2.1 Channel fabrication and characterisation

The channel was fabricated using polyacrylamide gel, as explained in appendix A. Three different compositions were used, resulting in gels with shear moduli 0.75 kPa, 2.19 kPa and 15.89 kPa. The gel with shear modulus 15.89 kPa is used as the hard-walled channel, since the shear modulus is sufficiently large that the soft-wall transition Reynolds number is higher than the maximum of approximately 3500 in the experiments.

Figure 1. The schematic (not to scale) of the top view (a), the cross-section of the undeformed channel (b) and the channel deformed due to a pressure gradient (c).

The channels are fabricated as a rectangular bore in a block of polyacrylamide gel with dimensions shown in figure 1. The channels consist of an upstream hard (development) section of length approximately 13 cm, made with gel of shear modulus 15.89 kPa, to damp out disturbances at the inlet. This is followed by a downstream soft section of length approximately 14 cm, where the gel has a lower elasticity modulus due to lower cross-linker concentration. The width of the channel is approximately 1.3 cm, while channels with two different heights, about 0.6 mm and 1.8 mm are fabricated. When there is a pressure difference applied across the channel, the walls deform in the test section, as shown in figure 1(c). The channel deformation is discussed in detail in appendix C.

2.2 Experimental configuration

The configuration and coordinate system used for analysing the flow are shown in figure 2. The channel is mounted on an optical breadboard with translation stages along the $x$ (flow) and $z$ (spanwise) directions, and goniometers with axes along the $x$ and $z$ axes. The inlet to the development section is connected to a pressurised tank through a needle valve for controlling the flow rate. The velocity profiles are measured using particle image velocimetry at the entrance to the test section, and it has been verified that the profiles are parabolic, as shown in appendix B.

The Reynolds number is defined based on the flow rate and the channel width,

(2.1) $$\begin{eqnarray}Re=\frac{\unicode[STIX]{x1D70C}Q}{W\unicode[STIX]{x1D702}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D702}$ are the fluid density and viscosity, $Q$ is the flow rate and $W$ is the channel width. For a channel of rectangular cross-section the above definition reduces to that based on the average velocity and channel height. However, the definition in (2.1) is advantageous because it is independent of height, and it does not change even when there is a downstream variation in height caused by the applied pressure.

Two different configurations, shown in figure 2, are used in the experiments. In the ‘symmetric’ configuration (figure 2 a), both the top and bottom walls of the channel are fixed to glass plates. In the ‘asymmetric’ configuration  (figure 2 b), the bottom wall is fixed to a glass substrate, while the top wall is unrestrained. In both cases, the soft-wall transition is observed at the same Reynolds number, to within experimental resolution. However, the ‘wall-flutter’ transition is not observed for the symmetric configuration where both walls are restrained; it is only observed for the asymmetric configuration where the top wall is unrestrained. Therefore, we report results only for the asymmetric configuration.

Figure 2. The configuration and coordinate system used for the particle image velocimetry measurements and the normal wall displacements (a) and the tangential wall displacements (b). Panel (a) also shows the symmetric configuration with a top glass plate, and panel (b) also shows the asymmetric configuration with an unrestrained top wall.

2.3 Fluid velocity measurements

The configuration shown in figure 2(a) is used to make particle image velocimetry (PIV) measurements using an IDT PIV system with a framing rate of 15 Hz. The laser sheet is directed along the centreline of the channel in the spanwise $z$ direction (vertical line in the cross-section figure 1 c). Glass beads with diameters in the range $8{-}14~\unicode[STIX]{x03BC}\text{m}$ were used for seeding the flow.

The PIV measurements are carried out at four different downstream locations, I, II, III and IV, shown in figure 1(a). There is very little variation in the velocity statistics between downstream locations III and IV. The PIV measurements for both laminar and turbulent flows are validated in appendix B.

For each channel configuration (wall shear modulus and height) and at each Reynolds number, two independent sequences of 2000 pairs of images each are recorded. These are separated into eight subsequences of 500 image pairs each. The mean value of a velocity moment is the average over the entire 4000 pairs of images. Eight averages over subsequences of 500 image pairs each are determined, and the standard deviation is calculated for these eight measurements. The error bars in the experimental results show one standard deviation above and below the mean value.

