3.1. Stability of the ferrofluid interface

As a first step in our analysis, we scrutinize the behaviour of the wall coating, with a focus on the dynamics of the interface between ferrofluid and water. In figure 2, we show a snapshot of the interface at different Reynolds numbers and fixed magnetic field intensity ($H=170\,{\rm Gs}$). When the Reynolds number is low (figure 2a), the interface appears rather flat. However, as the Reynolds number is raised, we observe the emergence of an interfacial instability, manifested in the form of travelling waves. These waves resemble those arising from a shear (i.e. Kelvin–Helmholtz) instability (Kögel et al. Reference Kögel, Völkel and Richter2020; Völkel et al. Reference Völkel, Kögel and Richter2020).

Figure 2. Instantaneous snapshot of the ferrofluid interface at $H=170\,{\rm Gs}$ for $Re=2780$, cf. experiment c.1 (a), $Re=7740$, c.3 (b) and $Re=12\,660$, c.5 (c).

To characterize the waves, we conduct an image postprocessing analysis as follows. The interface height is denoted as $h(x,t)$, where $x$ represents the streamwise coordinate and $t$ denotes the time. The interface elevation relative to its mean is $\zeta (x,t) = h(x,t) - \overline {h(x)}$, where $\overline {h(x)}$ is the time-averaged height of the interface. To characterize the unstable waves, we identify the zero upcrossings of the time signal $\zeta (x_{0},t)$ at various locations $x=x_{0}$ (10 points along the horizontal coordinate with equal spacing of $\approx 5.5\,{\rm mm}$). We then calculate the average wave amplitude, $a$, defined as the difference between the maximum and minimum of $\zeta (x_{0},t)$ between two adjacent zero upcrossings, and then average this value over all events and $x_{0}$ locations. Similarly, the wave period $T$ is computed as the average time span between zero upcrossings, and the wave frequency $\omega$ is defined as $\omega = 1/T$. Finally, to calculate the average wavelength $L$, we calculate the distance between the zero upcrossings of the spatial signal $\zeta (x,t_{0})$ at fixed times $t_{0}$ and average this measurement over all instances.

A synthesis of average wave characteristics across all experiments is outlined in table 2.

Table 2. Overview of wave characteristics, where $a$ is the wave amplitude, $L$ the wavelength and $\omega$ the wave time frequency. Letters ranging from ‘a’ to ‘d’ on the label represent the magnetic field intensity ($H$), while the numbers correspond to the Reynolds number. The colour-code depicts the magnetic field intensity while the symbol size varies with $Re$ number.

To contextualize the relative influence of the governing parameters, i.e. average water flow velocity $U$ and magnetic field intensity, we employ linear stability analysis. To this end, we compute the critical velocity curves, denoted as $U_{crit}$, which represent the average water velocity threshold for the onset of unstable travelling waves. These critical velocity curves are determined as a function of the magnetic field intensity $H$ and the wavenumber $k$ (Kögel et al. Reference Kögel, Völkel and Richter2020; Völkel et al. Reference Völkel, Kögel and Richter2020), defined as

(3.1)\begin{equation} U_{crit}(k,H) = \sqrt{\frac{\rho_{f}+\rho_{w}}{\rho_{f}\rho_{w}}\left( k \gamma+\tilde{\mu}H^{2}+\frac{g(\rho_{f}-\rho_{w})}{k} \right)}, \end{equation}

where $\rho _{f}$ is the ferrofluid density, $\rho _{w}$ is the ambient fluid density, $\gamma$ is the interface tension between the two fluids and $\tilde {\mu } = \mu _{0} \chi ^2 /(\chi +2)$ is proportional to the vacuum permeability $\mu _{0}$ and depends on the susceptibility $\chi$ (the ferrofluid properties are given in table 3). Figure 3 displays the critical velocity neutral curves for our experiments, along with the measured values of $U$ (the bulk average flow velocity) and $k=2{\rm \pi} /L$ (the average wavenumber of the interface). The neutral curves separate stable (below) from unstable (above) behaviour. The increase in the intensity of the magnetic field $H$ is a stabilizing factor for the ferrofluid interface (shifting the neutral curve upwards). Conversely, an increase in the average flow velocity of the ambient fluid is destabilizing. Experiments with a given magnetic field intensity show that as $Re$ (represented by colour-matched dots, ranging from larger to smaller) increases, there is a decrease in wavenumber ($k$). Remarkably, when the magnetic field is at its strongest, $H=460\,{\rm Gs}$, only one flow case exhibits instability ($Re=12\,380$, i.e. experiment a.5 in table 1). However, with a magnetic field intensity of $H=170\,{\rm Gs}$, nearly all flow cases exhibit instability, except for one experiment (experiment c.1 in table 1 corresponding to $Re=2780$). This is qualitatively consistent with figure 2 where the interface is rather smooth for c.1, whereas for c.3 and c.5 it is corrugated.

