## Abstract

The detrimental effects of surface roughness on the growth of turbulent boundary layers in turbines, and compressors, are well known. However, robust prediction of these effects can be problematic, especially for surfaces which are not similar to sand grains. Several publications have proposed that additional parameters such as effective slope, skew, etc. can be used to augment a wall-normal measure such as $Sa$ to correlate surface roughness. Here, we introduce a new roughness parameter, the mean feature separation, which explicitly measures the average separation between each local minima and its closest local maxima. Scans of turbine blade surfaces showed that they can have mean separations which are up to six times that of the sand grain surface which has the same wall-normal length scale. The availability of high resolution, typically < 0.1 µm, area scanning tools, and 3D printing techniques have enabled the development of a “Scan-Scale-Print-Measure” methodology. A surface scan is scaled-up, 3D printed, applied to a flat-plate, and then the turbulent boundary layer is measured. The surface roughness-loss is the additional momentum thickness, above that of the datum smooth surface. The Scan-Scale-Print-Measure methodology has enabled parametric studies to be undertaken where the mean separation was increased while keeping either the $Sa$ value or the feature size constant. Both studies demonstrated a surface roughness-loss that was strongly dependent on the mean separation and different to a Schlichting-based sand grain correlation. It also showed that for large mean separations, the roughness-loss decreased to zero. A parametric study into pits-and-peaks surface features showed that, at the same wall-normal length scale, peaks generated a surface roughness-loss twice that of pits. An engine representative surface produced only half the roughness-loss that would be attributed to a sand grain surface. The measured surface roughness-loss could be estimated within 5% using the aforementioned parametric studies.

## 1 Introduction

Europe's 2050 flightpath plan by the Advisory Council for Aeronautics Research [1] has challenged the aviation industry to develop more efficient engines with reduced specific fuel consumption and emissions. It is known that high levels of surface roughness have a detrimental effect on aeroengine efficiency. All methods used by the Gas Turbine Industry for prescribing an aerodynamic penalty to a rough surface are either directly or indirectly related to Nikuradse’s [2] 1933 data on sand grain surfaces. However, new manufacturing methods produce aerodynamic components which can have surface topographies which are different to traditional sand grain finishes.

Figure 1(a) shows a microscope image of P100 sandpaper, a commonly used datum roughness. Figures 1(b) and 1(c) show contour plots for the P100 surface and for a scaled turbine blade, BladeA. Both surfaces have a wall-normal roughness measure of $Sa=36\mu m$. Therefore, traditional correlations for the effects of surface roughness would predict similar aerodynamic penalty for both. However, the P100 surface feature size (wall-parallel extent) and feature separation shown in Fig. 1(b) are both smaller than those of BladeA, Fig. 1(c). Furthermore, the P100 surface has a skew of $+1.1$ while BladeA has a skew of $\u22120.9$, see Table 1.

ISO K (_{s}µm) | Sa (µm) | Mean (Dl) (µm) | Skew | ||
---|---|---|---|---|---|

Abtec P60 | 269 | 62.8 | 235.8 | 0.70 | |

Abtec P80 | * | 201 | 43.7 | 190.3 | 1.11 |

Abtec P100 | * | 162 | 36.0 | 141.7 | 1.05 |

Abtec P120 | 125 | 26.6 | 128.4 | 1.04 | |

Abtec P180 | * | 82 | 19.0 | 59.2 | 1.00 |

Abtec P220 | 68 | 16.6 | 48.4 | 0.74 | |

Abtec P240 | * | 58.5 | 15.7 | 48.7 | 0.71 |

Abtec P320 | 42.6 | 11.0 | 45.3 | 0.75 | |

Tooling-plate | * | – | 1.5 | 0.0 | |

BladeA | – | 35.7 | 338.1 | −0.88 |

ISO K (_{s}µm) | Sa (µm) | Mean (Dl) (µm) | Skew | ||
---|---|---|---|---|---|

Abtec P60 | 269 | 62.8 | 235.8 | 0.70 | |

Abtec P80 | * | 201 | 43.7 | 190.3 | 1.11 |

Abtec P100 | * | 162 | 36.0 | 141.7 | 1.05 |

Abtec P120 | 125 | 26.6 | 128.4 | 1.04 | |

Abtec P180 | * | 82 | 19.0 | 59.2 | 1.00 |

Abtec P220 | 68 | 16.6 | 48.4 | 0.74 | |

Abtec P240 | * | 58.5 | 15.7 | 48.7 | 0.71 |

Abtec P320 | 42.6 | 11.0 | 45.3 | 0.75 | |

Tooling-plate | * | – | 1.5 | 0.0 | |

BladeA | – | 35.7 | 338.1 | −0.88 |

Note: Those surfaces which have been aerodynamically measured to validate the facility are indicated by an asterisk.

The average feature size of a surface topography can be determined from an auto-correlation coefficient. However, there is no established surface roughness parameter which explicitly measures the average separation between a feature which is a local minima and its closest feature that is a local maxima.

### 1.1 In This Publication.

The focus is on boundary layers which are turbulent (i.e., ignoring transition and separation) and surfaces which have isotropic roughness (same in all directions). The research presented uses high-resolution, sub-micrometer, surface scans and high-quality three-dimensional printing to produce accurate, large-scale, versions of known surface roughness. A fully validated flat-plate facility is then used to measure the development of a turbulent boundary layer along the surface. This is referred to as “Scan-Scale-Print-Measure.”

The strategy here is to use the Scan-Scale-Print-Measure methodology to undertake carefully specified parametric studies of the effects of surface topography on the development of a turbulent boundary layer. The goal being to examine if there is a relationship between the mean separation and the aerodynamic penalty of an arbitrary, engine representative, surface roughness.

## 2 Background

In 1933, Nikuradse [2] studied the effects of sand grains, of diameter $ks$, on the inside of circular pipes and produced a correlation for the associated pressure drop. This forms the basis for the prediction of entropy generation associated with surface roughness in many aerodynamic and fluid dynamic fields. In 1937, Schlichting [3] introduced the concept of an equivalent sand grain diameter, $ksequiv$, to enable the production of correlations for specific families of rough surfaces.

Since then, many researchers have suggested refined correlations, many of which are, often linear, relationships between a wall-normal measure of roughness ($Sa$, $Sq$, etc.) and the equivalent sand grain diameter. Bons [4] listed more than 20 such roughness correlations used within the Gas Turbine Industry. These are all, essentially, one-parameter correlations.

The existence of so many specific one-parameter correlations suggests that there may be other important aspects of roughness that have not been included. Goodhand et al. [5], Abdelaziz et al. [6], and Vasilopoulos et al. [7,8] are a selection of publications where additional parameters (skew, wall-parallel length scale, effective slope, etc.) have been included to generate correlations using two, or more, parameters.

Many of the enhanced correlations that have been proposed have a common limitation: they are based only on surfaces which were available or produced by a specified process, e.g., “barreling” for a fixed time. This often meant that several topographic parameters were changed simultaneously, and thus it is unclear which might be the most important one(s).