It is important to note that due to the size of the particles, it is not possible to determine the velocity within a distance of approximately $15~\unicode[STIX]{x03BC}\text{m}$ from the wall. Therefore, in most of the profiles for the fluctuating velocities and Reynolds stress, the results are not shown within a distance of approximately $20~\unicode[STIX]{x03BC}\text{m}$ from the wall. In cases such as the mean velocity profile, where the results are shown at the wall, these results are extrapolated.

The particle size is approximately 3 % of the channel height for the smaller channel and approximately 1 % of the channel height for the larger channel. The uncertainties in the fluid velocity measurements due to the finite size of the particles can be estimated as follows.

1. (i) The particle size, approximately $10~\unicode[STIX]{x03BC}\text{m}$ , is much smaller than the two channel heights of 0.6 mm and 1.8 mm considered here. However, the smallest scales in a turbulent flow could be much smaller than the channel height. If we consider the Kolmogorov estimate for the smallest length scale as $Re^{-3/4}h$ , the smallest scale in a channel of height 0.6 mm at a maximum Reynolds number of approximately 900 is approximately $3.6~\unicode[STIX]{x03BC}\text{m}$ , and for a channel of height 1.8 mm at a maximum Reynolds number of 2500 is approximately $5.1~\unicode[STIX]{x03BC}\text{m}$ . (The Kolmogorov estimate is an underestimate for the low Reynolds numbers used here.) Thus, the particle diameter is 2–3 times larger than the smallest scales. Although the particle trajectories may not accurately capture the smallest scales, they most certainly capture the large-scale structures in the flow.

2. (ii) The effect of fluid inertia on the particle trajectories can be estimated from the particle Reynolds number based on the fluid fluctuating velocity, $(\unicode[STIX]{x1D70C}d_{p}v^{\prime }/\unicode[STIX]{x1D702})$ where $d_{p}$ is the particle diameter and $v^{\prime }$ is the fluid fluctuating velocity. (Here, the fluctuating velocity has been used as the basis for calculating inertial effects, because the particle mean velocity relaxes to the local fluid mean velocity within a time period estimated in the item (iii).) For the channel with height about 0.6 mm, the maximum of the streamwise fluctuating velocity are about $0.07~\text{m}~\text{s}^{-1}$ at the soft-wall transition and $0.3~\text{m}~\text{s}^{-1}$ at the wall-flutter transition, and the corresponding Reynolds numbers are 0.7 and 3 respectively. For the channel with height approximately 1.8 mm, the maximum of the streamwise fluctuating velocity is approximately 0.06 at the soft-wall transition and approximately 0.15 at the wall-flutter transition, and the corresponding Reynolds numbers are approximately 0.6 and 1.5 for $10~\unicode[STIX]{x03BC}\text{m}$ particles. When the particle Reynolds number is less than 3, there is no separation or formation of closed streamlines at the rear of the sphere, and the drag force exceeds the Stokes drag law by approximately 20 %.