Table 3. Properties of ferrofluid EFH1 (from Ferrotec USA corporate).

Figure 3. Stability maps for the experiments with $H=460\,{\rm Gs}$ (a), $H=250\,{\rm Gs}$ (b), $H=170\,{\rm Gs}$ (c) and $H=140\,{\rm Gs}$ (d). The colour-coding and bullet size are as specified in table 1. Empty symbols denote experiments featuring a stable interface, while filled symbols indicate cases characterized by an unstable interface.

To statistically characterize the wave frequency spectra, we conduct a Fourier analysis of the interface elevation. This involves computing the temporal power spectral density of the interface elevation, $\zeta (x,t)$. The power spectra are then averaged over all locations to obtain a single average temporal power spectrum density for each experiment. The power spectrum computation utilizes Welch's method through the ‘pwhelch’ function in MATLAB.

Figure 4 displays the temporal power spectrum density of the interface elevation for the experiments detailed in table 1. In this figure, the Reynolds number increases from left to right, while the magnetic field intensity ($H$) rises from the bottom to the top. Focusing on the first column (a.1–d.1), which pertains to experiments with the lowest Reynolds numbers, we observe that the spectra exhibit maxima near the origin of the abscissa, followed by a rapid decay as the frequency (${\kern0.8pt}f$) increases. This suggests that the interface remains mostly flat in these experiments.

Figure 4. Temporal power spectra of the interface elevation $\zeta$. The rows correspond to different magnetic field intensities, while the columns from left to right correspond to increasing Reynolds numbers.

Let us now consider the scenario with $H=460\,{\rm Gs}$ (figure 4 a.1–a.5). As the Reynolds number increases, the power spectra begin to fill the range of 0–100 Hz, with no distinct peak. That is, a range of waves starts to emerge without a dominant frequency. As the magnetic field decreases, a clear peak becomes evident in the power spectra at higher Reynolds numbers (as seen in, for instance, b.5, c.5, d.3). The prominence of this peak intensifies as the flow cases move upward from the corresponding instability curves depicted in figure 3. The presence of a distinct peak suggests the occurrence of monochromatic cylindrical waves at the interface (refer to cases c.4, c.5, d.3). In instances of multiple frequencies, the interface is characterized by a train of waves exhibiting different harmonics (see case a.5), indicating irregular wave patterns. Finally, increasing the magnetic field leads to a higher frequency manifestation of instability ($H$ acting as a stabilizing factor), indicated by a rightward shift in the peak (observe cases c.5 and b.5).

To characterize the spatio-temporal wave dynamics, in figure 5 we present space–time diagrams of the interface elevation $\zeta$. Observing figure 5, it is noticeable that the interface displays a relatively flat profile when the Reynolds numbers are low and the magnetic field intensities are high, as depicted in the panels of figure 5(a.1,a.2) and (b.1,b.2). However, as the Reynolds number increases and the magnetic field intensity decreases, clear wave crest–trough patterns become evident.

Figure 5. Space–time diagrams of the interface elevation $\zeta$ for the same experiments shown in figure 4.

