An alternative floating element design for skin-friction measurement of turbulent wall flows

Important error contributions arise from the necessary gap surrounding the FE, which makes it sensitive to (i) static pressure differences acting on non-parallel surfaces to the flow. Other sources of uncertainty are associated with (ii) misalignments of the FE relative to the boundary-layer plate and (iii) buoyancy forces resulting from uneven pressure distributions over the flow-exposed surface.

According to Brown and Joubert (1969), contribution (i) is the combination of the direct action of the freestream pressure gradient across the balance, \(F_{\alpha }\), and of the cavity-induced flow into and out from the test section, \(F_\mathrm{m}\). The resultant net force effectively masks out the action of the WSS. They suggest, as a first approximation, that \(F_{\alpha } \propto \lambda \rho \alpha S\) and \(F_\mathrm{m} \propto \rho \gamma U_\tau \alpha ^{1/2} S^{3/4}\), where \(\alpha =(1/\rho )\partial p/\partial x\) is the kinematic pressure gradient, \(\lambda\) is the lip thickness and \(\gamma\) is the nominal gap width, as indicated in Fig. 1. To minimise these loads, we engineered the fixed-to-floating surface joint to a tight tolerance, featuring an S-shaped labyrinth seal (0.5 mm thick and 15 mm long) to artificially tighten the gap. Baars et al. (2016) used instead a grooved-path labyrinth seal that is intrinsically more efficient, though it requires a thicker lip and it may prove difficult to implement. Alternatively, filling the gap with liquid could eliminate unknown pressure forces on the side walls of the FE (the surface tension can be easily quantified). This arrangement is particularly suitable for studies under conditions of non-zero pressure gradient (ZPG), such as those carried out by Frei and Thomann (1980) and Hirt et al. (1986), but was discarded for practical reasons. Note, the relationships mentioned above assume the spurious flow is solely driven by the static pressure gradient. They ignore the pressure difference between the flow-exposed surface and the underside of the FE, \(\Delta p\), with potentially greater significance. Enclosing the underside passively contributes towards reducing \(\Delta p\), but it is advantageous being able to actively control the pressure differential as well. In the present device, this is achieved by means of a centrifugal blower, as depicted in Fig. 2.

The vertical misalignment of the FE (ii) was initially investigated by O’Donnel (1964) using a single-pivot balance. Years later, Allen (1977, 1980) further considered the influence of varying parameters \(\lambda\) and \(\gamma\), and extended this analysis to parallel-shift designs (the reader is referred to Winter (1979) for a detailed description of both mechanisms). Although his empirical relations may not be directly transferable to other devices, the trends are expected to remain the same. So, they can still be used as guidelines to design balances that are essentially insensitive to misalignments. His results show that, for a fixed protruded or recessed position of the FE relative to the surrounding surface, wider gaps and thinner lips yield smaller lip forces. These act in the direction of a positive drag force and, therefore, should be minimised. Accordingly, the FE comprises a 1 mm-thick flange that extends 10 mm outward, effectively reducing the lip size, and its edges were rounded off to prevent flow separation. The gap, on the other hand, was primarily designed to negate contribution (i), so it is smaller than would be desirable in terms of sensitivity to misalignments. In any case, the relative position of the FE was optically verified to a resolution of 15 \(\upmu\)m before each test, by imaging the fixed-to-floating surface joint using a 16-MP camera with high magnification.

Allen (1977) additionally demonstrated how the static pressure distribution over the flow-exposed surface is affected for different protrusion values, causing the centre of pressure to move about. This phenomenon is commonly known as buoyancy (iii) and may have a similar impact to that of contributions (i) and (ii) depending on the design of the FE. If simply supported on a single-pivot point (Allen 1977, 1980; Acharya et al. 1985; Krogstad and Efros 2010), the pressure-induced moment is coupled with the moment produced by the surface shear stress—this effect becomes less important if the pivot point is moved farther away from the wall. In contrast, planar-supported elements are naturally insensitive in this regard. One of the first designs of this kind is the parallel-shift linkage, which is equivalent to extending the lever arm to infinity, thereby keeping the floating surface parallel to the flow (Kempf 1929; Dhawan 1951; Coles 1953). The present device is based on this arrangement, as shown in Fig. 1. Other alternatives have also been successfully employed, like placing the FE in a pool of liquid (Frei and Thomann 1980; Hirt et al. 1986) or using air bearings (Ozarapoglu 1973; Baars et al. 2016).

