$15 at $x/S0=1$⁠. It is interesting to note that despite the dimensions staying within 25 viscous units for a considerable streamwise extent, $Cf$ values remain higher than in the baseline USRe2 case as seen in Fig. 9(b).

To investigate this contrasting behavior of increasing $Cf$ for the URib2Re2 case, the vortical activity in the riblet ramp is carefully examined. Accordingly, Figs. 10(c)–10(e) compare the vorticity distribution and wall-normal velocity gradient, $∂u/∂y$⁠, over riblets under UHL for Re1 and Re2. $∂u/∂y$ is shown from a streamwise location approximately coinciding with the $Cf$ peak (see Fig. 9(b)). Each of the Figs. 10(c)–10(e) further shows a spanwise slice of the phase averaged vorticity magnitude contours. These slices are presented from a plane that intersects with the riblet trough. To generate the $∂u/∂y$ and contours in spanwise plane, the time-averaged flow field is further phase averaged over all the riblet elements to obtain the representative flow over a single riblet. The vorticity magnitude distribution for the URib1Re1 case features regions of reduced vorticity levels, except at the riblet tips for $x/S0<0.6$⁠. The corresponding velocity gradient contours show regions of reduced shear in the valley region. In the spanwise plane showing the ramp region, the riblet grooves exhibit an extended region devoid of any vortical activity along the riblet length. This is marked as a “quiescent region” in Fig. 10(c). For the URib1Re2 case, there are well identifiable pockets of high vorticity in riblet troughs as marked in Fig. 10(d). The phase averaged velocity gradients are correspondingly higher both in the riblet tip and valley regions. The entire ramp region facilitates the existence of intense vorticity with its magnitude being noticeably high in the riblet trough locations (marked in the figure). For the URib2Re2 case, when the riblet dimensions are more optimized, the well distinguishable pockets of high vorticity distribution seen in the range $0.58<x/S0<0.64$ for the URib1Re2 case have almost disappeared, signifying the efficacy of employing optimal riblet dimensions. However, the vorticity levels in the ramp, $0.54<x/S0<0.6$⁠, still remain high. Although the shear has partly reduced, the contours from spanwise plane in the ramp region show continuing presence of high vorticity. This suggests that at Re2, in addition to scaling down the viscous dimensions of riblets, the ramp length also has to be reduced accordingly to achieve a net skin-friction reduction. This aspect will be confirmed by the authors in future works.

To identify the effects of loading and Reynolds number on the streamwise Reynolds stress, the time and span averaged $⟨u′u′⟩$ are examined for the low-lift and ultra high-lift test cases in Figs. 11(a)–11(g). Associated with these cases, Fig. 11(h) shows the wall-normal variation of $<u′u′>$ at the hypothetical trailing edge, $x/S0=1$⁠. The Reynolds stress contours in Fig. 11(a) depict a characteristic two peak structure in the APG region. This is consistent with the observations of Vadlamani [42] and Aubertine and Eaton [43] that under APGs an extended plateau of elevated streamwise normal stress dominates the outer layer. Close to the smooth surface, the development of a near-wall turbulence peak is visible [42]. Comparing Figs. 11(a)–11(d), it is clear that the riblets reduce both the magnitude and wall-normal spread rate of the streamwise stress component under different loadings. This is further confirmed by the wall-normal variation of $⟨u′u′⟩$ for these cases in Fig. 11(h). These improvements are more pronounced under ultra high-lift loading. Over Rib1, both the near-wall and outer peaks are reduced. For the USRe1 case, as evident from Fig. 11(c), both the intensity and size of the $⟨u′u′⟩$ core have significantly increased compared to the LSRe1 case. It is worth noting that the outer peaks are intensified compared to the near-wall peaks under UHL loading. This is in line with the wall-normal variation of $⟨u′u′⟩$ in Fig. 11(h). The outer layer peak increases in magnitude with loading which is in agreement with the laser Doppler anemometry measurements of Aubertine and Eaton [43]. Figures 11(e)–11(g) compare the UHL test cases at Re2. For the USRe2 case, the $⟨u′u′⟩$ core shrinks in size compared to the USRe1 case which agrees with the $kmax$ variation in the range $0.7<x/S0<1$ in Fig. 9(d). For the URib1Re2 case, the $⟨u′u′⟩$ intensity becomes higher than the USRe2 case in the ramp region. Beyond this location, both the intensity and spread rate reduce downstream. Over the Rib2 surface, the Reynolds stress shows consistently lower values compared to the USRe2 case for the entire riblet length. From Fig. 11(h), it is clear that the outer peak depicts a diminishing trend from smooth to riblet surface for all cases. At Re2, the performance of Rib2 is better than that of Rib1. Moreover, with increase in Reynolds number, the outer peak shifts close to the wall for both smooth surface and riblet cases. The profiles also show that under ultra high-lift loading, the near-wall turbulence peak is not well developed and it is the outer peak that dictates the growth of $⟨u′u′⟩$⁠. This is likely attributable to the very strong wall-normal diffusion of turbulence under the influence of UHL. In accordance with the enhanced vorticity levels observed within the riblet troughs for the URib1Re2 and URib2Re2 cases, the $⟨u′u′⟩$ contours also show a higher stress in the valley region close to the riblet ramp. This persistent high stress manifests as a small bulge in the wall-normal $⟨u′u′⟩$ profile at $x/S0=1$ for the URib1Re2 case as marked in Fig. 11(h).

