2.1. Analytical solution for wall shear in channel flow

For two-dimensional steady-state fully developed channel flow – see figure 2, the Navier–Stokes equation for the streamwise velocity becomes

\begin{equation*} \mu \frac {\text{d}^2 u}{\text{d} y^2} = \frac {\text{d}p}{\text{d}x} , \end{equation*}

Figure 2. Channel set-up used for calibration of the hot-film sensor; figures are not drawn to scale: (a) top view of the channel showing the eight pressure ports, (b) exploded view of the inlet flow conditioner, (c) exploded view of section c–c of channel with gasket, polycarbonate sheet, aluminium plate and clamps and (d) outlet flow conditioner showing internal vanes.

where $p = p(x)$ is the pressure, $u=u(y)$ is the streamwise velocity and $\mu$ denotes the dynamic viscosity. On integrating the above equation with respect to $y$ once, and substituting the symmetry condition $d u/d y = 0$ at $y = h/2$ , we get

(2.1) \begin{equation} \mu \frac {\text{d} u}{\text{d} y} = \left (y - \frac {h}{2}\right )\frac {\text{d}p}{\text{d}x} . \end{equation}

Note that $h$ is the channel’s height. This result does not assume any particular flow regime, laminar or turbulent. From the above result, the shear stress $\tau$ can be written as

At the wall, where $y = 0$ , the wall shear stress $\tau _{\textrm {w}}$ is then given by

(2.2) \begin{equation} \tau _{\textrm {w}} = -\frac {h}{2}\frac {\text{d}p}{\text{d}x} ,\end{equation}

which implies that the wall shear stress in channel flow can be determined by measuring the pressure gradient. The channel shown in figure 2 was constructed to provide a linear pressure gradient over a large distance, so that the flow is fully developed and $({\text{d}p}/{\text{d}x})$ can be confidently estimated.

2.2. Working principle of hot-film anemometry

Hot-film anemometry is used to measure the velocity and turbulence properties of fluid flows by measuring the heat dissipated due to convection. The hot-film sensor used in our experiments is shown in figure 3. The H-shaped film at the tip of the sensor is a very thin electrical resistor, through which current is passed, generating heat. The heat transfer rate from the film into the fluid varies with flow velocity according to a modified form of King’s law (King, Reference King1914; Dantec, 2023).

(2.3) \begin{equation} q_{\textrm {conv}} = a+b u^n, \end{equation}

where $a$ , $b$ , $n$ are constants, and $u$ is the characteristic velocity near the sensor. The heat generated by electric current is

(2.4) \begin{equation} q_{\textrm {gen}} = \frac {E^2_{\textrm {a}}}{R_{\textrm {f}}} , \end{equation}

where $E_{\textrm {a}}$ is the voltage applied across the film, and $R_{\textrm {f}}$ is the film’s electrical resistance. A constant-temperature anemometer (CTA) holds $R_{\textrm {f}}$ constant by using a servo amplifier circuit to control the voltage across a Wheatstone bridge, of which the film forms part of one leg. Monitoring this bridge voltage yields $E_{\textrm {a}}$ when the rest of the bridge resistances are known. At constant temperature, energy conservation yields $q_{\textrm {conv}} = q_{\textrm {gen}}$ . Equations (2.3) and (2.4) can be combined and manipulated to yield the well-known relation

(2.5) \begin{equation} E^2_{\textrm {a}} = A+B\tau _{\textrm {w}}^n, \end{equation}

where $\tau _{\textrm {w}}$ is the wall shear stress (Tropea et al. Reference Tropea, Yarin and Foss2007). The calibration coefficients $A$ , $B$ , $n$ are determined by correlating the voltage output of the sensor with a known wall shear stress in fully developed channel flow. Because the heat transfer in laminar flow is significantly different than that found in turbulent flow, two sets of coefficients are required for flow fields which contain both regimes.