The existence of a fixed-to-floating surface joint entails a perturbation of the boundary-layer flow as it goes over, regardless of all design considerations. Different to contributions (i)–(iii), this error source is of random nature and primarily influences the signal-to-noise ratio (SNR) of the measurement. Here, we evaluate the extent of spurious flow over the FE using the scaling argument introduced by Baars et al. (2016). They argue that for small perturbations the extent of contaminated flow adjacent to the edges of the element is roughly proportional to its perimeter and the gap size or misalignment, whichever is higher. Correspondingly, the ratio of the total surface area to the edge affected area is defined as a proxy for the SNR; it follows from the schematic in Fig. 1 that \(\eta _\mathrm{A} \simeq l^2/(4 \gamma l)\). Large \(\eta _\mathrm{A}\) values are desirable to improve the quality of the signal. However, practical restrictions in size of wind-tunnel facilities frequently dictate the general dimensions of these devices, and increasing manufacture tolerances as a way to tighten the gap comes with an increase in cost of the project. As a consequence, \(\eta _\mathrm{A}\) is typically \({\mathcal{O}}(10)\) and it is highly challenging to achieve larger values. The FE developed by Baars et al. (2016) for the large-scale wind tunnel at the University of Melbourne is notable for having \(\eta _\mathrm{A} = 375\).

The unique design of the present device enables measuring not only the streamwise component of the aerodynamic load but also the induced pitching moment, as a mean to decouple extraneous loads from the measurement of WSS. Its assembly, shown in Fig. 1, consists of a closed aluminium frame with outer dimensions 220 mm\(\times 220\) mm\(\times 40\) mm (\(W\times L\times H\)), which holds all the mechanical and electronic components inside.

The working principle behind the WSS sensor is similar to that of a parallel-shift linkage. The streamwise load acting on the floating element (1) is directly transmitted to the floating frame below (5), which in turn is supported by a pair of flexures (4) that allow it to move freely along the streamwise direction and a pair of rigid blades (8). These are fixed to the upstream and downstream sides of the casing (3), respectively. The FE is supported by a set of holders (2) which rotate freely about the pivot point (7). The pitching moment generated around it is then counteracted by the pair of blades (6), mounted horizontally on each side of the floating frame. The pivot mechanism consists of a 0.05-mm flexure that is clamped on either sides with a free-standing length of less than 0.5 mm to prevent it from buckling. Both pairs of blades, (6) and (8), are instrumented with metal gauges for strain measurement, and their sensitivity can be adjusted by varying the effective length. In the current configuration, the vertical transducers are fixed-supported aluminium blades 15 mm long, 10 mm wide and 0.45 mm thick. Those for the pitching moment are instead simply supported, 15 mm long and 0.6 mm thick, resulting in greater sensitivity. The sensing element is a 200-mm-side square, small enough in relative terms to ensure local measurement of WSS. Its dimensions are such that the variation in friction coefficient \(C_\mathrm{F}\) is less than \(1\%\) between the leading and trailing edges. The 1/7th law skin-friction relationship was used to obtain estimates of the local \(C_\mathrm{F}\) for a smooth-wall boundary layer at \(Re_\mathrm{x} \simeq 2\times 10^6\). This corresponds to a freestream velocity of 10 ms\(^{-1}\) and a development length of approximately 3 m, according to the experimental setup described in Sect. 3. In view of error contributions (i) and (ii) (refer to Sect. 2.1), the fix-to-floating surface joint was carefully designed to a very tight clearance. Detailed in Fig. 1, the lip \(\lambda = 1\) mm \(\pm 0.1\) and the gap \(\gamma = 0.5\) mm \(\pm 0.1\), with a 15-mm-long labyrinth seal which further mitigates cavity-induced flow asymmetries. Finally, the size of the FE and the gap width yield an effective surface area coefficient of \(\eta _\mathrm{A}=100\). This value is much larger than average and falls in the same order of magnitude of the one from the large-scale TBL facility in Melbourne, though its dimensions are significantly smaller in comparison (the area ratio is 75).