We now inspect the turbulent kinetic energy distribution over the riblets by comparison with the corresponding smooth surface cases under UHL. Accordingly, Figs. 12(a)–12(d) show TKE contours extracted at a wall-normal plane immediately above the surface roughness (⁠$y/S0=0.02$⁠). Associated with each case are also shown the TKE distribution at the hypothetical TE, $x/S0=1$⁠. Figures 12(a) and 12(b) show the TKE distribution for the USRe1 and URib1Re1 cases. At $x/S0≈0.47$⁠, a TKE core could be spotted corresponding to the peak P3 observed in Fig. 9(c) for the ultra high-lift cases. Downstream of this location, corresponding to each roughness element, two small cores appear as a result of secondary instabilities associated with the hairpin vortices (correlates with peak P2 in Fig. 9(c)). Further downstream in the range $0.6<x/S0<0.7$⁠, large TKE cores develop signifying turbulent breakdown. This corresponds to peak P1 in Fig. 9(c). Each of the TKE cores associated with P1 is located midway between two consecutive roughness elements. Even though all these features exist in the TKE contours of the URib1Re1 case, the TKE levels over the riblets have reduced. Moreover, the cores associated with the peaks P2 and P1 have all diminished more than in the USRe1 case. At $x/S0=1$⁠, the boundary layer over the riblet surface registers a significant TKE reduction. For the USRe2 case, the higher TKE levels associated with these cores matches with the much larger peak P3 in Fig. 9(d). The two small TKE cores seen downstream of each roughness element at Re1 disappears at Re2. For the URib2Re2 case, the cores associated with turbulent breakdown have drastically reduced which is in line with the $kmax$ variation observed in Fig. 9(d). This configuration exhibits a superior performance by reducing the TKE at the hypothetical TE compared to the baseline USRe2 case.

To assess the overall performance of “rough-ribbed” LPT blades under different loading and Reynolds numbers, Figs. 13(a)–13(d) compare the percentage variation in flow properties that are of interest from a blade design point of view. In the present work, the Rib1 surface which was originally designed based on the $uτ,av$ of the HSRe1 case is tested across different loadings. As discussed earlier in Figs. 10(a) and 10(b), due to the variation of skin-friction velocity with loadings and Reynolds numbers, the viscous dimensions of the Rib1 surface vary accordingly. Accounting for this deviation, the viscous spacing and height of riblets are recalculated for the Rib1 surface.