Additionally, in scenarios featuring intermediate Reynolds numbers and higher magnetic field strengths, exemplified by figure 5, e.g. panel (a.3), the interface maintains a relatively flat profile. In contrast, when considering the same Reynolds number but with lower magnetic field intensity, as exemplified in figure 5(d.3), significantly elevated interface profiles become apparent. It is worth noting that for flow cases displaying distinct peaks in the power spectrum diagrams (e.g. figures 4c.4,c.5 or 4d.3), figure 5 shows a regular wave train (monochromatic waves). In cases dominated by a spectrum of frequencies, as seen in figure 4(a.5) numerous harmonics emerge (colours alternate irregularly without a discernible trend) in the space–time diagrams. This observation corroborates our findings from the Fourier space analysis. Furthermore, it is important to highlight that these observations align well with the theoretical predictions shown in figure 3.

3.2. Impact of ferrofluid interface stability on the flow

To correlate our observations regarding the characterization of the interface (§ 3.1) with drag reduction, we employ the velocity measurements obtained from 2D-PTV. For clarity, we focus on three distinct flow cases, namely a.5, b.5 and c.5 detailed in table 1, which correspond to the scenarios with the highest Reynolds number ($Re \approx 12\,500$) under varying magnetic field intensities: $H=460\,{\rm Gs}$, $H=250\,{\rm Gs}$ and $H=170\,{\rm Gs}$.

In the subsequent part of this section, we normalize the vertical coordinate (${\kern0.8pt}y$) using the channel fluid height ($D-h$) and mirror the bottom half of the velocity profiles, which is bounded by the uncoated wall (figures 6–8, black lines), to ease the comparison with the upper half, which is bounded by the coated wall (figures 6–8, coloured lines). As a result, the origin $y=0$ corresponds to both the ferrofluid interface (figures 6–8, coloured lines) and the rigid-wall location (figures 6–8, black lines).

Figure 6. Mean streamwise velocity profiles of a.5, b.5 and c.5 (columns) in large scale units (a–c), wall units (d–f) and wall units with the slip velocity subtracted (g–i). For the colour of the lines see table 1. The corresponding rigid-wall profiles are in black. Inset (g–i): zoom of the rigid wall.

Figure 6(a–c) illustrates the mean streamwise velocity profiles. We note a distinct slip velocity $U_{slip}$ at the ferrofluid interface for all three experiments, contrasting with the classical no-slip condition observed at the rigid wall. This slip condition at the interface, in conjunction with the zero net mass flow of the ferrofluid (§ 2.2), implies a recirculation within the ferrofluid layer, as shown in Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024) and illustrated schematically in figure 1. We also note differences away from the origin implying an asymmetry of the velocity profile in the bulk. While for the strongest magnetic field (figure 6a) the average velocity is always higher on the coated side with respect to the rigid-wall side, for the lowest magnetic field intensity experiments (figure 6c) the flow is generally faster in the lower half of the channel. In the case of $H=250\,{\rm Gs}$ the ‘no-slip’ flow is locally faster at $y/(D-h)\approx 0.05$. This difference in velocity profiles might be associated with a difference of turbulence intensity in the interface proximity (see below).

Figure 6(d–f) shows the average velocity profiles in wall units $y^+ = y u_{\tau }/ \nu$ and $U^+ = U/u_{\tau }$, with $u_{\tau } = (\tau _{0} / \rho )^{1/2}$ where $\tau _{0}$ is the space and time averaged wall shear stress. For the rigid-wall side, the wall shear stress is $\tau _{0} = \tau _{w} = \mu \,{\rm d}U/{{\rm d} y}|_{wall}$ evaluated at the wall location, while for the coated wall $\tau _{0}$ is evaluated as the total shear stress $\tau _{0} = \tau _{c} = \mu \,{\rm d}U/{{\rm d} y} - \rho \langle u'v'\rangle$ at the interface location, where $\rho$ is the water density, $u'$ and $v'$ velocity fluctuations in the streamwise and wall-normal directions. Here, $\langle {\cdot }\rangle$ denotes averaging over time and the streamwise direction $x$, given the statistical stationarity and homogeneity of the flow in $x$. Also depicted in figure 6 is the theoretical law of the wall, comprising the linear law $u^+ = y^+$ of the viscous sublayer and the logarithmic law $u^+ = 1/\kappa \ln (y^+) + C^+$ characterizing the log-law region (with $\kappa =0.4$ and $C^+=5$). As is evident from figure 6(d–f), rigid-wall profile cases are in good agreement with both laws (see also inset in figure 6g–i). Although the presence of the ferrofluid layer makes the velocity profile slightly asymmetric in the bulk (figure 6a–c), this asymmetry is not noticeable in the near-wall region. For the coated-wall profiles, a markedly distinct behaviour is observed. In experiments a.5 and b.5 (figure 6d,e), there is a significant deviation between the experimental velocity profiles and the law of the wall, occurring in both the viscous sublayer and the log-region. In contrast, experiment c.5 (figure 6f) is closer to the no-slip profile. Generally, an upward shift in the velocity profile indicates drag reduction. As reported in Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024) concerning an undulating wall with slip velocity, the departure from the law of the wall within the viscous sublayer primarily results from the slip, while the deviation in the log-layer (${\kern0.8pt}y^+>30$) is linked to interface waviness. To disentangle the two contributions, in figure 6(g–i), we present the same average velocity profiles in wall units after subtracting the contribution of the slip velocity, denoted as $U_{slip}$. A significant difference persists for flow cases a.5 and b.5 (figure 6g,h), particularly evident for $y^+>30$, whereas the profile of case c.5 nearly conforms to the no-slip profile (figure 6i). This implies that a significant part of the deviation for the c.5 case can be attributed to the slip velocity, while for the other two cases, the primary influence stems from interface waviness.