3. (iii) The fidelity with which the particles follow streamlines depends on the particle Stokes number, which is the ratio of the viscous relaxation time of the particle and the fluid integral time scale. Based on Stokes drag law, the viscous relaxation time of the particle, $\unicode[STIX]{x1D70F}_{v}=(m/3\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}d_{p})$ , is $4.3\times 10^{-5}~\text{s}$ for glass spheres with density $2500~\text{kg}~\text{m}^{-3}$ and diameter $10~\unicode[STIX]{x03BC}\text{m}$ . The fluid integral time scales as the inverse of the macroscopic strain rate, $(h/\bar{v}_{x})$ , where $h$ is the channel height and $\bar{v}_{x}$ is the mean velocity. Based on the maximum velocity of $2~\text{m}~\text{s}^{-1}$ for the channel of height 0.6 mm and $1.4~\text{m}~\text{s}^{-1}$ for the channel of height 1.8 mm, the fluid time scales for the largest eddies are $\unicode[STIX]{x1D70F}_{I}=3\times 10^{-4}~\text{s}$ (approximately $7\unicode[STIX]{x1D70F}_{p}$ ) for the smaller channel and $1.3\times 10^{-3}$ s (approximately $30\unicode[STIX]{x1D70F}_{p}$ ) for the larger channel. Both of these are much larger than the particle relaxation time. The turnover times for the smallest eddies in the flow are, of course, much smaller than those for the large-scale structures. If we use the Kolmogorov estimate, the time scales for the smallest eddies are $Re^{-1/2}$ smaller than the large-scale flow. (The Kolmogorov time scale underpredicts the smallest relaxation time for low Reynolds numbers up to approximately 2500 used here.) For a the channels of height 0.6 mm, the maximum Reynolds number is approximately 900, and so the smallest eddy turnover time is about 30 times smaller than that for the largest eddies, which is about $0.25\unicode[STIX]{x1D70F}_{p}$ . For the larger channels of height 1.8 mm, the maximum Reynolds number is approximately 2500, and so the smallest eddy turnover time is about 50 times smaller than that for the largest eddies, which is about $0.6\unicode[STIX]{x1D70F}_{p}$ . Thus, the particle relaxation time is somewhat larger than the Kolmogorov estimate of the smallest eddy turnover time by a factor of 2–4, but is certainly much smaller than the turnover time of the largest eddies.

4. (iv) Close to the wall, a lift force could be generated due to the particle rotation and the particle motion relative to the wall. The Magnus lift force on the particles scales as $C_{L}((\unicode[STIX]{x03C0}/8)\unicode[STIX]{x1D70C}v_{p}(\bar{v}_{x}d_{p}^{2}))$ , where $v_{p}$ is the particle velocity relative to the fluid, $\bar{v}_{x}d_{p}^{2}$ is a measure of the circulation of the fluid around the particle due to the mean flow and $C_{L}$ is the lift coefficient. If we consider the fluid fluctuating velocity $v^{\prime }$ as the characteristic particle velocity, the lift force is $C_{L}((\unicode[STIX]{x03C0}/8)\unicode[STIX]{x1D70C}\bar{v}_{x}v^{\prime }d_{p}^{2})$ . The ratio of the lift and the drag force $3\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}d_{p}v^{\prime }$ is, $(C_{L}\unicode[STIX]{x1D70C}\bar{v}_{x}d_{p}^{2}/24\unicode[STIX]{x1D707}h)\sim (C_{L}Re(d_{p}/h)^{2}/24)$ , where $Re$ is the Reynolds number based on the mean velocity. Zang, Balachandar & Fisher (Reference Zang, Balachandar and Fisher2005) report that $C_{L}$ is in the range $0.1{-}1.1$ when the distance of the particle centre from the wall is $0.75d_{p}$ to $d_{p}$ , and when the particle Reynolds number is less than 10. For the channel with height 0.6 mm and a maximum Reynolds number of approximately 900, this ratio of the lift and drag forces is approximately 0.01, while for the channel with height approximately 1.8 mm at a maximum Reynolds number of approximately 2500, this ratio is approximately 0.003. Thus, the lift force is likely to be much smaller than the correction to the drag force due to inertial effects.

2.4 Wall displacement measurements

The normal and tangential wall displacement in the soft wall were also measured by embedding glass beads of diameter $8{-}14~\unicode[STIX]{x03BC}\text{m}$ within the wall during the fabrication process. The glass beads were mixed into the prepolymer before gelation, and they were rigidly fixed in the solid due to the formation of cross-links in the polymer after gelation. A camera connected to a zoom tube was used to record the displacement of a bead close to the surface. The displacement was determined from the correlation peak in the Fourier transform of the grey scale intensity matrix, as discussed in Srinivas & Kumaran (Reference Srinivas and Kumaran2015). In order to determine the tangential displacement of the wall in the $x$ and $z$ directions, the configuration in figure 2(b) was used, and images of the beads at both the top and bottom surfaces were captured. Due to the clear line of view from above, relatively high magnification was achieved for the tangential displacement of the surface – an area of dimension 0.375 mm $\times$ 0.222 mm was imaged with a resolution of 804 $\times$ 484 pixels, resulting in each pixel occupying a linear dimension of approximately $0.466~\unicode[STIX]{x03BC}\text{m}$ . For the normal wall displacement in the $y$ direction, the configuration in figure 2(a) was used, where images of an embedded bead is captured from the side. In this case, the magnification achieved was relatively less due to the relatively large depth of view required, as indicated in the cross-section in figure 1(b,c). An area of 7 mm $\times$ 4.24 mm was imaged with a resolution of 804 $\times$ 484 pixel, resulting in each pixel occupying a linear dimension of approximately $8.7~\unicode[STIX]{x03BC}\text{m}$ .