For a sand grain surface, Nikuradse [2] observed that there are voids between the individual grains which arise due to their imperfect packing and, possibly, the quality of coverage associated with the number of grains per unit area. The microscope image of Abtec P100 sandpaper shown in Fig. 1(a) is similar to a microphotograph included in Ref. [2]. The presence of these voids introduces the possibility that boundary layer development over sand grain surfaces may have a dependence on not just the grain diameter but also the quality of coverage, possibly via a mean separation.

Flow over a rough surface would be expected to respond not only to the changes in wall-normal height but also to the wall-parallel distance between a local minima and maxima. This has led several authors to suggest that, in addition to a wall-normal length scale, correlations for the effects of surface roughness ought to include some measure of the wall-parallel length scale. Many have indirectly included this by incorporating an effective slope of the roughness. However, effective slope is the average value of the modulus of the surface slope, Eq. (A3), so must be combined with another length scale, usually wall-normal, to obtain a wall-parallel length scale.

## 3 Quantification of a Rough Surface

Roughness can be interpreted as the discrepancy, in the wall-normal direction, between the actual and the ideal surface. There are many types of surface roughness: e.g., those associated with as-manufactured parts, in-service deterioration (including deposition, erosion, corrosion, etc.), and the introduction of new materials. In many cases, roughness appears as, usually random, features which are distributed across a surface, see Fig. 1. In this publication, the size (wall-parallel extent), height (wall-normal), and the separation of roughness features are statistical in nature.

To develop an objective assessment of a rough surface, it is necessary to choose parameters to characterize it. Based on the above discussion, a wall-normal length scale and a wall-parallel length scale will be adopted in this publication.

### 3.1 Wall-Normal Length Scale.

In the above, the first term is the integral of all the points above the mean, $\Delta zabove$, but averaged over the whole area *A*; similarly for the second term (the minus sign arises because $\Delta zbelow\u22640$). Thus, $Sa$ is the difference between the average value of the points above the mean and the average value of the points below the mean (each average is over the whole area).

The above form is useful in cases such as deposition or pitting, where the features are sparsely distributed (of a wall-normal length scale given by $habove$ or $hbelow$) but the $Sa$ value is close to zero because the corresponding fractional area becomes very small.

### 3.2 Wall-Parallel Length Scale.

Because it relates to the distance between a local minima and maxima, the mean feature separation will be adopted as the wall-parallel length scale. The wall-parallel distance, $Dl$, between a local minima and its closest local maxima is

Figure 1 shows that $Dl$ has a range of values so $mean(Dl)$ will be used as a wall-parallel length scale, full description later.

### 3.3 Wall-Normal Versus Wall-Parallel Chart.

It is useful to compare the measured $Sa$ and $mean(Dl)$ values for a selection of sand grain surfaces (sandpapers produced by Abtec). These are listed in Table 1 along with the ISO6344 sand grain diameters. The sand grain data are plotted (black circles) on the wall-normal, $Sa$, versus wall-parallel, $mean(Dl)$, chart shown in Fig. 2. Also included is the best-fit line through the sand grain data which passes through the origin.

The agreement between the sand grain data and the best-fit line, $Sa/mean(Dl)=0.25$, confirms that correlating against sand grain surfaces is a one-parameter family. The slight variation (data versus best-fit) might be associated with different grain packing densities. This might also explain why the measured skew values cover a range (0.7–1.11).

Also plotted in Fig. 2 are the $Sa$ and $mean(Dl)$ for approximately 20 turbine blade surfaces. All the turbine blade data are plotted at the flat-plate facility scale (determined by scaling the stagnation point to suction trailing edge distance to match the reference length, *L*, of the flat-plate). The turbine blade data are split into those which are primarily isotropic (filled diamonds) and those which are non-isotropic (open diamonds).

Clearly, many aeroengine turbine blade surfaces are not *sand grain like* because their surface roughness wall-normal and wall-parallel length scales do not match the *sand grain line*.

### 3.4 Turbine BladeA.

One of the isotropic turbine blade surfaces in Fig. 2 (BladeA, green-filled diamond) has been selected because it has the same $Sa$ value as P100 but a larger $mean(Dl)$ (i.e., not sand grain like). A comparison of the roughness parameters for BladeA and P100 are given in Table 2. BladeA has mean separation and feature size that are both approximately a factor of 2.5 greater than P100. This is consistent with the observed differences between Figs. 1(b) and 1(c).

Wall-normal, $Sa$ (µm) | Wall-parallel, $mean(Dl)$ (µm) | Feature size, $d70$ (µm) | |
---|---|---|---|

P100 | 36.0 | 141.7 | 69.5 |

BladeA | 35.7 | 338.1 | 173.3 |

Ratio | 1.0 | 2.4 | 2.5 |

Wall-normal, $Sa$ (µm) | Wall-parallel, $mean(Dl)$ (µm) | Feature size, $d70$ (µm) | |
---|---|---|---|

P100 | 36.0 | 141.7 | 69.5 |

BladeA | 35.7 | 338.1 | 173.3 |

Ratio | 1.0 | 2.4 | 2.5 |

Because BladeA has roughness parameters, $Sa$ and $mean(Dl)$, which are not on the sand grain line, its aerodynamic penalty due to roughness will be studied later in this publication.

## 4 Parametric Investigation of Roughness

Figure 3 is a generic version of the wall-normal versus wall-parallel length scale chart. It has already been observed that all sand grain surfaces lie on a radial line from the origin. Changing the sand grain diameter is equivalent to moving along this radial line. This is effectively a simple geometric scale of the surface topography (with a corresponding adjustment to the boundary layer velocity to maintain the required Reynolds number). At all points along the sand grain line, the ratio of the wall-normal length scale to wall-parallel length scale, $Sa/mean(Dl)$, is the same. Consequently, at a fixed roughness Reynolds number,^{2} the fluid dynamic mechanisms that generate the aerodynamic penalty for a sand grain surface are the same.

The turbine surfaces in Fig. 2 show that, in general, surface roughness can be anywhere on the wall-normal versus wall-parallel chart. Therefore, any surface which is not on the sand grain line can have different fluid dynamic mechanisms that generate the aerodynamic penalty (even at the same roughness Reynolds number). This is probably why many researchers have had to develop enhanced correlations to account for surfaces which are not on the *sand grain line*.

The P100 and BladeA comparison, Table 2, suggests that for a given value of $Sa$, investigating different values of both the mean separation, $mean(Dl)$, and feature size, $d70$, are necessary to understand which surface roughness parameters might be appropriate for producing robust correlations.

### 4.1 Stretch Study.

A simple way to vary $Sa/mean(Dl)$ is to apply a wall-parallel stretch factor to the *x* and *y* values but to keep the *z* values unchanged. All the wall-normal metrics, $Sa$, $Sq$, minimum to maximum *z*-height, etc., will be unchanged but the mean separation, $mean(Dl)$, will increase. The feature size, $d70$, will also increase by the same factor. Thus, the stretch study will also keep the value of $d70/mean(Dl)$ constant.