This balance also features an external centrifugal blower used to modify the pressure inside, \(p_\mathrm{b}\). There are two nozzles mounted at the bottom, one on each of the two sides of the acquisition module, as depicted in Fig. 1. This is to ensure uniform pressure distribution on the underside of the FE and to minimize airflow inside the balance. \(p_\mathrm{b}\) is monitored via two pressure taps, included in the same schematic, mounted on opposite corners away from the nozzles. The circuit is detailed in Fig. 2. Both the pressure taps and the ports to the blower inlet are externally connected. The blower is driven by a 24 VDC motor that is controlled through pulse-width modulation and is able to generate up to 1400 Pa of head pressure. The intensity of the blower can further be adjusted by regulating a three-way valve placed upstream, which lets air from the room to be sucked in.

The ability of the balance to measure WSS depends, amongst other factors, on the integrated wall drag, the sensitivity of the force transducers and the effective resolution of the analog-to-digital converter (ADC). Since practical restrictions in size of the FE limit the magnitude of the force it experiences (\(F_{\mathrm{w}} = \tau _\mathrm{w}S\)), an acquisition module able to provide a stable low-noise signal is paramount to ensure the quality of the measurements. In this section, we briefly explain the signal conditioning system especially designed for this application, which involves a four-channel acquisition module and two sets of active strain sensors, one for each of the two load components (drag and pitching moment).

Hot-wire measurements of the streamwise velocity profile were taken 3.3 m downstream of the contraction, at the centreline of the FE, as depicted in Fig. 4. A total of 9 freestream velocities equally spaced between 10 and 26 ms\(^{-1}\) were considered.

The boundary-layer probe is an Auspex A55P05 with 10-mm-long prongs spanned by a 3-mm-long and 5-\(\upmu\)m-diameter tungsten wire. The central active region is approximately 1 mm long and is shouldered on either side by copper-plated sections, yielding an effective length-to-diameter ratio of 200. The wire was operated by a Dantec StreamLine Pro constant temperature anemometer (CTA) system set to an overheat ratio of 1.8. We used the National Instruments USB-6212 BNC ADC to sample the signal at 20 kHz. Pre- and post-calibrations were conducted using a Pitot-static tube, mounted at the same streamwise location, that was connected to a Furness FCO510 micromanometer. The temperature was constantly monitored throughout calibration and acquisition stages using a T-type Omega thermocouple. Each velocity profile is populated by 30 points taken for a sampling period of \(T=40\) s, which in the worst case (\(U_0 = 10\) ms\(^{-1}\)) is equivalent to \(T\delta /U_0 = 8000\) eddy turnover times.

The mean skin friction was estimated using the method proposed by Rodríguez-López et al. (2015). Accordingly, the mean velocity profile is compared with a canonical description of the boundary layer following Musker (1979), for the inner and logarithmic layers, and using the exponential wake by Chauhan et al. (2009). The method allows adjusting the wall-normal coordinate to overcome any potential uncertainty in determining the initial relative position of the wall probe. The skin friction is then estimated as that which minimises the error between the experimental and canonical profiles. For further details on this method the reader is referred to the original paper (Rodríguez-López et al. 2015). The accuracy in determining \(U_\tau\) is better than 1% for perfectly accurate velocity profiles and better than \(0.5\nu /U_\tau\) for the initial relative position of the wall probe. The method was originally validated against DNS and experimental databases, under presence of mild pressure gradients, wall probe interference in the inner layer and poorly converged data. Additionally, it has been shown to perform adequately in highly disrupted flows past strong tripping conditions (Rodríguez-López et al. 2016), porous fences (Rodríguez-López et al. 2017) and TBL under freestream turbulence (Esteban et al. 2017).