Figure 3. A schematic of the hot-film sensor 55R46 by Dantec dynamics (Dantec, 2023).

3.1. Design of channel

The channel used for calibration of the hot-film sensor is motivated by prior work reported in the literature (Sun et al. Reference Sun, Wang, Luo, Deng, Guo and Ma2018). A schematic of the channel is shown in figure 2(a) and a sectional view through a pressure port is shown in figure 2(c). The channel was fabricated using aluminium and its cross-section ( $w = 50$ mm, $h = 5$ mm) was chosen to target a Reynolds number of $\textit{Re} = 10,\!000$ Footnote ¹ with the available pump.

The length of the channel was chosen to be $L = 1,\!200$ mm to ensure that the flow would be fully developed over a significant distance. The top of the channel is covered with a clear polycarbonate sheet and leakage is prevented through the use of a gasket along the length of the channel. An aluminium plate is placed over the polycarbonate sheet and clamps are used to apply uniform pressure on the gasket along the length of the channel to make it leak proof – see figures 2(c) and 4(a).

A submersible utility pump was used to circulate water through the channel. The flow rate is controlled with a gate valve installed upstream of the channel’s inlet. The flow is conditioned at the inlet of the channel with a series of baffles, followed by a honeycomb, and then a series of three meshes of decreasing hole size – see figures 2(b) and 4(b). To reduce the developing length, the boundary layer is tripped with a coarse-grit sand paper strip at the inlet of the channel. The outlet of the channel is connected to a flow conditioner with four internal vanes to minimise end effects – see figure 2(d).

Figure 4. Assembled view of the channel set-up in figure 2: (a) top view of the channel showing sensor mount (without sensor) and eight pressure ports (see figure 2) connected via three-way valves, (b) inlet flow conditioner, (c) hot-film sensor, (d) magnified view of sensor mount with sensor, (e) sectional view of channel set-up through the sensor mount and sensor.

Figure 5. Pressure at the eight different ports of the channel (see figure 2) for the mass flow rate of 0.422 kg s⁻¹, computed based on pressure differential measurements relative to port $\unicode{x2460}$ and assignment of an arbitrary pressure to port $\unicode{x2467}$ . Note that the straight line fit was obtained by using the data from ports $\unicode{x2463}$ though $\unicode{x2467}$ .

To measure pressure along the channel, eight pressure taps are placed on the side of the channel along its length. The ports are $150$ mm apart and drilled with a size of $0.8$ mm. The ports are connected by a three-way valve – see figure 4(a), to a DP15 differential pressure transducer (a product of Validyne Engineering (Validyne 2023), not shown. The pressure differential between ports $\unicode{x2461}$ through $\unicode{x2467}$ relative to $\unicode{x2460}$ are recorded at steady state; the data are then used to compute the pressure at all the ports by assigning an arbitrary pressure to port $\unicode{x2467}$ – see figure 5. We observe a linear pressure gradient over the entire measurement section, which indicates that the flow is fully developed downstream of port $\!\unicode{x2460}$ . We mounted the hot-film sensor between ports $\unicode{x2463}$ and $\unicode{x2464}$ on the polycarbonate sheet, as shown in figure 4(a). A closeup view of the sensor mount is provided in figure 4(d). A sectional view of the assembled channel set-up through the sensor mount and sensor is shown in figure 4(e). In addition to making it easier to place our sensor, this mount also provides the same conductive environment between the probe and the wall in both the calibration and the experiment.