The spectra of the displacement fields were determined by taking the Fourier transform of the time series of the displacement data,

(2.2) $$\begin{eqnarray}\displaystyle \tilde{\star }(\unicode[STIX]{x1D714})=\frac{1}{T}\int _{0}^{T}\,\text{d}t\exp (\imath \unicode[STIX]{x1D714}t)\star (t), & & \displaystyle\end{eqnarray}$$

where $\star (t)$ is the time series of the relevant component of the displacement of a bead embedded in the wall. In our experiments, the maximum framing rate is 1000 Hz, and consequently the Nyquist frequency is 500 Hz. Since oversampling by a factor of 5–10 is required for accurately capturing the spectra, the maximum frequency reported here is approximately $500~\text{rad}~\text{s}^{-1}$ . The time period $T$ used was usually 1 s, although longer sequences of up to 15 s were used to verify that there are no systematic low-frequency signals. The frequency spectrum is calculated only for the tangential displacement recorded from above as shown in figure 1(b), because the resolution is much higher than that for the normal displacement recorded from the side as shown in figure 1(a).

For the wall displacement measurements, three sequences of 1000 frames (1 second) each are recorded. The average and the root mean square of the displacement are calculated as averages over all three sequences (3000 frames). The averages for each subsequence of 1000 frames is also calculated, and the standard deviation is determined using the averages for the three subsequences.

3.1 Soft-wall transition

3.1.1 Flow characteristics

The mean velocity profiles across the channel with walls made of shear modulus 0.75 kPa at the downstream location III (figure 1 a) are shown in figure 3. The bottom wall of the channel is at $y=0$ , while the location of the top wall is shown by the dashed vertical lines on the right in figure 3. The height increase is quite small at the downstream location III at Reynolds numbers up to 500 – the height increases by only 8 % when the Reynolds number increases from 278 to 448. However, the qualitative features reported here are also observed at the other locations.

Figure 3. The cross-stream variation of $\bar{v}_{x}$ (a); $v_{x}^{\prime }$ (b); $v_{y}^{\prime }$ (c); $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ (d); the stress $\unicode[STIX]{x1D70F}_{xy}$ (solid line) and viscous stress $\unicode[STIX]{x1D70F}_{xy}^{v}$ (red dashed line) (e); and $(\bar{v}_{x}/v_{\ast })$ versus $(yv_{\ast }/\unicode[STIX]{x1D708})$ (f), at location III (figure 1 a) when the wall is made of shear modulus 0.75 kPa at Reynolds number $278$ (○), $301$ (▵), $335$ (▿), $392$ (◃) and $448$ (▹) for a channel with height approximately 0.6 mm in the undeformed state. The vertical dashed lines show the location of the top wall and the dashed curves in panel (a) show parabolic velocity profiles with equal average velocity. In panel (f), the dotted curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ , and the dashed line is $(\bar{v}_{x}/v_{\ast })=3.45(yv_{\ast }/\unicode[STIX]{x1D708})-1.8$ .

In figure 3(a), the dashed curves show the parabolic profiles that have the same average velocity as the experimentally measured profiles. It is observed that the velocity profiles are parabolic up to a Reynolds number of 301. When the Reynolds number increases to 335, there is a distinct departure from the parabolic profile, and the difference is significantly higher than the experimental standard deviation. The experimental velocity profiles are clearly flatter at the centre and steeper at the walls in comparison to the parabolic profile. At $Re=335$ , the root mean square of the streamwise fluctuating velocity $v_{x}^{\prime }$ shows a significant increase in magnitude (figure 3 b). More importantly, the shape of the $v_{x}^{\prime }$ profile exhibits the near-wall maxima which are characteristic of a turbulent flow in a channel. A significant increase in the cross-stream root mean square velocity, $v_{y}^{\prime }$ is also observed in figure 3(c) when the Reynolds number increases from $301$ to $335$ . Figure 3(d) shows that the correlation $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ , which is zero to within the experimental standard deviation for $Re\leqslant 301$ , increases significantly in magnitude at $Re=337$ and exhibits the characteristic shape of the Reynolds stress profiles in channel flow.