The balance was statically calibrated in situ using a set of weights and the wire–pulley arrangement pictured in Fig. 5. The pulley is an ME-9450 from Pasco with 50-mm-diameter string and an effective coefficient of friction of \(7\times 10^{-3}\). It was attached to a 50-mm linear traverse which enabled adjusting its vertical position to within 0.05 mm. For calibration purposes, a 40-mm-long stud with machined-in grooves was mounted perpendicularly to the FE at the spanwise centerline, above the pivot point. A 0.1-mm nylon wire was then looped around it and strung over the pulley. A small mass was suspended to tension the wire, so that its alignment could be adjusted. First, by sliding the traverse system across the spanwise direction and second, by turning the knob to adjust the vertical position of the pulley to a predetermined height.

Prior to any acquisition, the device was turned on and let to warm up for 2 h. The calibration involved recording the response of the strain sensors to \(n=24\) load combinations using M1-class weights. The output voltages were sampled for 20 s to average out noise and the mean values of each pair of transducers were summed up. For the smooth wall, six different weights including the zero reading (0, 1, 3, 5, 10 and 15 g) were applied at four different heights (0, 9.6, 19.2 and 28.8 mm). This process was repeated 5 times to achieve statistical significance, so a grand total of 120 calibration points were taken. Note the loading direction of the balance over the course of the calibration changed to account for potential hysteresis. The values of force \(F_\mathrm{w}\) (N) and pitching moment \(M_\mathrm{y}\) (Nm) were obtained by multiplying the mass of the weights by the local gravity acceleration \(g = 9.81084\) ms\(^{-2}\), from the land gravity survey data by the British Geological Survey.

We consider the second-order response surface model in two variables:By letting \(x_3 = x_1^2\), \(x_4 = x_2^2\), \(x_5 = x_1x_2\), \(\beta _3 = \beta _{11}\), \(\beta _4 = \beta _{22}\) and \(\beta _5 = \beta _{12}\), Eq. 3 is rewritten in matrix form aswhere\(Y_{n\times 2}\) is the calibration design matrix; its rows i refer to different loading conditions and the columns j are for the individual components of the applied load, that is, \(F_\mathrm{w}\) and \(M_\mathrm{y}\), respectively. The model matrix, \(X_{n\times 5}\), contains information about the response of the force traducers to each load combination (in units of mV/V), as well as information about the model form—the number of columns corresponds to the number of terms in the right-hand side (RHS) of Eq. 3. The regression coefficients B are estimated via weighted least squares (WLS) fit to account for the uncertainty in the predictors. This is particularly important as the size of the dataset is fairly small. From Strutz (2016),where \({\hat{B}}\) is the least squares estimator of B, \(W^j\) is the weighting matrix for a particular degree of freedom (DOF), j, and the overbar indicates averaged quantities over all calibration runs (five in total). As suggested by Reis et al. (2013), the weighting matrix is computed as the sum of the error contributions due to weights and the wire–pulley setup \(V_\mathrm{w}\), such as misalignments and friction forces, and the variance in the readings of the strain sensors \(V_\mathrm{r}\). Accordingly,The second term on the RHS of Eq. 6 should be thought of as the projection of the variance in the readings onto the estimated load. \({V_\mathrm{r}}_{2n\times 2n}\) is diagonally symmetric and is computed as the covariance between the readings of the force traducers, whereas \(D^j_{n\times 2n}\) is the sensitivity matrix whose diagonal contains the partial derivatives of Eq. 3 with respect to a given DOF. Finally, \({V_w^j}_{n\times n}\) is a diagonal matrix and its values are statistically approximated by the sum of squared residuals (\(\hbox {SS}_{E}\)) of the fit by replacing W for the identity matrix.