3.2. Calibration procedure

The hot-film sensor 55R46, shown in figures 3 and 4(c), was calibrated using the constant-temperature anemometer MiniCTA 54T42, also a product of Dantec Dynamics (Dantec 2023). The calibration was performed using 21 different flow rates through the channel; a gate valve was installed between the prime mover and the inlet of the channel to control the flow rate. For each flow rate, the pressure transducer and hot-film sensor measurements (voltages) were recorded. The pressure measurement (voltage) provides the pressure drop between ports $\unicode{x2463}$ and $\unicode{x2464}$ ; transducer calibration data are used to express it in Pa and the wall shear stress $\tau _{\textrm {w}}$ is then computed using (2.2). The variation of the square of the hot-film sensor voltage ( $E_{\textrm {a}}^2$ ) with the wall shear stress $\tau _{\textrm {w}}$ is plotted in figure 6 with the objective of computing the calibration coefficients in (2.5). It can be seen that the wall shear stress increases monotonically with increase in the mass flow rate.

Figure 6. Calibration data showing the variation of $E_{\textrm {a}}^2$ with $\tau _{\textrm {w}}$ . The fitted curves are the same as those shown in figure 7. The error bars indicate the propagation of pressure transducer error (0.5 % of full range) through (2.2). That sensor’s full range is 8.6 kPa.

The variation of $E_{\textrm {a}}^2$ with $\tau _{\textrm {w}}$ depicts three distinct behaviours corresponding to three distinct flow regimes: turbulent, critical and laminar. The decision to use separate calibrations for the laminar and turbulent regimes, while not the most common approach in the literature, is scientifically justified based on documented thermal regime-dependent effects. Although Reichert and Azad (Reference Reichert and Azad1977) and Haselbach and Nitsche (Reference Haselbach and Nitsche1996) demonstrate that a 1/3 power law between heat transfer and wall shear stress theoretically holds for both laminar and turbulent flows, these studies show that while both regimes follow the same functional form, the fitting coefficients in this relationship are not identical between regimes.

Specifically, Haselbach and Nitsche (Reference Haselbach and Nitsche1996) note that ‘significant differences between the two temperature fields of the test cases occur mainly on the flow side’ and demonstrate that laminar flows exhibit ‘much more distinct temperature wake’ compared with turbulent flows with their ‘smaller temperature wake due to increased turbulent heat transport’. These thermal environment differences directly affect the calibration coefficients in the 1/3 power law relationship. This is further supported by Haselbach et al. (Reference Haselbach, Swoboda, Wald and Nitsche1995), who provide experimental visualisation showing that ‘the surface temperature wake is much greater in the laminar flow compared with the turbulent flow’ and emphasise that improved calibration methods should ‘take into account the convective as well as the conductive heat loss of a surface hot-film sensor’.

While universal calibrations represent an engineering compromise that uses averaged coefficients across both regimes and works adequately for many applications, the literature evidence suggests this approach prioritises practicality over precision. Even in studies demonstrating universal calibrations, observable scatter and systematic differences between laminar and turbulent data points indicate that regime-dependent coefficient variations exist but are typically considered acceptable rather than negligible. For applications requiring maximum accuracy, particularly in transitional flows where both regimes may be present, separate calibrations with regime-specific coefficients account for the documented thermal differences and provide superior measurement precision.

Table 1. Calibration coefficients and exponent for laminar and turbulent flow regimes

Figure 7. Linear calibration curves for laminar and turbulent regimes.

The data points corresponding to the laminar and turbulent regimes are used to find a linear fit between $E_{\textrm {a}}^2$ and $\tau _{\textrm {w}}^n$ , where the value of the exponent $n$ should lie in the range $[0.1, 0.5]$ – see Tropea et al. (Reference Tropea, Yarin and Foss2007). A linear fit is found by choosing $n=0.4$ for the data in the laminar regime and $n=0.2$ for the data in the turbulent regime. The fitted curves are provided in figures 6 and 7 and the calibration coefficients in (2.5) are provided in Table 1; these coefficients were obtained with an $R^2$ value of 0.99 and a confidence level of 95 %.

This calibration channel achieved a maximum pressure difference of 4.95 kPa between adjacent ports with the pump running at maximum capacity, resulting in a maximum shear stress of 12.29 Pa. The range of the calibration channel restricts our results to a single operating point (mass flow rate) because of the large variation in shear stress under the Bernoulli pad.