The stretch study is indicated by the horizontal stretch line shown in Fig. 3. As the stretch factor is increased, the “steep mountains” discussed by Goodhand et al. [5] will become “rolling hills” but the value of $Sa$ will be unchanged. If the wall-parallel stretch is large enough, the rolling hills would be expected to become hydraulically smooth.

### 4.2 Spacing Study.

An alternative way to vary the mean separation is to keep the feature sizes and heights fixed and increase the spacing between them (thus increasing $mean(Dl)$ but keeping $d70$ constant). This is indicated by the space line in Fig. 3. The $Sa$ value will decrease to zero as the spacing is increased, see Eq. (5), but the features do not. The spacing study will vary the value of $d70/mean(Dl)$. The spacing study might correspond to varying amounts of deposition and pitting.

### 4.3 Pits-Peaks Study.

Although P100 and BladeA have similar wall-normal, $Sa$, values, they have, almost, equal and opposite skew values. Therefore, the negative version (+*z* to −*z*) of the P080 surface will be produced and tested. Furthermore, the P080 surface will also be decomposed into two surfaces: the peaks (portions above the mean) and the pits (portions below the mean). This will add information to the discussion about local maxima being more damaging than local minima.

## 5 Analysis of High-Resolution Surface Scans

A key enabler of this investigation is the access to high-resolution scans of engine component surfaces. Typically, the resolution is ∼1 *µ*m in the wall-parallel direction and <0.1 *µ*m in the wall-normal direction. The scanned area is a rectangle of several millimeters squared.

### 5.1 Preprocessing Surface Scans.

To remove long length scale curvature effects, all the surface scans are first best-fitted, using orthogonal Legendre polynomials of degree up to seven, and the resulting function subtracted from the raw data. The resulting “leveled” surface is then analyzed for uniformity by comparing its statistical properties ($Sa$, $Sq$, probability density function for the height, etc.) determined for the whole region to those values for the four “quarter-surfaces,” nine “ninth-surfaces,” and 16 “sixteenth-surfaces.” An example of this is shown in Fig. 4 where the probability density function for the post-leveled surface is evaluated for five different regions, denoted by red and green regions. The probability density functions are similar indicating that statistics evaluated from the whole (red) or any the sub-regions (green) are representative.

### 5.2 Feature Size.

For surfaces where the features are randomly distributed, the auto-correlation coefficient will decay from the value one, at zero offset, toward zero at large offsets. This is also true for randomly distributed features of identical size. The rate of decay with increasing offset is related to the feature size (not the feature separation—unless the features are periodically aligned).

Based on several studies, the offset at which the auto-correlation coefficient has decreased to 0.7 has empirically been chosen as specifying a characteristic feature size, denoted $d70$. Because the auto-correlation coefficient is non-dimensionalised by the variance, the feature size, $d70$, measures only the wall-parallel *x*, *y* extent.

### 5.3 Feature Separation.

The high-resolution scans allowed the identification of all the local maxima, minima, and saddle points of the surface. The nature and position of the stationary points were determined by best-fitting multiple rectangular regions using a second degree polynomial in both *x* and *y*. The size of the rectangular region was chosen to correspond to the $d70$ feature size, so the stationary points are for the characteristic features.

Once all the minima, maxima, and saddle points were identified, a Balltree [9] sorting algorithm was used to find the nearest local maxima to each local minima. The statistics for the wall-normal, $Dz$, and the wall-parallel, $Dl=Dx2+Dy2$, distances between the minima-to-maxima pairs were recorded.

### 5.4 Surface Triangulation and Scale-Up.

Once the scan had been pre-processed and the region to be printed is identified, it needed to be triangulated—the process where triangles were used to represent the surface. Three methods were investigated:

*Simple*: A Cartesian mesh was specified, the average surface height in each cell was determined, and then the two triangles were formed by dividing each cell along one diagonal.

*Curvature*: The local curvature was estimated, and a point cloud was generated whose density was related to the curvature. Delaunay triangulation was used to produce a triangulated surface.

*Optimized*: An initial regular triangular mesh was distributed across the rectangular region with a given average triangle size. The (*x*, *y*, *z*) position of each vertex was then optimized to minimize the rms error between each triangle and the $z(x,y)$ surface that it spanned. The four corner vertices of the region were only moved in the *z* direction and vertices along the edges of the rectangular region could only be moved in the (*x*, *z*) or (*y*, *z*) directions.

Typically, a $9160\mu m\xd79160\mu m$ region of the scanned surface would be triangulated and then chequerboarded to form a 18.32 mm square region. By chequerboarding, there were no discontinuities where the scanned region was joined together and the larger square was periodic in both the *x* and *y* directions. This was then duplicated by 18 times in the *x*-direction and 6 times in the *y*-direction to produce a $330mm\xd7110mm$ surface which then had a base and four sides added to produce a tile with a nominal thickness of $1mm$ to produce an unformatted stl file.

### 5.5 Three-Dimensional Printing.

The stl files were printed using an Objet 350 polyjet printer with a maximum volume of 340 mm × 340 mm × 200 mm. The machine has a *z*-layer height of $16\mu m$ (confirmed by Alicona scanning) and *xy* step size of $42\mu m$ (600 dpi). After several trials, the optimization approach was found to be the best but the greatest limitation was that the 3D printer had a limit of $590\xd7106byte$ file size. This restricted the maximum number of triangles that could be used.

### 5.6 Statistical Accuracy.

Comparisons will be made between the original and the 3D-printed version of the surface topography such as the feature size, feature separation, and wall-normal length scale. It must be remembered that any scale-up factor will reduce the number of features per unit area by $(1/scale\u2212up)2$. So, for a fixed printed area, the number of features will be reduced and, by the standard error of the mean, the standard deviation will increase to $\sigma scale\u2212up=scale\u2212up\xd7\sigma original$.

## 6 Experimental Methodology

This experiment has been designed to determine the impact of current aerodynamic component surface topographies (roughness) on the development of a turbulent boundary layer and thereby on aeroengine efficiency.

Here, the dominant boundary layer term is $2\theta te/w$ where $\theta te$ is the momentum thickness at the trailing edge and *w* is the staggered spacing of the turbine blades.

*U*, and the local skin friction coefficient, $cf$, which depends on the local boundary layer properties and the local surface roughness.

From above, either the local skin friction coefficient, $cf$, or the downstream momentum thickness, *θ*, could be used to investigate the effects of surface roughness. The momentum thickness was chosen as it is directly related to blade row loss coefficient and requires only one traverse to determine it (whereas local skin friction coefficient requires two traverses).

### 6.1 Experimental Facility.

The flat-plate facility is a parallel duct in which there is a large tooling-plate at mid-height spanning the entire width and extending a total of 1.6 m upstream of the exit, see Fig. 5. The basic parameters are listed in Table 3. The tooling-plate has a removable elliptic leading edge, Fig. 6, and identically shaped upper and lower liners can be fitted to impose a specified pressure gradient along the plate [11].