The parameters of the linear regression are listed in Eq. 7. They also appear divided by the respective uncertainty \(|{\hat{B}}/u_{\mathrm{B}}|\) to highlight the relative importance between them. \(u_B\) is the positive square root of the main diagonal elements of \(({\bar{X}}^TW^j{\bar{X}})^{-1}_{5\times 5}\), which is the covariance matrix of the model coefficients.The large magnitude of \(B_{11}\) in comparison to the remaining terms reveals a strong linear behaviour of the drag sensor, independent of the pitch moment acting on the FE. On the contrary, the latter appears to be equally dominated by the linear and nonlinear terms of both force transducers. The corresponding values of \(|{\hat{B}}/u_\mathrm{B}|\), however, indicate that the relative importance of the second-order terms is negligible as the associated variance is significantly high. This is likely due to lack of orthogonality in the design of the response model (Johnson et al. 2010). The variance inflation factor (VIF) is a statistical indicator that quantifies the degree of multicollinearity in a least squares regression. As a rule of thumb, the value of VIF is ideally kept under ten, but in this particular case is \({\mathcal {O}}(10^2)\), which is explained by the large cross correlation value between the readings, \(x_{i1}\otimes x_{i2} =0.953\). Still, the standard error of the fitting is relatively small: \({\hat{\sigma }}_{F_\mathrm{w}}=0.057\) mN and \({\hat{\sigma }}_{M_\mathrm{y}}=0.006\) mN m, where \({\hat{\sigma }}^2=\hbox {ss}_{\mathrm{E}}/(n-5)\).

A series of pre- and post-calibrations were carried out over the course of 1 month to assess the long-term stability of the balance to environmental changes and setup. The difference between measurements of WSS obtained using linear estimators from calibrations performed weeks apart is shown in Fig. 6. Apart form the data point taken at the lowest Reynolds number, discrepancies between calibrations are generally smaller than 0.5% with a standard deviation of just over 0.8%, or 0.1 mN in absolute terms.

The balance is first turned on and let to warm up for 2 h, unless a pre-off-tunnel calibration was performed before the measurement. In the meantime, the tunnel is ran for an extended period to ensure steady flow conditions. The acquisition sequence is fully automated; it starts with a shakedown period of 3 min during which the wind tunnel is set at the maximum freestream velocity. It is then turned off for half an hour before zeroing the balance. This initial kick guarantees the very first data point is taken under the same conditions as the first data point of subsequent runs. A total of nine drag values were acquired over a range of Re for both the smooth and the rough wall. The evaluation method is illustrated in Fig. 7. For each freestream velocity the pressure inside the balance \(p_\mathrm{b}\) is progressively varied using the blower such that the pitching moment passes through the zero-crossing. This experiment was conducted in a suction wind tunnel, so the static pressure in the test section is naturally lower than \(p_\mathrm{b}\). Consequently, given the location of the pivot point, the pitching moment is initially positive but gradually becomes negative with increasing suction of the blower. The values of wall drag are the Y-intercept of the linear regression through each set of data points acquired at fixed Re, where the pitching moment is null. This is a necessary step since the extraneous contribution of the vertical load to the measurement of WSS is minimised when the moment it induces is cancelled out, as explained in Sect. 2.2.1. Acquisitions last 30 s at 150 Hz, equivalent to 2500 boundary-layer turnover times at the lowest operating velocity. Between them, a readjustment period of 3 min is given for the response of the system to reach a nearly constant value. The SNR of these measurements is found within the range \(1<\sigma /\mu <5\), where \(\sigma\) and \(\mu\) are the standard deviation and the mean value of the signal, respectively. Once the run is over, the wind tunnel is turned off and the balance is set to idle for another period of half an hour between runs. The aforementioned procedure is repeated five times for the results to achieve statistical significance.

This analysis follows the rules established by the International Bureau of Weights and Measures for evaluating and expressing uncertainty in measurement (BIPM et al. 2008). The complete uncertainty budget for the friction velocity \(U_{\tau }\) is given in Fig. 8 as a function of the Reynolds number. It factors the most important contributions, including (a) calibration errors, (b) the inference method of WSS, (c) the inclination of the FE, (d) determination of air properties, temperature and pressure, and (e) precision of the measurements.