Subject to experimental uncertainties, the mean velocity and the root mean square of the fluctuating velocities are symmetric, and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ is antisymmetric, about the centreline of the channel. The transition at the Reynolds number of $335$ is also significantly different from those for a rigid channel in some important aspects. Due to experimental limitations (the seed particles are $8{-}14~\unicode[STIX]{x03BC}\text{m}$ in diameter), it is not possible to resolve the region within a distance of $15~\unicode[STIX]{x03BC}\text{m}$ from the wall of the channel. The data in figures 3(b) and 3(c) suggest that the root mean square of the fluctuating velocities could be non-zero at both walls although the data could plausibly be extrapolated to zero velocity at the wall. The data in figure 3(d) do suggest that $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ is non-zero at both walls. It should be noted that there are uncertainties in the measurement of the fluctuating velocities close to the wall. However, figure 21(c) in appendix B for a Reynolds number of approximately 3500 indicates that it is possible to capture the Reynolds stress accurately within approximately $50~\unicode[STIX]{x03BC}\text{m}$ of the wall, and a steep decrease in the Reynolds stress is observed at the wall. In contrast, in figure 3(d), a reasonable polynomial extrapolation predicts a non-zero Reynolds stress at the wall, although a sharp decrease in the Reynolds stress close to the wall cannot be ruled out by our measurements. More work is required to resolve this issue.

Figure 3(e) shows the variation in the total stress

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{xy}=\unicode[STIX]{x1D702}(\text{d}\bar{v}_{x}/\text{d}y)-\unicode[STIX]{x1D70C}\langle v_{x}^{\prime }v_{y}^{\prime }\rangle ,\end{eqnarray}$$

as well as the variation in the viscous stress $\unicode[STIX]{x1D70F}_{xy}^{v}=\unicode[STIX]{x1D702}(\text{d}\bar{v}_{x}/\text{d}y)$ , as a function of the cross-stream distance. For $Re\leqslant 301$ , the viscous stress is a linear function of distance (because the mean velocity profile is parabolic), and the difference between the total and viscous stress is smaller than the experimental uncertainty in the stress measurement. However, for $Re\geqslant 335$ , there is a significant difference between the viscous and the total stress, indicating that the Reynolds stress provides a substantial contribution to the total stress, even at the wall of the channel. Moreover, the viscous stress profiles are not a linear function of the cross-stream distance. The total stress profiles are expected to be nearly linear, since the wall slope is small at the location III (figure 1 a), as shown in figure 23 in appendix C. The linear stress profile can be obtained only when the Reynolds stress (which is non-zero at the wall) is added to the viscous stress, suggesting that the fluctuating velocities play an important role in the cross-stream transport of momentum.

The near-wall variation in the mean velocity $\bar{v}_{x}$ , scaled by the friction velocity $v_{\ast }=(\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D70C})^{1/2}$ , is shown as a function of the scaled distance $yv_{\ast }/\unicode[STIX]{x1D708}$ in figure 3(f). Here, $\unicode[STIX]{x1D70F}_{w}$ is the wall shear stress which is the sum of the viscous stress and the Reynolds stress at the wall and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. The velocity profile satisfies the linear relationship $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ close to the wall for the laminar profile for $Re\leqslant 301$ . In the turbulent regime for $335\leqslant Re\leqslant 448$ , there is no evidence of a viscous sublayer to within the experimental resolution for $(yv_{\ast }/\unicode[STIX]{x1D708})\leqslant 2$ . However, in the narrow range $3\leqslant (yv_{\ast }/\unicode[STIX]{x1D708})\leqslant 20$ , there is clear evidence of a logarithmic layer, $(\bar{v}_{x}/v_{\ast })=A\log (yv_{\ast }/\unicode[STIX]{x1D708})+B$ , at both the top and bottom walls.