From figure 6(a–c), we noted that the ferrofluid layer introduces a slip condition at the upper wall of the channel, affecting the shear rate in the channel flow. In figure 7, we further explore the influence of the ferrofluid layer on both viscous and turbulent shear rate profiles. In figure 7(a–c), we present the profiles of the viscous shear rate $\mu \,{\rm d}U/{{\rm d} y}$, where $\mu$ is the dynamic viscosity of the water and $U$ the space ($x$ direction) and time average velocity profile. As expected, the profiles for the rigid wall exhibit a peak precisely at the wall location. Conversely, a distinct behaviour is observed for the coated wall. Close to the interface (at approximately $y/(D-h) = 0.02$), $\mu \,{\rm d}U/{{\rm d} y}$ reaches its maximum value, and notably decreases as it approaches the interface due to the slip condition, resulting in a significantly lower value at the interface. This reduction is connected to the change in the gradient of the velocity profile as observed in figure 6, which may be associated with the presence of waves with a moderate amplitude, inducing higher turbulent stresses near the interface as compared with stronger magnetic field cases.

Figure 7. Mean shear rate profiles (a–c) of a.5, b.5 and c.5 (columns) experiments, mean Reynolds shear stress profiles (d–f) and mean Reynolds shear stress profiles in wall units (g–i). For the colour of the lines see table 1. The corresponding rigid-wall profiles are in black.

In figure 7(d–f), we present the profiles of the turbulent shear stress $-\rho \langle u'v'\rangle$. It is evident that, in the case of the most intense magnetic fields (d–f), the turbulent shear stress is comparable to that observed on the rigid-wall side. In contrast, for lower magnetic intensities, $-\rho \langle u'v'\rangle$ is higher especially around the peak location $y/(D-h)\approx 0.08$. This implies that additional velocity fluctuations are caused by waves, which were observed to have a higher amplitude $a$ (at $Re\approx {\rm const.}$) for lower magnetic fields (§ 3.1).

In figure 7(g–i), we present the Reynolds shear stresses in wall units. Remarkably, for cases with the most intense magnetic field, the peak is closer to the coated wall as compared with the rigid wall (figure 7g–h). A different behaviour is evident in cases of lower magnetic field (figure 7i), where the peak of $-\rho \langle u'v'\rangle$ aligns with the rigid-wall case. Interestingly, in this case, $-\rho \langle u'v'\rangle$ remains high and positive in the bulk. This sustained non-zero $-\rho \langle u'v'\rangle$ value evidences a detrimental impact of the interface instability, i.e. interface waviness causes an additional turbulent shear stress in close proximity to the coated wall.