By definition \(U_{\tau } = \sqrt{\tau _\mathrm{w}/\rho _0}\); the WSS is expressed as \(\tau _\mathrm{w} = F_\mathrm{w}/S\) and the air density is obtained via the ideal gas law \(\rho _0 = p_0/(RT_0)\), where \(R = 287.05\) J kg\(^{-1}\) K\(^{-1}\) is the specific gas constant, \(p_0\) is the atmospheric pressure and \(T_0\) is the room temperature. Hence, \(U_{\tau }(F_\mathrm{w},T_0,p_0)\) is a function of three independent variables. Assuming all input quantities are uncorrelated, the combined uncertainty of \(U_{\tau }\) is the positive square root of the variance \(u^2_{U_{\tau }}\), written aswhere \(u_{\Phi }\) is the standard uncertainty of the arbitrary variable \(\Phi\).

Contributions (a) and (b) The first term on the RHS of Eq. 8 is the combination of sources (a), (b) and (c), which are inherent to the working principle of the balance. Particularly, (a) and (b) cannot be decoupled; the calibration error \(u_{\mathrm{(a)}}\) propagates through the linear regression and is reflected in the uncertainty estimate of the Y-intercept, from here on denoted \(u_{(\mathrm{a}+\mathrm{b})}\). \(u_{\mathrm{(a)}}\) is determined by multiplying the readings of the force transducers by the covariance matrix of the calibration coefficients \(({\bar{X}}^TW^j{\bar{X}})^{-1}_{5\times 5}\), defined in Sect. 3.2. Accordingly,where \({u_{\mathrm{(a)}}}_j\) is the uncertainty associated with the load component j (drag or pitch), \({\bar{X}}\) is the averaged model matrix of the calibration, \(W^j\) is the weighting matrix for j and X is the vector containing the readings of the force transducers, written according to the model form expressed in Eq. 3. Potential errors induced by wire misalignments, friction of the pulley and weight application are all factored into the covariance matrix \(W^j\). Further details can be found in Reis et al. (2013). \(u_{(\mathrm{a}+\mathrm{b})}\) amounts to approximately 97% of the total variance at low Re but only 30% at higher values. In absolute terms, its magnitude is fairly independent of the loading conditions, so its relative significance diminishes with Re as opposed to that of \(u_{\mathrm{(c)}}\), which increases monotonically to become the primary source of uncertainty.

Contribution (c) Surface inclination causes the pressure-based wall-normal load \(F_\mathrm{n}\) to be projected in the streamwise direction. Similar to WSS, this error contribution is a flow-driven phenomenon which manifests as an offset in skin-friction coefficient (Baars et al. 2016). Provided a constant tilt angle, it can be corrected for, but the additional DOF of the present device rules out this possibility. It thus stands as a source of bias error that must be included in the uncertainty budget, \(u_{\mathrm{(c)}}\). The streamwise projection \(F^{'}_{\mathrm{w}} = F_\mathrm{n}\sin (\alpha )\), where \(\alpha\) is the tilt angle of the FE. Then it follows thatThe flow-exposed surface is mounted parallel to the principal flow direction, thus \(\sin (\alpha ) = 0\) and \(u_{\mathrm{F}_\mathrm{n}}\) can readily be ignored. In turn, the accuracy of the digital spirit level (\(0.1^{\circ }\)) in addition to the maximum predicted rotation of the FE (\(0.075^{\circ }\)) yields \(u_\alpha = \sqrt{{0.1}^2+{0.075}^2} \simeq 0.125^{\circ }\). The second term of Eq. 10 can be rewritten as \([\Delta p_\mathrm{d} \,S\, \cos (\alpha ))]^2\,u^2_{\alpha }\), where \(\Delta p_\mathrm{d}\) is the pressure difference at which wall drag is evaluated. Considering that buoyancy forces, no matter how small, entail the existence of a net vertical force when the pitching moment is cancelled out, \(\Delta p_\mathrm{d}\) will depend on the flow conditions. Specifically, the latter increases with Re, hence the trend \(u_{\mathrm{(c)}}\) is exhibits in Fig. 8.