The percentage variation in $Cf$ realized for both the riblet surfaces under different loading and Re is consolidated in the form of a typical drag reduction curve as shown in Fig. 13(a). The experimental data available in the literature for comparable riblet dimensions and shape (also shown in the figure) are overlaid for comparison. It is to be noted that these data from Walsh and Lindemann’s research [8] are based on ZPGTBLs. To the best of our understanding, suitable data of scallop-shaped riblets for transitional boundary layers under APGs are unavailable in the literature. For the same physical geometry, the skin-friction reduction increases with loading. The skin-friction reduction increases from $ΔCf=3.4%$ under low-lift loading to $8%$ under ultra high-lift conditions. This is a very favorable outcome in the context of improving the performance of modern LPT blades. It is also noticeable that under APGs, the riblets result in superior skin-friction reduction than under ZPG and these trends concur with the experimental findings of Sundaram et al. [10] and Debisschop and Nieuwstadt [9]. At Re2, the URib1Re2 and URib2Re2 configurations result in a $Cf$ increase of $5%$ and $6.4%$⁠, respectively. This is attributed to the fact that the riblet ramp designed for Re1 performs poorly at Re2 as it facilitates the penetration of vortical motions within the grooves (as observed in Figs. 10(d) and 10(e)). The major contribution (⁠$≈$90%) to the suction surface profile loss comes from the mixed out loss of the boundary layers and $θTE$ is directly proportional to this loss component [42]. Accordingly, Fig. 13(b) compares the reduction in trailing edge momentum thickness over riblets relative to the baseline. It is encouraging to note that the $ΔθTE$ over riblets reduces from $10.5%$ under low-lift loading to $15.4%$ under UHL. This behavior confirms that the riblet efficacy can be harnessed further under ultra high-lift conditions. Interestingly, at Re2, the Rib2 surface registers a $ΔθTE$ reduction as high as $30%$ despite an overall skin-friction rise. To clarify these contrasting trends, when span averaging the $Cf$ curves at Re2, normalization is performed with respect to the spanwise running length of the riblet surface instead of using $Lz$⁠. The corresponding variations are shown in Fig. 13(e). It is interesting to note that when normalized using spanwise running length, $Cf$ values over riblets stay consistently lower than in the baseline USRe2 case. This demonstrates that the Rib1 and Rib2 designs both have contributed positively in reducing the near-wall turbulence compared to the USRe2 case (consistent with an overall reduction in $θ$ and $kmax$ for the URib1Re2 and URib2Re2 cases) and the residence of vortical motions within the riblet grooves at Re2 brings out the effect of net surface area increase with riblets compared to a smooth surface.

Figure 13(c) compares the $kmax$ reduction at the hypothetical TE for all the riblet cases. The TKE at the trailing edge is consistently lower for all the riblet cases and the percentage reduction increases with both loading and Reynolds numbers. For the URib1Re1 case (⁠$ΔkmaxTE≈22.4%$⁠), reduction is achieved. This confirms that the riblets reduce the amplitude level of fluctuations at the trailing edge. In a turbine stage, the fluctuations from the upstream blade rows impact the development of downstream boundary layers. Thus a reduction in $kmaxTE$ might aid in mitigating the boundary layer losses of downstream blade rows. To study the flow blockage introduced by the application of riblets, the blockage factor, B, is defined following the flow analysis of Zheng and Yang [37] on blade passage flows. The method of estimating B is explained in our earlier work [5]. This factor is an indication of the extent to which a physical flow passage area gets reduced due to the development of boundary layers. Figure 13(d) compares the percentage change in the blockage factor at the hypothetical trailing edge due to the application of riblets. For the low-lift and ultra high-lift cases, incorporation of riblets in the flow domain further constricts the passage flow area by $≈$20% and $≈$3.6%, respectively, compared to the corresponding smooth surface cases. The boundary layer integral parameter that dictates the flow blockage is the displacement thickness. It is clear from Fig. 8(a) that both the LRib1Re1 and HRib1Re1 cases suffer from increased $δTE*$ values compared to their baseline cases. For these loadings, the reverse flow generated in the grooves close to the riblet ramp leads to an increase in $δTE*$⁠, despite the riblet cases having lower growth rate of $δ*$ relative to the smooth surface cases (see Fig. 8(a)). The blockage factors for the ultra high-lift loading for both the Reynolds numbers have reduced compared to the corresponding baseline cases. This is because under UHL, the wall-normal boundary layer growth acts as the major contributor to flow blockage and the ramp generated reverse flow is not strong enough to increase it further. Thus, under UHL, the ramped riblets reduce the flow blockage and this effect is enhanced at higher Re. This riblet action is another desirable quality as it shows the potential to minimize deviation from the design exit flow angle by reducing blockage in LPT blades. This could help mitigate additional losses [44].