Plate streamwise length | 1.6 m |

Plate transverse width | 0.6 m |

Leading edge ellipse (6:1 ratio), semi-major | 45 mm |

Trip: 30 < x < 50 mm from leading edge | P60 |

Reference length (including leading edge), L | 1.3 m |

Upstream traverse position | 0.2L |

Downstream traverse position | 1.0L |

Exit velocity (zero pressure gradient), U | <40 m/s |

Reynolds (based on U and L) | <3.5 × 10^{6} |

Plate streamwise length | 1.6 m |

Plate transverse width | 0.6 m |

Leading edge ellipse (6:1 ratio), semi-major | 45 mm |

Trip: 30 < x < 50 mm from leading edge | P60 |

Reference length (including leading edge), L | 1.3 m |

Upstream traverse position | 0.2L |

Downstream traverse position | 1.0L |

Exit velocity (zero pressure gradient), U | <40 m/s |

Reynolds (based on U and L) | <3.5 × 10^{6} |

### 6.2 Instrumentation.

Four upstream Pitot probes, 10 static pressure tappings along the centerline of the upper side of the tooling-plate, and a further 10 static pressure tappings on the back wall, 10 mm above the plate (for checking pressure distribution when a test surface has covered the pressure tappings on the plate).

### 6.3 Reference Reynold's Number.

*x*=

*L*, traverse plane location are used to define the reference velocity:

### 6.4 Control of Operating Point.

In addition to adjusting the wind-tunnel operating point to achieve the required reference Reynolds number, it was also essential to control the leading edge incidence to ensure repeatability of the measured boundary layer growth. This is because the pressure distribution around the leading edge is very sensitive to incidence and this affects the early development of the boundary layer and thereby the overall growth of the momentum thickness. Zero incidence is determined by examining four static pressure tappings on the elliptic leading edge, two on the top and two on the bottom at ±200 mm either side of the centerline. Zero incidence is maintained by adjusting the vertical position of the gauze at the exit of the lower flow channel, see Fig. 5. The gauze at the exit of the upper flow channel was always positioned to cover the entire upper channel, thus minimizing any flow redistribution which might affect the pressure distribution toward the trailing edge of the plate.

### 6.5 Measuring the Pressure Distribution.

The tooling-plate has static pressure tappings along the upper surface which are used, in conjunction with those around the elliptic leading edge, to determine the pressure distribution. When a test surface has been installed on the tooling-plate, the static pressure tappings in the rear wall, just above the surface, are used and compared against the back wall values measured previously.

For all the investigations discussed in this publication, a zero pressure gradient distribution was used and it is shown in Fig. 7. Because of the finite thickness of the flat-plate, there is an overspeed around the elliptic leading edge. The static pressure tappings, used to determine the zero incidence case, are located in the vicinity of the overspeed. The measured surface pressures agree well with a MISES [13] calculation, for a hydraulically smooth surface.

Repeatability was verified by re-measuring a test geometry two years after the initial tests—the results were identical.

### 6.6 Boundary Layer Profile.

This was measured by traversing a calibrated, using Kings' law with temperature correction (to account for changes in flow temperature), hotwire toward the flat-plate. At each location from the wall, the traverse software examined the time average velocity, compared it with the freestream reference velocity and then determine the size of the next step toward the wall (suggested by Dr. B. J. Crowley, private communication). By using this automated step-sizing approach, the traverse could be taken down into the laminar sublayer without time consuming geometric positioning (which was found to be problematic when measuring a rough surface).

### 6.7 Reynolds Limit for Hydraulically Smooth.

The key topic of this investigation is how the boundary layer development changes between a hydraulically smooth surface and when a rough surface is fitted. Therefore, it is essential to confirm that both the tooling-plate and the 3D-printed tiles without any roughness features (i.e., 3D-smooth) remain hydraulically smooth at all Reynolds numbers of the flat-plate facility.

*Tooling-plate*: This has a precision machined finish with a measured surface roughness of $Sa\u223c1.5\mu m$. Schlichting et al.'s [14] criteria for sand grain roughness to not affect the growth of a turbulent boundary layer (hydraulically smooth) is

For the operating range, the tooling-plate is hydraulically smooth.

*Three-dimensional printed smooth tile*: The printed 3D-smooth tile has a measured surface roughness of $Sa\u223c1.8\mu m$, with a standard deviation of $\u223c2.2\mu m$ and a surface height distribution which is approximately Gaussian. Thus, the majority of the surface would lie within $\xb16.6\mu m$. This is consistent with the Object 350 printer which has a wall-normal step size of $16\mu m$ (verified via GOM scanner), so an upper limit of $Sa\u223c8\mu m$. Although this upper limit is approximately five times larger than the tooling-plate, the smooth tile surface would be expected to be hydraulically smooth up to $ReL<5\xd7106$.

## 7 Validation of Experimental Measurements

Three independent exercises were undertaken to validate the experimental measurements. First, for both the tooling-plate and the 3D-printed smooth tiles, the measured momentum thickness at the traverse plane was compared with MISES [13] calculations. Second, for the tooling-plate, the measured boundary layer profiles were compared with direct numerical simulation (DNS) calculations. Third, a range of sandpapers were measured for a range of Reynolds numbers.

### 7.1 Validation of Hydraulically Smooth Cases.

The momentum thickness measured at the downstream traverse plane for the hydraulically smooth cases is compared in Fig. 8. The cases shown are the tooling-plate surface; the surface, including the joints, when the 3D-printed “smooth” tiles are fitted and the calculated MISES boundary layer growth. The good agreement between all three across the entire range of operating Reynolds numbers confirms that both the tooling-plate and the 3D-printed smooth tiles can be considered hydraulically smooth.

### 7.2 Comparison With DNS.

To verify that the measured boundary layer traverses (time mean velocity and turbulence level) for the hydraulically smooth tooling-plate are reliable, they are compared in Fig. 9 with DNS calculated profiles from Ref. [15]. There is excellent agreement for both the time mean profile and the turbulence level. The turbulence level within the flat-plate facility is not zero, and that is why there is a slight difference in Fig. 9, right hand, outside the boundary layer.

### 7.3 Validation Using Sand Grain Surfaces.

Demonstrating repeatable and reliable measurements of the turbulent boundary layer development along a rough surface is crucial to the current investigation. Here we will examine four different grades of Abtec sandpaper (marked by asterisk in Table 1).

The measured boundary layer momentum thickness at the downstream traverse plane for four different sand grain surfaces is shown in Fig. 10. The comparisons are good for the surfaces with larger sand grains. For the smaller sand grain sizes, the trend is continuous but the measurements are slightly less than the Schlichting correlation. This is consistent with the observation concerning the hydraulically smooth case being affected by the overspeed around the elliptic leading edge.