Contribution (d) Air density is function of the atmospheric pressure and room temperature, \(\rho _0(p_0, T_0)\). The first was acquired at the start and finish of the experiment, using a weather station accurate to within 100 Pa. The mean value was then assigned to \(p_0\), so it follows that its uncertainty includes not only the accuracy of the measurement system but also the range of the fluctuations; \(u_{p_0}=\sqrt{{(P_{0_{\mathrm{acc}}})}^2 + {(P_{0_{\mathrm{max}}}-P_{0_{\mathrm{min}}})}^2}\). The freestream temperature, on the other hand, was continuously monitored using a T-type Omega thermocouple with an accuracy of \(0.5^{\circ }\). A similar system was dedicated to measure temperature inside the balance.

Differential pressures were monitored via three Furness FCO510 micromanometers (model 2) with full-scale (FS) ranges of 200 and 2000 Pa. These have a reported accuracy of \(\pm 1\) digit below 20 Pa and 0.25% of the reading above this threshold. The freestream dynamic pressure \(q_0\) is acquired via a Pitot-static tube positioned above the FE, aligned with its leading edge. Following the assumption of a thin boundary layer, potential wall-normal pressure gradients are neglected and the pressure over the flow-exposed surface of the FE is taken to be equal to the local static pressure. The static port is then shared with a second manometer, by means of a T-junction, to monitor the pressure difference between the top and underside of the FE \(\Delta p\). Additionally, the streamwise pressure gradient \(\partial p/\partial x\) is estimated by measuring the pressure drop between the aforementioned Pitot-static tube and another Pitot probe mounted far upstream, assuming a linear growth of the boundary layer. Note that from these quantities solely \(q_0\) represents a source error in \(C_\mathrm{f}(F_\mathrm{w}, q_0)\). Despite playing a role in the uncertainty estimation of \(F_\mathrm{w}\), that of \(\Delta p\) is not factored in this analysis (refer to Eq. 10).

Values of WSS inferred from hot-wire anemometry of the boundary-layer profile were used as the standard of comparison for the FE measurements. The results, given in Fig. 9, show an excellent agreement between these techniques, especially for \(Re_{\theta }\) over \(5\times 10^3\) (\(U_0 = 14\) ms\(^{-1}\)) where the relative difference between them falls below 2%. For the most part, the disparity of the data at low freestream velocities may be explained by the increased uncertainty in the measurement; with the exception of the first data point, whose error bar does not encompass the power curve fit through the hot-wire values. In absolute terms, the magnitude of the force discrepancy is approximately 0.92 mN, which makes it remarkably difficult to identify its cause. This could either be an indication of additional error sources which were not accounted for, or of poorly estimated uncertainty contributions. A similar behaviour is observed on the skin-friction measurements of Baars et al. (2016), who suggested that it could be related to secondary loads acting on non-parallel surfaces to the flow and to the projection of the pressure-based force in the streamwise direction that could not be fully eliminated. From these potential contributors, the first was not considered in the uncertainty analysis presented in Sect. 3.4. Additionally, it is plausible to assume some residual coupling of the vertical net force, despite the measures to work around this limitation.

The smooth-wall skin-friction relationship of Österlund et al. (2000) is also included in Fig. 9 for comparison. While the values of \(C_\mathrm{F}\) obtained from hot wire share the same trend, they are generally lower by 2%. This is also true for the FE measurements at higher Reynolds numbers, plausibly for the reasons discussed in Sect. 3: the streamwise pressure gradient in the contraction and tripping conditions.