To further characterize turbulence, in figure 8(a–c), we depict the profile of turbulent kinetic energy ($tke=\langle u'^2 + v'^2\rangle$). Note that the full $tke$ also entails $w'^2$ ($w'$ is the velocity fluctuation in the span-wise direction), which, however, was not accessible in our measurements. Notably, $tke$ tends to be smaller in flow cases with higher $H$ (figure 8a,b) as compared with rigid-wall side, while higher values can be observed for lower magnetic field intensity (figure 8c). To analyse the individual components, we present in figure 8(d–i) the contributions of the streamwise and wall-normal components. The presence of the coating consistently diminishes the peak value of the streamwise component. However, in the case of lower magnetic field intensity, the wall-normal component experiences a significant increase. This is consistent with the increase in wave amplitude with decreasing $H$, as qualitatively noted in figure 5. In essence, the coating proves instrumental in reducing turbulent kinetic intensity up to a point where the waves generated at the interface reach an amplitude substantial enough to markedly elevate the wall-normal velocity fluctuation components.

Figure 8. Turbulent kinetic energy profiles (a–c) of a.5, b.5 and c.5 experiments (columns), streamwise velocity (d–f) and wall-normal (g–i) component contribution. For the colour of the lines see table 1. The corresponding rigid-wall profiles are in black.

3.3. Drag reduction efficiency in relation to ferrofluid interface stability

We calculate the Fanning friction factor, denoted as $f = {2\tau _{0}}/({\rho _w U^2})$, for both the bottom rigid wall and the interface, where $\rho _{w}$ is the water density. These results are illustrated in figure 9(a). The figure also includes the theoretical solutions for laminar flow, represented by $f = {c}/{Re}$ with $c = 14.2$ (Hartnett, Kwack & Rao Reference Hartnett, Kwack and Rao1986; Kakaç, Shah & Aung Reference Kakaç, Shah and Aung1987), and turbulent flows, depicted as $f = 0.079(c_{d} {\cdot } Re)^{-1/4}$ with $c_{d} = 1.125$ (Jones Reference Jones1976; Choi & Cho Reference Choi and Cho1999). The figure incorporates friction factor data (depicted in grey) from Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024) in which the authors used a similar set-up to the present one. In those experiments, the channel had smaller dimensions, measuring $10\times 10 \,{\rm mm}^2$ in cross-section, while an array of magnets directly affixed to the upper wall of the channel generated a magnetic field intensity of approximately $H=90\,{\rm Gs}$ at the ferrofluid interface. Their experimental conditions encompassed both stable and unstable ferrofluid interface scenarios. However, due to the smaller cross-section in their set-up, instability in the ferrofluid interface occurred at relatively low $Re$ numbers ($Re<4000$) compared with our current configuration, given the significantly higher bulk average velocity $U$ at the same $Re$.

Figure 9. (a) Friction factor $f$ against the Reynolds number $Re$. The filled and the open symbols represent, respectively, the results for the top and the bottom of the channel. For the colours of the symbols, refer to table 1. Grey symbols represent data from Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024). Dash-dotted line the solution for turbulent flows, while the dashed line is the solution for laminar flows. (b) Difference between the experimental velocity profile and the theoretical log-law ${\rm d}U_{50}^+$ at $y^+=50$, plotted against the normalized wave amplitude $a$ with respect to the viscous length scale $y^+_{5}$. (c) Ratio between the slip velocity $U_{slip}$ and the bulk average velocity $U_{av}$ against the Reynolds number $Re$.

From figure 9(a), it is evident that the friction factor values for the bottom wall (open symbols) align closely with the turbulent curve. In contrast, the majority of the values at the top interface are significantly below the turbulent curve, in some cases even below the laminar curve. That is, the drag experienced at the interface between the water and ferrofluid is generally much lower compared with that at the bottom of the channel.