## 8 Scan-Scale-Print-Measure

There are two aspects to the assessment of the Scan-Scale-Print-Measure methodology: first, the geometric accuracy and second, the boundary layer development.

### 8.1 Accuracy of the Printed Geometry.

The $Sa$ values of the 3D-printed surfaces are compared with the original scan, see Table 4. Using a printer scale-up factor of three, with P100, achieved a wall-normal $Sa$ ratio of 0.88 when the optimized triangulation method used triangles of average size $80\mu m$. However, using a printer scale-up factor of three, allowed more triangles, average size of $30\mu m$, and the feature size was increased by printing P080. The resulting $Sa$ ratio was 1.02 which is as required.

Printed surface | $Sa/Saoriginal$ | xcc | RLF |
---|---|---|---|

3D-1P100-simple | 0.60 | 76% | 40% |

3D-1P100-curvature | 0.71 | 76% | |

3D-1P100-opt ($80\mu m$) | 0.74 | 83% | 40% |

3D-2P100-opt ($80\mu m$)^{a} | 0.75 | 75% | 80% |

3D-3P100-opt ($80\mu m$) | 0.88 | 80% | 90% |

3D-3P080-opt ($30\mu m$) | 1.02 | 96% | 100% |

Printed surface | $Sa/Saoriginal$ | xcc | RLF |
---|---|---|---|

3D-1P100-simple | 0.60 | 76% | 40% |

3D-1P100-curvature | 0.71 | 76% | |

3D-1P100-opt ($80\mu m$) | 0.74 | 83% | 40% |

3D-2P100-opt ($80\mu m$)^{a} | 0.75 | 75% | 80% |

3D-3P100-opt ($80\mu m$) | 0.88 | 80% | 90% |

3D-3P080-opt ($30\mu m$) | 1.02 | 96% | 100% |

Scan was of low quality.

The exact profile of the printed surfaces was examined using the cross-correlation coefficient between a scan of the printed surface and the original scan (the two scans were aligned by identifying the position which gave the maximum correlation coefficient, xcc). For 3D-3P080-opt ($30\mu m$), a 96% cross-correlation coefficient was achieved, see Fig. 11, which confirms that surface roughness can be accurately printed.

### 8.2 Boundary Layer Development.

The momentum thickness measured at the downstream traverse plane for a range of Reynolds numbers matches very well with the Schlichting correlation using three times the ISO sand grain diameter for P080 (see Fig. 12). This confirms the Scan-Scale-Print-Measure methodology.

### 8.3 Parametric Studies.

Many of the parametric studies were undertaken using a printer scale-up factor of two, which only yielded $\u223c80%$ of the roughness-loss. However, the discrepancy was due to the geometrical printing limitations not the aerodynamic measurements. Therefore, in the following parametric studies, the data analysis will be done focusing on the measured geometry of 3D-printed surface, rather than the design intent.

## 9 Wall-Parallel Stretch Study

To investigate how important the mean separation is in determining the boundary layer growth, the 3D-2P100 surface will be stretched in the wall-parallel direction by stretch factors between 1 and 18. The wall-normal distances are unchanged. Because this is a comparative study, it is sufficiently accurate to print the surfaces at twice size and focus on the surface topography of the printed geometries.

As discussed earlier, in Fig. 3, all these stretched surfaces should have the same wall-normal length scale values, i.e., all should have exactly the same $Sa$. It is confirmed, by the data in Table 5, that the $Sa$ values are within $2\mu m$ which is small compared to the $16/2=8\mu m$ effective layer height of the 3D printer (noting that a printer scale-up factor of 2 had been used).

Surface | $Sa(\mu m)$ | $d70(\mu m)$ | $mean(Dl)(\mu m)$ |
---|---|---|---|

3D-2P100 | 26 | 72 | 142 |

3D-2P100-3xy | 28 | 151 | 500 |

Ratio | 2.1 | 3.5 |

Surface | $Sa(\mu m)$ | $d70(\mu m)$ | $mean(Dl)(\mu m)$ |
---|---|---|---|

3D-2P100 | 26 | 72 | 142 |

3D-2P100-3xy | 28 | 151 | 500 |

Ratio | 2.1 | 3.5 |

Note: Printer scale-up factor of two has been removed.

For the wall-parallel stretch by a factor of three, the ratio of the values in Table 5 for the feature size and the feature separation for the two surfaces are 2.1 and 3.5, respectively. Ideally, these would be expected to equal three, the wall-parallel stretch factor. The discrepancy is thought to be associated with the increased standard deviation due to the printer scale-up factor of three and the small area scanned. Evaluation of the cross-correlation coefficient would be a more robust assessment.

The measured non-dimensional boundary layer momentum thickness at the downstream traverse plane is shown as a function of Reynolds number and wall-parallel stretch factor in Fig. 13. As the stretch factor is increased from 1 to 18, the measured momentum thickness at the downstream traverse plane decreases toward the hydraulically smooth value. For a stretch factor of 2, there is virtually no change in the roughness-loss (difference between 3D-2P100 and 3D-smooth) while for a stretch factor of 18, the roughness-loss is decreased to about 10% of the original value.

For all the stretch study cases shown in Fig. 13, the non-dimensional momentum thickness curves are monotonically distributed between the original, 3D-2P100, and the hydraulically smooth case, 3D-Smooth. This suggests that plotting the roughness-loss fraction (RLF) (Eq. (18)) might be informative. Furthermore, all these stretched surfaces have the same wall-normal $Sa$, but with values of the wall-parallel $mean(Dl)$ proportional to the stretch factor. Therefore, the ratio of $Sa/mean(Dl)$ is inversely proportional to the stretch factor and corresponds to the amplitude-to-wavelength ratio of the “average” surface topography. The roughness-loss fraction versus $Sa/mean(Dl)$ is shown in Fig. 14.

### 9.1 Stretch Study Roughness-Loss Fraction.

This is shown in Fig. 14 and is plotted against $Sa/mean(Dl)$ because all the stretch study surfaces have the same $Sa$ value. The Reynolds dependence is much weaker for the range of values Reynolds numbers measured (0.5–3 million). Furthermore, for $Sa/mean(Dl)<0.1$ (stretch factors greater than two), the roughness-loss fraction is almost linearly related to the reciprocal of the stretch factor. This suggests that the projected area argument associated with form (pressure) drag is appropriate for low values of $Sa/mean(Dl)$, which corresponds to long-wavelength roughness.

This wall-parallel stretch study has shown that an entire family of surfaces has the same wall-normal, $Sa$, value but a diminishing roughness-loss. This confirms the hypothesis that a wall-normal length scale alone is not a robust measure of roughness. Furthermore, an interesting observation is that all these surfaces have the same Sa value but become hydraulically smooth at large wall-parallel stretch factors. This suggests that the criterion for admissible roughness cannot be only a function of roughness height to viscous length scale.