The von Kármán plots for the mean velocity are shown in more detail in figure 4 at different downstream locations when the walls are made of shear modulus 0.75 kPa and 2.19 kPa, only for the Reynolds numbers where soft-wall turbulence is observed. It is clear that all the velocity profiles, scaled by the friction velocity, do follow a logarithmic law for $3\leqslant (yv_{\ast }/\unicode[STIX]{x1D708})\leqslant 20$ . There is no visible viscous sublayer observed even for $(yv_{\ast }/\unicode[STIX]{x1D708})$ as low as $1$ . The best fits for the von Kármán constants do change when the shear modulus is changed, and the constants are also very different from those in the turbulent flow past a rigid surface.

Figure 4. The near-wall velocity profiles when the wall is made of shear modulus 0.75 kPa (a) and 2.19 kPa (b) for a channel with height about 0.6 mm in the undeformed state. In panel (a), ○ location II, $Re=340$ ; ▵ location II, $Re=441$ ; ▿ location II, $Re=539$ ; ◃ location III, $Re=335$ ; ▹ location III, $Re=392$ ; ♢ location III, $Re=448$ ; $\times$ location IV, $Re=335$ ; $+$ location IV, $Re=467$ , the dotted curve shows the relation $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ ; and the dashed line shows the relation $(\bar{v}_{x}/v_{\ast })=3.45\log (yv_{\ast }/\unicode[STIX]{x1D708})-1.8$ . In panel (b), ○ location II, $Re=599$ ; ▵ location II, $Re=668$ ; ▿ location II, $Re=758$ ; ◃ location III, $Re=600$ ; ▹ location III, $Re=674$ ; ♢ location III, $Re=757$ ; $\times$ location IV, $Re=599$ ; $+$ location IV, $Re=665$ ; ● location IV, $Re=779$ ; the dotted curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ ; and the dashed line is $(\bar{v}_{x}/v_{\ast })=3.67\log (yv_{\ast }/\unicode[STIX]{x1D708})-2.53$ .

Figure 5. The measures $\bar{v}_{diff}$ (a) $(\max (v_{x}^{\prime })/\bar{v})$ (b), $(\max (v_{y}^{\prime })/\bar{v})$ (c) and $(\max (|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |)/\bar{v})^{2}$ (d), as a function of the Reynolds number for a channel with height about 0.6 mm in the undeformed state at locations II (○), III (▵) and IV (▿) shown in figure 1(a), when the channel is made of gel of shear modulus 0.75 kPa (solid line) and 2.19 kPa (dashed line). The symbol ♢ shows the respective measures for the flow in a rigid channel. The Reynolds number for the soft-wall and wall-flutter transitions are shown using the labels SW and WF respectively.

Quantitative measures of the departure from the parabolic profile and the velocity fluctuations, shown in figure 5, are now analysed. A quantitative measure of the difference between the actual profile and the parabolic velocity profile is,

(3.2) $$\begin{eqnarray}\displaystyle \bar{v}_{diff}=\sqrt{\frac{1}{h(\bar{v}_{x}^{(p)})^{2}}\int _{0}^{h}\,\text{d}y(\bar{v}_{x}(y)-\bar{v}_{x}^{l}(y))^{2}}, & & \displaystyle\end{eqnarray}$$

where $\bar{v}_{x}^{(p)}$ is the profile-averaged mean velocity defined in (C 1) and $\bar{v}_{x}^{l}(y)$ is the parabolic profile with average velocity equal to $\bar{v}_{x}^{(p)}$ . The other measures used are the scaled maxima of the root mean square velocity fluctuations, $\max (v_{x}^{\prime })/\bar{v}$ and $\max (v_{y}^{\prime })/\bar{v}$ , and the maximum of $|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |/\bar{v}^{2}$ , where $\bar{v}$ is the average velocity (ratio of the flow rate and the channel cross-section).