Based on the present research, the key findings are listed below:

• The efficacy of riblets in reducing profile loss increases under UHL loading. An improvement in skin-friction reduction, $ΔCf$⁠, from 3.4% under LL to 8% under UHL loading is realized at cruise conditions. The corresponding reduction in trailing edge momentum thickness, $ΔθTE$⁠, also improved from 10% to 15%.

• Under UHL loading, $θTE$ reduction with riblets improves from 15% at cruise Re to 25% at a higher Re (off-design) with optimized riblet dimensions. Thus, the effect of riblets in reducing the mixing losses improves with increasing Re.

• The tendency of riblets to trigger early transition diminishes with increasing Re.

• Riblets reduce flow blockage due to boundary layers under UHL loading for both design and off-design Re.

• Riblets consistently reduce the TKE levels under different loadings and Reynolds numbers.

• To achieve $Cf$ reduction across a range of Re, the ramp length of riblets requires further optimization. A practical design choice would be to optimize the ramp length for cruise Re where the LPTs operate for the most duration.

• The widely varying skin friction under UHL loading suggests that better performance might be achievable by varying the riblet dimensions along the flow direction.

In this paper, the efficacy of a “rough-ribbed” LPT blade in mitigating the profile loss is tested under various streamwise loadings and Re. The suction surface boundary layer is investigated by employing the configuration of a flow developing over a flat plate under streamwise varying pressure gradients and freestream turbulence. Three different loading types, namely, low-lift, high-lift, and ultra high-lift, are chosen to investigate the performance of riblets at a low Reynolds number, Re1 = 83,000, typically encountered during aircraft cruise of small business jets. To test the off-design performance of blades during take-off and landing, a higher Reynolds number, Re2 = 166,000, is also considered. To the author’s knowledge, the present work forms the first set of direct numerical simulations that explores the efficacy of riblets under design and off-conditions in LPT applications.

First, a thorough grid refinement study is established for the high Reynolds number simulations and the streamwise varying pressure gradient setup is validated with experimental data for different loading. Subsequently, an additional set of nine scale-resolving simulations are performed. By employing surface roughness downstream of the peak suction location, the separation bubble is eliminated under all loadings and Re. For each loading and Re combinations, a pair of simulations with and without scallop-shaped riblets in the APG region is carried out. Thus the efficacy of different riblet cases are compared with their respective baseline simulations. In line with the previous studies, the present investigation shows that riblets promote early transition under all loading conditions in relation to a smooth surface. However, this effect diminishes with increasing Re. It is perceived that the development of part-span rollers over riblets might affect the transition process by hastening it. The present study at low Reynolds number shows that appropriately sized riblet elements reduce the blade profile loss under all loading conditions. Specifically, the scallop-shaped riblets designed for HL loading with $h+,s+=22$ resulted in a $Cf$ reduction of 3.4%, 4.6%, and 8% for the LL, HL, and UHL loadings, respectively. This enhancement of $Cf$ reduction with loading is a very favorable outcome that can positively impact the design of modern ultra high-lift blades. At low Re, since the streamwise variation of riblet dimensions is limited in the range $16<s+,h+<23$ across different loadings, the streamwise averaged value of $s+$ and $h+$ from the turbulent regime could be used to design the riblets.