## 10 Feature Spacing Study

The above, wall-parallel stretch study has demonstrated the importance of the feature separation on roughness-loss. However, in the wall-parallel stretch study, both the feature size and the feature separation are changed (but the wall-normal is unchanged). There is another way of investigating the importance of the mean separation: keep the feature size fixed but increase the spacing between the features, see Fig. 3.

For the spacing study, it is necessary to decompose a surface into features which could be re-positioned according to the required spacing. This was achieved using the positions, heights, and depths of the stationary points from the scan of Abtec P100 sandpaper. A *synthetic* surface was then generated by locating a Gaussian peak or pit of the appropriate height at each of the maxima and minima positions. At the original, datum, spacing, the $Sa$ value was slightly lower than the original P100, so a simple wall-normal scale factor was applied to ensure that the datum synthetic surface, S100-0100 (original, 100%, spacing), has the same wall-normal length scale as the original P100.

The wall-normal length scale, $Sa$, and feature size, $d70$, for the two surfaces are listed in Table 6. Although the definition for both of these surfaces has the same $Sa$ values, the 3D-printed versions do not. The synthetic has approximately 35% higher value. However, the key aspect of this spacing study is the effect of increasing the spacing between the features without changing the feature size.

Surface | $Sa(\mu m)$ | $d70(\mu m)$ |
---|---|---|

3D-2P100 | 26 | 72 |

3D-2S100-0100 | 35 | 76 |

Surface | $Sa(\mu m)$ | $d70(\mu m)$ |
---|---|---|

3D-2P100 | 26 | 72 |

3D-2S100-0100 | 35 | 76 |

Note: Printer scale-up factor of two has been removed.

The measured boundary layer momentum thickness for the synthetic surface with different feature spacings is shown in Fig. 15. Some of the trends with Reynolds number are not as expected, so, first, the geometric aspects of the 3D-printed surfaces will be compared (Table 7).

Spacing | $Sa(\mu m)$ | $d70(\mu m)$ | ||
---|---|---|---|---|

Expected | 3D print | Expected | 3D print | |

100% | 37 | 35 | 72.7 | 76 |

150% | 20 | 21 | 72.9 | 85 |

200% | 12 | 17 | 73.0 | 80 |

300% | 6 | 73.0 | ||

600% | 1.5 | 73.0 | ||

900% | 0.7 | 73.0 |

Spacing | $Sa(\mu m)$ | $d70(\mu m)$ | ||
---|---|---|---|---|

Expected | 3D print | Expected | 3D print | |

100% | 37 | 35 | 72.7 | 76 |

150% | 20 | 21 | 72.9 | 85 |

200% | 12 | 17 | 73.0 | 80 |

300% | 6 | 73.0 | ||

600% | 1.5 | 73.0 | ||

900% | 0.7 | 73.0 |

Note: Printer scale-up factor of two has been removed.

The wall-normal length scale, $Sa$, agrees well between the expected (as determined from the surface definition file) and the measured 3D-printed values. The discrepancy in $Sa$ at the 200% scale factor, $12\mu m$ and $17\mu m$, is probably due to the effective layer height of the 3D printer ($8\mu m$, scale-up of two).

The expected feature size, $d70$, determined from the surface definition files is virtually identical. This is because, as discussed earlier, the auto-correlation coefficient “measures” the feature size and that is unchanged in this study. The 3D-printed values for the feature size range from $76to85\mu m$ which are reasonable given that the 3D printer has an effective *x* and *y* step size of $21\mu m$ (with printer scale-up factor of two).

The effect of different spacings of features of a fixed size for various Reynolds numbers is shown in Fig. 15. At feature spacings of 150%, there appears to be a strong Reynolds numbers effect. Increasing the feature spacing, to 200%, has a much larger effect than the stretch factor of two (shown in Fig. 13). At 900% feature spacing, the behavior is similar to the hydraulically smooth case.

### 10.1 Spacing Study Roughness-Loss Fraction.

Because all the spacing study surfaces have the same feature height but different mean separations, it is convenient to plot the roughness-loss fraction against $habove/mean(Dl)$, see Fig. 16. The roughness-loss fraction is 0.5–0.6 for a spacing-factor of two which is a more pronounced decrease than that observed for the wall-parallel stretch study (0.7–0.9). There is also a noticeable Reynolds dependence for spacing factors between 1 and 3.

For $habove/mean(Dl)<0.1$ (spacing factors 3 and above), the roughness-loss fraction becomes quadratic with the reciprocal of spacing-factor. A best-fit quadratic is shown in Fig. 16. This is consistent with the aerodynamic penalty due to roughness being proportional to the number of features when they are relatively sparse. In the spacing study, the features are of fixed size and height, so it is also proportional to the windward projected area.

The feature spacing study at $habove/mean(Dl)\u22480.05$ (i.e., 600% spacing) has a roughness-loss fraction of approximately 0.15, i.e., not hydraulically smooth. This is unexpected because the surface has $Sa\u223c2\xd71.5=3\mu m$ (see Table 7) which is smaller than the estimated hydraulically smooth limit of $Sa\u223c8\mu m$ for the 3D-printed smooth tile.

Also shown in Fig. 16 is the expected roughness-loss fraction, for the range of Reynolds numbers measured, assuming the mapping of $ks/Sa=4.5$ with the Schlichting correlation for sand grains. Not only are the expected values significantly lower than those measured but the Reynolds number trend is in the wrong direction: the measured roughness-loss fraction decreases with increasing Reynolds number whereas the expected value increases. This may indicate that the fluid dynamic processes involved do not scale with $Sa$ alone.

## 11 Pits-Peaks Study

The measured momentum thicknesses as a function of Reynolds number for both the 3D-3P080-Peaks and the 3D-3P080-Pits are shown in Fig. 17 (blue line, triangles upward and downward, respectively). The peaks-alone surface has a roughness-loss which is approximately four times the pits-alone value.

However, the peaks and pits have different amplitudes. For P080, the above–below decomposition, Eqs. (4) and (5), are given in Table 8. For the peaks, $habove=51.1\mu m$ whereas for the pits $hbelow=\u221238.1\mu m$ which is approximately 75% of $habove$. If the peaks were only 75% of the height, the Schlichting correlation would estimate ∼60% of the roughness-loss. Therefore, the roughness-loss for peaks-alone might only be $4\xd70.6=2.4$ times that for pits-alone of similar wall-normal length scale.