5.1. Procedure

The STS was placed into a water tank with dimensions of $2120$ $\boldsymbol{\times }$ $749$ $\boldsymbol{\times }$ $762$ mm. The frame suspended the workpiece in the centre of the tank’s footprint and $356$ mm from the bottom. The dimensions of the tank and the location of the workpiece were chosen to ensure that the radial outflow between the pad and workpiece was not affected by the water splashing from the tank walls. Shims were placed between the frame and tank to level the STS before it was secured to the tank with clamps. The pump draws water from beneath the workpiece, and this water is at the ambient temperature ( $17.25^\circ$ C) before data collection. The density and viscosity of water at this temperature are $\rho = 998.87$ kg m $^{-3}$ and $\mu = 0.001073$ Pa $\,$ s.

In this experiment, the gap height between the pad and the workpiece was chosen to be $H = 1.3$ mm – see figure 8 (a). Shims were placed in the gap to ensure that the gap height was uniform; they were removed after ensuring uniform desired gap height. The working mass flow rate was $\dot m = 0.046$ kg s⁻¹. This operating point, defined by a combination of $H$ and $\dot m$ , was chosen to ensure that the maximum shear stress experienced by the hot-film sensor was within our achievable calibration range.Footnote ² At lower flow rates, the relative error at large $r$ increases significantly, along with a tendency of the flow to become visibly non-axisymmetric.

Once the mass flow rate from the pump was at steady state, the linear stage was used to move the Bernoulli pad from its nominal position, and hot-film voltages are measured at each position. At large radii of the Bernoulli pad, we used steps of $5$ mm, which is approximately twice the diameter of the hot-film sensor. The step size was reduced to $1.0$ – $2.0$ mm close to the neck of the Bernoulli pad, where large variations in the shear stress are expected. The total distance of travel was approximately $\ell = 83.0$ mm, which ensured that the sensor was never exposed to air.

The voltage outputs from the hot-film sensor were first corrected for temperature difference between calibration and wall shear stress measurement conditions using (Bruun, Reference Bruun1995)

\begin{equation*} E_{\textrm {corr}} = \left (\frac {T_{\textrm {f}} - T_0}{T_{\textrm {f}} - T_{\textrm {a}}}\right )^{0.5} E_{\textrm {a}} , \end{equation*}

where $E_{\textrm {corr}}$ is the corrected voltage, $T_0 = 25.49^\circ$ C is the water temperature during calibration, $T_{\textrm {f}} = 40^\circ$ C is the film temperature and $T_{\textrm {a}} = 17.25^\circ$ C is the water temperature during data acquisition. This film temperature represents an overheat ratio of 0.1. The temperature difference between calibration and experiment is larger than we would have preferred, but this variable was difficult to control between our set-ups. The corrected voltage values were used to obtain the wall shear stress values using (2.5) with the calibration coefficients provided in Table 1.

5.2. Experimental results and comparison with simulation

Numerical simulations were carried out using the Spalart–Allmaras model based on its suitability for adverse pressure gradients (ANSYS, 2022) and the transition-shear stress transport (SST) model due to its extensive use in our prior work with Bernoulli pads (Kamensky et al. Reference Kamensky, Hellum and Mukherjee2019, Reference Kamensky, Hellum, Mukherjee, Naik and Moisander2020; Kamensky, Reference Kamensky2020). The latter work and the work in Tomar et al. (Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022) present PTV data which validate the computational approach used in those works and the present paper over a large section of the flow field. The approach used there did not have sufficient resolution near the wall to resolve the velocity gradient accurately, which pushed us toward the present approach for measuring the wall shear stress. These simulations were carried out for identical flow domain and boundary conditions in the experiments – see figure 10. The domain is axially symmetric without variations in the azimuthal directions. Therefore, we used a two-dimensional axisymmetric model to reduce computational time. Assuming incompressible flow, we imposed a mass flow rate of $\dot {m} = 0.046$ kg/s at the inlet, and exit pressure $p = p_{\text{atm}}$ at the pad’s outlet. No-slip conditions were imposed on all solid walls. The computational domain is meshed with quadrilateral dominant elements. The conservation equations are solved using Semi-Implicit Method for Pressure Linked Equations (SIMPLE) algorithm and the pressure and momentum terms are discretised using PREssure STaggering Option (PRESTO)! and second-order discretisation, respectively. The convergence is said to be obtained when the residual is less than $1 \times 10^{-6}$ for all the variables.