Upon closer examination of the coated-wall cases, figure 9(a) illustrates a consistent trend where the friction factor decreases with increasing Reynolds number ($Re$), particularly evident in experiments conducted with higher magnetic field intensities, namely $H=460\,{\rm Gs}$ and $H=250\,{\rm Gs}$. This holds true also for experiments with lower magnetic field intensities ($H=170\,{\rm Gs}$ and $H=140\,{\rm Gs}$), except for cases at higher Reynolds number where an inversion in this trend occurs (refer to experiments c.4, c.5, and d.3). In these cases, the friction factor increases with $Re$ at a given $H$, albeit remaining below the values observed for a rigid wall. A noteworthy characteristic of experiments c.4, c.5 and d.3 is the high wave amplitudes formed at the interface of the ferrofluid layer compared with other experiments (refer to figure 5). This may be linked to an increase of the turbulence intensity in the interface proximity due to additional velocity fluctuations connected to the interface instability. Generally, both the slip condition and interface waves contribute to drag reduction. However, when the waves reach high amplitudes, they appear to transition from being advantageous to becoming detrimental. This notion gains additional support through a direct comparison of our experimental data with those presented by Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024) (depicted as grey symbols in figure 9a). Although there is a general consistency in trends between the two datasets, Stancanelli et al. (Reference Stancanelli, Secchi and Holzner2024) noted smaller friction factors at smaller Reynolds numbers ($2000< Re<4000$). This difference could stem from the noted earlier instability of the interface at these Reynolds numbers in their experiments, which augmented drag reduction. In contrast, in the current scenario, the interface remains flat at these Reynolds numbers, with the sole contributing factor being the slip condition.

To analyse a possible relationship between drag reduction and the kinematics of the interface, we illustrate in figure 9(b) the disparity between the experimental velocity profile with subtracted slip velocity (see e.g. figure 6g–i) and the log-law ${\rm d}U_{50}^+$ at $y^+=50$, plotted against the wave amplitude $a$ normalized by a viscous length scale, namely 5 wall units (${\kern0.8pt}y^+_{5}$). The choice of this viscous length scale is motivated by the fact that the viscous shear stress dominates near the ferrofluid interface (see e.g. figures 8a,b and 8d,e), i.e. even though there is a slip condition there is a viscous layer at the interface in analogy to the viscous sublayer near the solid wall. In figure 9(b), ${\rm d}U_{50}^+$ quantifies the effect of the wall waviness on the shear stress. That is, high positive values indicate a lower shear stress at the interface location, signifying drag reduction. Across all experiments, an increase in wave amplitudes corresponds to an increase in ${\rm d}U_{50}^+$ as long as $a$ remains small compared with the viscous length scale. However, when the wave amplitudes approach the viscous length scale (experiments c.4, c.5 and d.3), the advantage derived from surface waviness diminishes significantly. This observation further substantiates our hypothesis that the positive effect of the waves on drag reduction diminishes with wave amplitude and may even become detrimental.

Finally, to complete our analysis of the observed frictional behaviour at the coated wall, we present in figure 9(c) the percentage of slip velocity ($U_{slip}$) relative to the average bulk velocity ($U_{av}$). Notably, $U_{slip}/U_{av}$ exhibits nearly linear growth with increasing Reynolds numbers in the range covered by our experiments. This is consistent with the Reynolds number trend observed in the friction factor (figure 9a), where, in general, there is a decrease in $f$ with $Re$, except for cases c.4, c.5 and d.3. Nonetheless, these latter cases are characterized by an elevated slip velocity (figure 9c), mitigating to some degree the effects of extra velocity fluctuations brought about by the ferrofluid interface waves.

A final element in assessing the effectiveness of the drag reduction technique is the so-called drag reduction coefficient. In figure 10, this coefficient, expressed as $DR(\%) = (\tau _{w} - \tau _{c})/\tau _{w}{\cdot } 100$, is depicted against the Reynolds number $Re$, with $\tau _{w}$ denoting the wall shear stress at the rigid wall and $\tau _{c}$ representing the corresponding value at the ferrofluid interface. It is evident that, in general, drag reduction tends to increase with the Reynolds number for a given $H$, except in cases where the interface wave amplitude is particularly elevated compared with the viscous length scale $y^+_{5}$ (flow cases c.4, c.5 and c.5). It is, however, noteworthy that remarkably high values of $DR$, up to $95\,\%$, can be achieved, exemplified in scenarios characterized by $Re \approx 12\,500$ and $H=460$ or $H=250$.

Figure 10. Drag reduction $DR(\%)$ against the Reynolds number $Re$. The colour-coding and symbol size are as specified in table 1. Not shown in the figure, experiment c.1 in which the drag coefficient is slightly negative (drag increase).