$fbelow$ | $fabove$ | h_{below} (µm) | h_{below} (µm) | Sa (µm) | |
---|---|---|---|---|---|

P080 | 0.573 | 0.427 | −38.1 | 51.1 | 43.7 |

P080-neg | 0.427 | 0.573 | −51.1 | 38.1 | 43.7 |

BladeA | 0.434 | 0.566 | −41.2 | 31.6 | 35.8 |

P100-neg | 0.426 | 0.574 | −42.2 | 31.3 | 36.0 |

P100 | 0.574 | 0.426 | −31.3 | 42.2 | 36.0 |

$fbelow$ | $fabove$ | h_{below} (µm) | h_{below} (µm) | Sa (µm) | |
---|---|---|---|---|---|

P080 | 0.573 | 0.427 | −38.1 | 51.1 | 43.7 |

P080-neg | 0.427 | 0.573 | −51.1 | 38.1 | 43.7 |

BladeA | 0.434 | 0.566 | −41.2 | 31.6 | 35.8 |

P100-neg | 0.426 | 0.574 | −42.2 | 31.3 | 36.0 |

P100 | 0.574 | 0.426 | −31.3 | 42.2 | 36.0 |

### 11.1 Negative Surface.

Although P100 and BladeA have the same $Sa$, it was observed that they have almost equal and opposite skew values, see Table 1. Looking at the above–below decomposition for P100 and BladeA (Table 8), the values for the fractional areas $fabove$ and $fbelow$ interchange as do those for $habove$ and $hbelow$. Thus, P100 and BladeA could be described as negatives of each other (points above the mean are transformed to points below the mean, +Δ*z* to −Δ*z*).

Therefore, the “negative” surface 3D-3P080-neg was generated to compare with the 3D-3P080 measurements. The results (Fig. 17, green line) show that the negative form of the surface produces, at most, only 63% of the roughness-loss of the original surface. This reduction would apply to all sand grain surfaces at the appropriate roughness Reynolds number.

BladeA has a probability density function that is more similar to P080-neg than P080, see Fig. 18. Furthermore, the ratio of the $Sa$ values for P080-neg and BladeA is $43.7/35.8=1.2$, so the probability density function for BladeA would be expected to be 20% higher and 20% narrower than those for P080-neg. This is consistent with the figure. Therefore, BladeA might be better modeled by the negative of a sand grain surface.

## 12 Measured Engine Surface (BladeA)

To remove the issues associated with the accuracy of the 3D printer, the comparisons are done using the measured values of the 3D-printed surfaces. 3D-2BladeA has $Sa=52\mu m$ and 3D-2P100 has $Sa=2\xd726\mu m$, see Table 5. Therefore, both printed surfaces have wall-normal length scale of $Sa=52\mu m$ when using a printer scale-up factor of two. The original scanned surfaces both have $Sa=36\mu m$.

The measured momentum thicknesses for the scaled turbine blade, 3D-2BladeA, along those for 3D-2P100 and 3D-smooth are shown as a function of Reynolds number in Fig. 19. The measured roughness-loss for 3D-2BladeA is about half of that measured for 3D-2P100.

The measurements shown in Fig. 19 demonstrate that engine representative surfaces can have a significantly different aerodynamic penalty due to roughness than that which would be expected using just a wall-normal measure of roughness.

### 12.1 Schlichting Model for BladeA.

Using the measured wall-normal length scale, $Sa=52\mu m$, for 3D-2BladeA with $ks=4.5Sa$, the Schlichting model, shown as blue line in Fig. 19, agrees well with the measured momentum thickness for the sand grain surface 3D-2P100. In essence, this validates the wall-normal mapping $ks/Sa=4.5$.

The first and last bars in bar chart shown in Fig. 20 represent the Schlichting model and the measured non-dimensional momentum thickness for $ReL=3\xd7106$. Each bar is split into the hydraulically smooth and roughness-loss portions.

### 12.2 Wall-Parallel Stretch for BladeA.

### 12.3 Distribution Model for BladeA.

From Table 8, the surface height distribution of BladeA is similar to a negative version of P100, in terms of the above–below decomposition. So, based on the negative surface study, Fig. 17, the expected effect would be that the roughness-loss fraction would be 63%. This is the third bar in Fig. 20.

### 12.4 Stretch and Distribution Combined.

Neither of these two reduced roughness-losses on their own match the measured value. However, both are known to be present, so combining them, by multiplication, gives 53% which is represented by the fourth bar in Fig. 20. This agrees well with the observed 48%. (The 5% roughness-loss discrepancy discussed above for $ReL=3\xd7106$ becomes a −5% discrepancy at $ReL=1\xd7106$.)

## 13 Discussion

In this publication, the roughness-loss is the increase in the non-dimensional momentum thickness above the hydraulically smooth value (same Reynolds number) for turbulent flow over a rough surface. It is a direct measure of the aerodynamic penalty due to roughness. A similar decomposition, where a roughness correction is applied to the hydraulically smooth friction velocity, is used in computational fluid dynamics.

Examination of a selection of turbine blades has shown that, when characterized in terms of their $Sa$ (wall-normal) and mean feature separation (wall-parallel) length scales, engine representative surfaces can be significantly different to sand grains. Taking the wall-normal length scale as the primary parameter, engine surfaces can have a wall-parallel length scale between one half and six times that of a sand grain topography.

### 13.1 Effects of the Wall-Parallel Length Scale.

These have been investigated using two parametric studies, one where the surface is stretched in the wall-parallel direction and one where the feature size and height are held constant and the spacing between them increased. In both cases, the roughness-loss fraction approximately halved when the mean separation had been increased by six. However, the wall-parallel stretch study had a more gradual effect than the spacing study and also exhibited little Reynolds number dependence.

It should be noted that for the wall-parallel stretch study, the roughness-loss was smaller than the unstretched case. For the spacing study, the measured roughness-loss fraction was greater than what would be expected based on the $Sa$ value.

### 13.2 Long Wall-Parallel Wavelength Roughness.

For large stretch or spacing factors, the roughness-loss became proportional to the windward face area and decreased to zero, effectively becoming hydraulically smooth.

- In the stretch study, when the $Sa$ values are small compared to the mean separation, $Sa/mean(Dl)<0.1$, the RLF is almost linearly related to the windward projected area:(21)$RLFstretch\u224810.3\xd7windward$
- In the spacing study, using the synthetic surface based on Abtec P100, the roughness-loss fraction again became proportional to the windward projected area when $habove/mean(Dl)<0.1$:(22)$RLFspacing\u224818.8\xd7windward$

Although in both studies the roughness-loss fraction became proportional to the windward area, the constants were significantly different. This suggests that the fluid dynamic mechanisms involved for long-wavelength stretch studies (expected to be primarily pressure, form-drag) are different to those for the spacing studies at large spacings. This suggests that the discrete features of the spacing study might involve similar fluid dynamics to those used by Braslow [16] in the spacing of discrete roughness features to promote transition.

### 13.3 Concept of Hydraulically Smooth.

The concept of a hydraulically smooth surface needs to be carefully interpreted. In the stretch study, the wall-normal, $Sa$, length scale remained constant. In the spacing study, the feature height $habove$ remains constant (but $Sa$ decreased to zero). In both these studies, as the mean separation is increased, the roughness-loss fraction decreases to zero. However, in both, a meaningful measure of the wall-normal length scale remains constant.