Figure 10. A schematic of the Bernoulli pad used in simulations.

In our experimental set-up, we were careful to choose materials and surface finishes which will allow us to describe the pad and workpiece as ‘hydraulically smooth’. This condition is met when $Rz/\delta _l \lt 5$ (Nikuradse, Reference Nikuradse1950), where $\delta _l$ is the thickness of the laminar sublayer. The representative thickness of our laminar sublayer is $\delta _l = 9.9$ $\mu$ m based on the maximum measured value of $\tau _{\textrm {w}}$ . Given that $Rz \lt 1.4$ $\mu$ m for polished aluminium and less for cast acrylic, we satisfy the hydraulically smooth descriptor. In the simulation, we fix surface roughness for the pad and workpiece at $\varepsilon _{\text{p}} = \varepsilon _{\text{w}} = 1$ $\mu$ m.

In our experiments, the hot-film sensor voltage is obtained for the measurement region of length $\ell = 83.0$ mm shown in figure 10. A comparison of these voltages with the voltage values in figure 7 indicate that the flow is laminar for $0 \leq r \lt 12$ and $14 \lt r \leq 83$ mm and turbulent for $12 \leq r \leq 14$ mm. The change from laminar to turbulent flow close to the neck of the pad is expected. In this region, the mean velocity is high because of the reduced area relative to the stem. This is amplified by a recirculation region resulting from the sharp corner (Shi & Li, Reference Shi and Li2016) which further reduces the area through which the flow is moving in the $+r$ direction. The wall shear stress and radial distance are non-dimensionalised using the expressions (Tomar et al. Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022)

(5.1) \begin{equation} \bar \tau _{\textrm {w}} = \left (\frac {\pi \rho d^3}{4 \dot m \mu }\right ) \tau _{\textrm {w}}, \qquad \bar r = \left (\frac {2}{D}\right ) . \end{equation}

The variation in the non-dimensional wall shear stress $\overline {\tau }_{\text{w}}$ with non-dimensional radial distance $\overline {r}$ is shown in figure 11. Contour plots of the radial velocity and pressure are presented in figure 12. There is a separation bubble attached to the corner of the pad at $r=d/2$ . This separation bubble reduces the effective cross-sectional area of the flow field under the pad, increasing the local radial velocity and thereby increasing the shear stress on the workpiece – see the inset of figure 11. As the radial flow moves past the separation bubble, there is momentum transferred from the radial direction to the wall-normal direction, yielding a sharp drop in wall shear stress. The momentum recovers its purely radial direction, and since the boundary layer is still very thin, the shear stress increases even though the mean velocity through the cross-section at $r$ is decreasing $\propto r$ . However, further downstream, the decreasing mean velocity yields monotonically decreasing wall shear stress. In order to discuss the differences between experiment and simulation, it is helpful to divide the discussion of the data and simulation results into regions of the wall shear stress field.

Figure 11. Variation of $\bar \tau _{\textrm {w}}$ with $\bar r$ : a comparison of simulation and experimental results. The experimental results, obtained at discrete values of $\bar r$ , are shown using ‘ $*$ ’ marks. The turbulent calibration coefficients in Table 1 are used from $\bar r \in [0.12, 0.14]$ , the laminar values are used in the rest of the domain. The top figure shows the recirculation region and the flow around it (Shi & Li, Reference Shi and Li2016) – see figure inset. A magnified view of the dotted portion of the top figure is shown in the bottom figure for comparison of the two turbulence models with experimental results.