It is encouraging to note that the parameters dictating the mixed out losses in boundary layers, such as the trailing edge momentum thickness, Reynolds stresses, and TKE, are all reduced by scallop-shaped riblets both under different loadings and Reynolds numbers. The trailing edge momentum thickness reduced from 10.4% under LL to 15.4% under UHL loading at Re1. These values are further increased to 25% at Re2. The present investigation shows that riblets having viscous dimensions between 18 and 28 units are all effective in reducing the TKE and $θTE$⁠. Furthermore, the riblets effectively alters the flow by reducing both the near-wall turbulence peak and the outer layer peak, which is prominent under strong APGs. However, a contrasting behavior of increasing $Cf$ at Re2 is observed and it is found that careful optimization of riblet ramp in accordance with changing Re is required to prevent the increase in skin friction. The phase averaged vorticity magnitude contours in the riblet ramp indicates that the ramp length is to be shortened with increasing Re to prevent the vortical motions from penetrating the riblet troughs. The riblets are effective in reducing the passage flow blockage caused by boundary layers under ultra high-lift loading at both Re1 and Re2. This might potentially mitigate the deviation of exit flow angle during off-design operating conditions.

The current research has demonstrated the beneficial effects of riblets in reducing the suction surface profile loss under various operating conditions. It remains to be explored how the relative spacing between the idealized roughness and the riblet onset location affects the boundary layer development over the riblets. Furthermore, the effects of realistic roughness [45] in such a flow environment need to be investigated. Also, due to a wide variation in the skin-friction velocity under UHL loading, it turns out that the riblet design should account for this variation by altering the dimensions along the streamwise direction for achieving maximum benefits. It will be interesting to explore the performance of such a configuration in future. In addition, the effect of incoming wakes on riblet performance will have to be investigated.

This research was supported by The University of Melbourne’s Research Computing Services and the Petascale Campus Initiative. The authors gratefully acknowledge the PARAM Siddhi cluster under the National Supercomputing Mission, India, for providing computing resources. Dr. Vadlamani acknowledges the financial support through the SPARC project via grant No. “SPARC/2024-2025/CAMS/P1806” and the Science and Engineering Research Board (SERB) under the MATRICS Scheme (MTR/2022/000807).

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.

$h$ =

riblet height

$k$ =

height of hemispherical roughness

$s$ =

riblet spacing

B =

blockage factor

$kmax$ =

maximum turbulent kinetic energy

$uτ$ =

friction velocity

$uτ,av$ =

averaged value of friction velocity in the turbulent regime

$Cf$ =

time and span averaged skin-friction coefficient

$Cfxr$ =

skin-friction coefficient over the riblet surface

$Lz$ =

spanwise extent of the domain normalized by $S0$

$Lr$ =

spanwise length along the riblet surface in terms of $S0$

$S0$ =

suction surface length

$Ue$ =

time-averaged velocity at the boundary layer edge

$Uin$ =

inflow velocity

$UTE$ =

trailing edge velocity

$Upeak$ =

velocity at the suction peak

$Zw$ =

Zweifel lift coefficient

$d+$ =

diameter of streamwise vortices in viscous units

$h+$ =

riblet height in viscous units

$s+$ =

riblet spacing in viscous units

$y+$ =

wall-normal coordinate normalized using $ν$ and $uτ$

$⟨u′u′⟩$ =

streamwise Reynolds normal stress

APG =

adverse pressure gradient

DF =

diffusion factor

HL =

high-lift

LL =

low-lift

Re =

Reynolds number (based on $UTE$ and $S0$⁠)

$Rein$ =

Reynolds number (based on $Uin$ and $S0$⁠)

TE =

trailing edge

TBL =

turbulent boundary layer

TKE =

turbulent kinetic energy

UHL =

ultra high-lift

$U,u$ =

streamwise velocity (time averaged)

$δRe99$ =

boundary layer thickness over the flat surface at Re

$θ$ =

momentum thickness

$μ$ =

time-averaged dynamic viscosity

$ν$ =

time-averaged kinematic viscosity

$ρe$ =

time-averaged density at the boundary layer edge

$τw$ =

time-averaged wall shear stress