\(C_\mathrm{F}\) measurements of the rough-wall boundary layer are shown in Fig. 10. Values obtained using the present FE take a nearly constant value (to within \(1\%\)) across the entire range of Re, which indicates the flow reaches the fully rough regime farther upstream of its location. Here, we determine the equivalent sand-grain roughness \(k_\mathrm{s}\), normalised to viscous units \(k_\mathrm{s}^+=k_\mathrm{s}U_{\tau }/\nu\), based on the boundary-layer thickness reported by Cheng and Castro (2002) for the same flow conditions, \(\delta = 0.12\) m at \(U_0 = 10\) m/s. Assuming the functional form for the viscous-scaled mean velocity profile \(U^+\) of Coles (1956), the downward shift relative to the smooth-wall case \(\Delta U^+ = 13.8\). This yields \(k_\mathrm{s}^+ = 1070\) that is well above the empirical threshold \(k_\mathrm{s}^+ \ge 80\) from which the flow/surface becomes fully rough (Jiménez 2004). The test–retest variability of the measurements appears to have worsened compared to the smooth wall, revealed by the spread of individual (grey) data points around the mean. At high Re, it becomes in fact the largest uncertainty contribution. Additional estimates of \(C_\mathrm{F}\) are included in the same figure, specifically that inferred from cross-wire anemometry of the boundary-layer profile by Cheng and Castro (2002), and those obtained by Claus et al. (2012) using a pivot-type FE and pressure-tapped roughness obstacles. Overall, the wall drag over the staggered array of cubes is substantially higher compared to the smooth wall and the disparity between the estimates is in the order of 25%.

Claus et al. (2012) highlighted the marked discrepancy between their measurements and advanced the hypothesis that, while trends with Re are correct, the uncertainty in the pressure data could undermine the validity of the form-drag estimate. They further argue, by comparing the results from both techniques, that a lower \(C_\mathrm{F}\) of the FE would imply a negative friction drag contribution. On these grounds, and given the quoted uncertainty of their FE (Krogstad and Efros 2010), they concluded the absolute pressure-drag value is likely overestimated. The present measurements support this observation, as the mean skin-friction coefficient \(\overline{C_\mathrm{F}}=8.45\times 10^{-3}\) is still \(8\%\) lower than that of the pressure data. However, they do not agree with the FE estimate of Claus et al. (2012) which is \(17\%\) lower in comparison. To explain this discrepancy, it would be required to carry out a comprehensive uncertainty analysis of their measurements, otherwise we can only speculate. Two plausible sources of error are contemplated here. The first is related to the streamwise pressure gradient, which could be neglected in terms of flow development (based on the acceleration parameter), but still stands as a source of buoyancy effects. Naturally, the suction wind tunnel where these experiments were conducted imposes a favourable pressure gradient in the freestream and over the flow-exposed surfaces. Since the distribution on the underside of the FE is likely to be uniform, the pressure difference across \(\Delta p\) becomes larger downstream. As small as it may be, this effect induces a moment opposite to the action of the WSS that could result in an apparent drag reduction considering the setup of Claus et al. (2012). Another potential source of uncertainty has to do with roughness-induced pressure inhomogeneities within the canopy. When the roughness length scale is comparable to the size of the FE, the integrated vertical pressure force is likely to lie off-centre, thus creating a spurious moment around the pivot point with similar consequences to that mentioned above depending on the obstacle distribution.

Cheng and Castro (2002) reported wind tunnel measurements over geometrically identical staggered arrays of cubes, 10 and 20 mm tall. They evaluate \(C_\mathrm{F}\) for both surfaces from spatially averaged shear stress profiles, using cross-wire anemometry. For the largest case, they further provide an estimate based on the pressure distribution over a unit area, and verified that the shear stress in the inertial subrange is roughly 21% lower in comparison—it possibly does not reflect the spatial anisotropy within the canopy. In view of this observation, Reynolds and Castro (2008) defined a correction factor for the WSS of staggered arrays of cubes based on the shear stress profile (\(U_{\tau } = 1.12\sqrt{-\overline{uv}}\)), which Placidi and Ganapathisubramani (2015) extended to arbitrary cubical arrays with varying frontal and plan solidities. The underlying assumption to this approximation is that the ratio between the shear stress above the canopy and the total surface stress remains constant, or it changes marginally with fetch. Accordingly, we can use the value of \(C_\mathrm{F}\) obtained by Cheng and Castro (2002) taken at the same fetch as the present experiment, shown in Fig. 10, to obtain a corrected estimate of \(C_\mathrm{F}\) to compare against the FE measurement. This yields a skin-friction coefficient of \(8.8\times 10^{-3}\) that is only \(4\%\) higher. A slight overestimation could either be explained by the uncertainty associated with the form-drag data, which was not provided, or by limitations of the aforementioned assumption. Nevertheless, the agreement is remarkable.