Before reattachment ( $\bar r \leq 0.14$ ): there is good agreement between both models and experiment on the values of peak shear. This also suggests that the models are accurately capturing the velocities and extent of the recirculation region. From an application perspective, this is promising; the cleaning efficacy of a Bernoulli pad is characterised by the maximum shear it produces (Kamensky et al. Reference Kamensky, Hellum and Mukherjee2019), while the deformation and fracture limits on a workpiece are likewise set by the maximum local shear and normal forces. The maximum non-dimensional wall shear stress $\tau _{\textrm {w, max}}$ for all cases, including a power law estimation from Tomar et al. (Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022), is shown in Table 2. In the region where the incoming flow impinges on the wall, the simulations underpredict the experiment. However, this region of the flow is the region which matches the calibration conditions the least well because this is an impingement region, in which the flow is deflected from purely axial to predominantly radial. As opposed to the fully developed boundary layer with zero wall-normal velocity in the calibration channel, this impingement region presents a very strong gradient of wall-normal velocity with a boundary layer that is just starting to develop. In consequence, the correlation between heat dissipation and shear stress would not necessarily follow the same functional form in the impingement region as in the channel.

Fully radial flow ( $ \bar r \geq 0.32$ ): qualitatively, the gradual decay of wall shear in this region is easily explained by the $1/r$ proportional decay in radial velocity as the cross-sectional area of the gap increases with radius (Guo et al. Reference Guo, Shan, Xie, Li, Xu and Zhang2017). While the experiments and the simulations both show this behaviour, the models substantially over-predict the experiments. In this region of the flow, the experiment predicts that the flow is laminar. In addition to the two models presented in figure 11, we also ran several other turbulence models with no transition model whatsoever, and these all showed over-prediction of the wall shear stress in this region. The tendency to over-predict was also observed in our prior work (Tomar et al. Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022), where a numerical model was compared with PTV data. We tentatively suggest that the over-prediction in this region is driven by a failure to accurately characterise the flow in this region as laminar, and that better predictions at large $r$ will require a model tuned to capture this phenomenon.

Near reattachment ( $0.14 \lt \bar r \lt 0.32$ ): in this region, there is substantial difference between each model and experiment. All realisations indicate a local minimum followed by a net rise before $r=0.3$ . The experiment shows an immediate rise to a local maximum near $r=0.2$ . The Spalart–Allmaras model was chosen as a candidate model because it was designed to perform well in wall-bounded flows at moderate to low Reynolds numbers under adverse pressure gradients (ANSYS, 2022). The Spalart–Allmaras model indicates a similar local minimum, slightly downstream of experiment. Likewise, the local maximum is further downstream than observed experimentally, consistent with a separation bubble which is larger in the streamwise direction. The four-equation transition-SST model shows a broad region of near-zero shear stress followed by a rise to $r=0.3$ . The $\tau _w\approx 0$ region is likely an indicator of laminar separation (Tomar et al. Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022), followed by another transition to turbulence. The cited work found transition SST to perform well in a fully separated flow. However, there are empirical correlations used in the development of the transition-SST model (Menter et al. Reference Menter, Langtry, Likki, Suzen, Huang and Völker2006) which were performed on a flat plate. It may be that the current flow field, characterised by sharp radial pressure gradients, presents an out-of-sample challenge for the correlations.

Table 2. Dimensionless maximum wall shear stress values obtained from experiment, numerical models and power law (Tomar et al. Reference Tomar, Kamensky, Mejia-Alvarez, Hellum and Mukherjee2022)

Figure 12. Contour plots of (a) total pressure and (b) radial velocity in the neck region of the computational domain using the Spalart‐Allmaras model.