### 13.4 Pits-and-Peaks Study.

It has been demonstrated, by the negative surface study, that surfaces with positive skew are likely to generate a larger aerodynamic penalty than surfaces with negative skew. However, the choice of $Sa$ as the wall-normal length scale cannot capture this effect (because Eq. (1) contains a modulus). This has led many authors to explicitly include skew into their correlations. However, skew is a “broad brush” and it may be more appropriate to explicitly evaluate $habove$ and $hbelow$, Eq. (4), and develop a correlation where the aerodynamic penalty is an asymmetric combination of the above and below quantities. Note: $Sa$ can be evaluated using Eq. (5) (with $fabove$ and $fbelow$).

*Characterization of surface topography*: A given surface topography could have *z*-heights non-dimensionalised by the wall-normal length scale, $Sa$. Similarly, the *x* and *y* coordinates by the wall-parallel length scale, $mean(Dl)$. It might then be possible to determine roughness parameters using an approach similar to the above–below decomposition. This might enable robust criteria to be developed for a broad range of surfaces.

### 13.5 Scan-Scale-Print-Measure Methodology.

A rough surface can be scanned, scaled, 3D printed, and then the boundary layer growth measured in the flat-plate facility. The validation process identified that for the current range of 3D printing technologies available, a further printer up-scale factor is required. This means that for the analysis of specific engine configurations, either a large wind-tunnel working section is necessary or values of the local skin friction coefficient need to be used.

### 13.6 Existing Correlations.

The observation is that at large stretch factors, the roughness-loss fraction is proportional to $Sa/mean(Dl)$ and is consistent with the proposal from Refs. [7,8] that $ksequiv$ scales with the effective slope. However, the current work shows that the linearity assumption is only valid for long wavelengths.

Other researchers, for example Ref. [5], have suggested skew as an important correlating parameter but they were only able to investigate negative skew. Thus, it is not possible to draw any comparisons with the single point positive and negative skew study included here.

### 13.7 Predictive Capability.

The scaled turbine blade, BladeA, has demonstrated that an engine representative surface can have a significantly different roughness-loss (only ∼50% in this case) than a sand grain surface. Furthermore, it has been shown that the stretch, spacing, and pits-and-peaks studies can be used to predict the roughness-loss fraction using a hydraulically smooth model and a sand grain model such a Schlichting.

## 14 Relevance to Industry

The Scan-Scale-Print-Measure methodology provides an opportunity to process high resolutions scans to determine the boundary layer development.

These studies have shown that for surface roughness that is *not sand grain like*, the behavior of a turbulent boundary layer can be quite different. In this publication, “sand grain like” was interpreted by examining how the feature size, $d70$, and the mean separation, $mean(Dl)$, were related.

## 15 Conclusions

The Scan-Scale-Print-Measure methodology has been successfully developed and enables carefully controlled investigations of the effects of surface roughness on a turbulent boundary layer.

The proposed mean separation (wall-parallel length scale) is an explicit measurement of the average separation between a local minima and the closest local maxima of a rough surface. Therefore, when combined with a wall-normal length scale, correlations are more likely to capture the fluid dynamic mechanisms associated with flow over rough surfaces.

Many surfaces are not sand grain like, in that their feature size to feature separation ratio is different. Therefore, the application of a pure sand grain correlation is unlikely to be robust in determining the aerodynamic penalty.

The stretch and spacing studies have demonstrated that a wall-parallel length scale is crucial in understanding the likely aerodynamic penalty of surface roughness.

The investigation of the pits-and-peaks study has shown that surface peaks have a greater effect on the roughness-loss than surface pits.

For the turbine blade surface examined, BladeA, the measured roughness-loss was approximately half which would be expected based on a sand grain correlation. When combined with the parametric studies, the predicted surface roughness-loss was within 5% of that measured.

There is evidence that the conditions for hydraulically smooth walls may need improving. For both the wall-parallel stretch and the feature spacing studies, the roughness-loss becomes zero in the long-wavelength limit. However, in both these studies, a meaningful measure of the wall-normal length scale remains constant.

## Acknowledgment

The authors are grateful for the financial support provided by Rolls-Royce, EPSRC, the ATI Programme, and Innovate UK, for CORe Design Intelligent TEchnology (CORDITE, project 75107). The authors gratefully acknowledge the numerous discussions with Prof. Rob Miller, Dr. Andy Wheeler, Dr. Chris Clark, Dr. James Taylor, Prof. Nick Cumpsty, and Dr. John Coull, alongside Matt Wang and the technician team at the Whittle Laboratory for supporting the experimental work. The authors would like to thank Mr. Frederic Goenaga, Dr. Raul Vazquez, Mr. Dougal Jackson, Dr. Masha Folk, and Dr. Marcus Meyer from Rolls-Royce for their advice, patience, and encouragement.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## Nomenclature

*f*=fractional area

*r*=radius

*s*=specific entropy

- $t$ =
trailing edge thickness

- $w$ =
staggered spacing

- $z\xaf$ =
mean height of roughness

*A*=total area

*H*=boundary layer shape factor

*L*=reference length (LE to downstream traverse plane)

*R*=gas constant

*U*=reference velocity (at

*x*=*L*)- $cf$ =
local skin friction coefficient, $\tau /(1/2)\rho U2$

- $ks$ =
sand grain diameter

*u*=_{τ}frictional velocity

- $Cpb$ =
trailing edge base pressure coefficient

*V*_{te}=trailing edge velocity

- $ksequiv$ =
equivalent sand grain diameter

- acc =
auto-correlation coefficient, Eq. (7)

- $d70$ =
offset at which auto-correlation coefficient decreased to 70%

*p*,*p*_{0}=static, stagnation pressure

- $pdf$ =
probability density function

- skew =
distribution skew, Eq. (A2)

- $x,y,z$ =
streamwise, transverse, and wall-normal (metal-on)

- xcc =
cross-correlation coefficient

- $Dz$ =
wall-normal distance

- $Dl$ =
wall-parallel length scale

- ES
=_{x} effective slope, Eq. (A3)

- LE =
upstream point of elliptic leading edge

- Re
=_{l} Reynolds number based on length

*l*- Sa =
centerline averaged roughness height, Eq. (1)

- Sq =
root mean square roughness height, Eq. (A1)

*T*,*T*_{0}=static, stagnation temperature

- $\delta 99$ =
boundary layer thickness at 99% freestream velocity

- $\delta te*$ =
boundary layer displacement thickness

- $\Delta s$ =
specific entropy

- $\Delta ht$ =
work transfer

*η*_{turbine}=turbine efficiency

- $\theta $ =
boundary layer momentum thickness

- $\rho $ =
density

*σ*=standard deviation

*τ*=wall shear stress

### Subscripts and Superscripts

## Appendix

*x*-direction, is

It must be noted that the definition of effective slope does not distinguish between positive and negative slopes and is a non-dimensional quantity.

*x*-direction (mainstream), the windward projected area, divided by the total plan area

*A*, is given by

*x*and

*y*components of the surface area associated with a small region $\Delta A$ of the plan area. Including the maximum function ensures that the leeward area is excluded. The windward value corresponds to the total windward projected area non-dimensionalized by the total plan area.

## Footnote

Reynolds number based on roughness height.