The measure $\bar{v}_{diff}$ is shown as a function of Reynolds number in figure 5(a). There is relatively little variation in $\bar{v}_{diff}$ downstream of location III, indicating that the flow has reached a fully developed state downstream of location III (figure 1 a). There is a sharp increase in $\bar{v}_{diff}$ at a Reynolds number of approximately 335 when the shear modulus of the walls is 0.75 kPa, and at approximately 480 in the shear modulus of the walls is 2.19 kPa. The increase in $\bar{v}_{diff}$ takes place at the same Reynolds number as the steep increase in the maximum values of $v_{x}^{\prime }$ , $v_{y}^{\prime }$ and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ in figure 5(b–d). This Reynolds number is labelled as SW (for soft-wall transition) in figure 5. The root mean square of the fluctuating velocities in the streamwise and cross-stream directions initially increase rapidly when the transition Reynolds number is exceeded, and then saturate at approximately 12 % and 8 % of the mean velocity for soft-wall turbulence. Despite the sharp nature of the transition, we did not detect any hysteresis in the transition Reynolds number while increasing and decreasing the flow velocity, suggesting that this transition is supercritical.

An important observation in figure 5 is that the magnitudes of the turbulent velocity fluctuations (when scaled by suitable powers of the average velocity) after the soft-wall transition are significantly higher than those observed at the hard-wall transition at a Reynolds number of approximately 1000 in a rigid channel shown by the ♢ symbols.

3.1.2 Wall dynamics

The mean displacement of the top and bottom walls in the streamwise $(x)$ direction are shown as a function of the Reynolds number at three different downstream locations in figure 6 when the wall is made of gel with shear modulus 0.75 kPa. In figure 6, there is no indication of a sharp change in the mean displacement at the Reynolds number for the soft-wall transition or the wall flutter, where there is a striking change in the flow dynamics. Similar results were obtained when the wall is made of shear modulus 2.19 kPa and for the channels with height 1.8 mm; these are not shown here.

Figure 6. The mean displacement $\bar{u}_{x}$ on the top wall (a) and the bottom wall (b) at locations II (○), III (▵) and IV (▿) in figure 1(a) when the wall is made with shear modulus 0.75 kPa for a channel with height approximately 0.6 mm in the undeformed state. The Reynolds number for the soft-wall and wall-flutter transitions are shown using the labels SW and WF respectively.

There is, however, a discontinuous change in the root mean square of the displacement fluctuations tangential to the surface at the soft-wall transition, as shown in figure 7(a). There is a sharp increase in the root mean square of the tangential displacement to approximately $4{-}5~\unicode[STIX]{x03BC}\text{m}$ when there is the soft-wall transition (labelled SW in figure 7) at a Reynolds number of approximately 300 for walls with shear modulus 0.75 kPa, and at a Reynolds number of approximately 480 for walls of shear modulus 2.19 kPa. The spanwise root mean square of the displacement fluctuations, $u_{z}^{\prime }$ (not shown for conciseness) is approximately 0.5–0.75 times that of $u_{x}^{\prime }$ , and $u_{z}^{\prime }$ also exhibits a sharp increase at the soft-wall transition Reynolds number. In both the streamwise and spanwise directions, the fluctuations are symmetric, and the magnitude of the fluctuations on the top wall is comparable to that on the bottom wall.

Even though there is a sharp increase in the tangential displacement fluctuations at the surface, a remarkable observation is that there are no discernible displacement fluctuations in the direction perpendicular to the surface. This is shown in figure 7, where $u_{y}^{\prime }$ (measured using the configuration in figure 2(a) using a side camera) is shown on an inverted right vertical axis for clarity. Subject to the experimental uncertainties, (the minimum dimension that can be resolved is approximately $3~\unicode[STIX]{x03BC}\text{m}$ ) there is no measurable wall motion perpendicular to the surface. This is consistent with the observations of Verma & Kumaran (Reference Verma and Kumaran2013) and Srinivas & Kumaran (Reference Srinivas and Kumaran2015) for the flow in a micro-channel. The frequency spectra of the streamwise displacement fluctuations show low-frequency structure below a frequency of approximately $200~\text{rad}~\text{s}^{-1}$ ; these are not shown here for conciseness.

Figure 7. The variation of $u_{x}^{\prime }$ on the top wall (open symbols, red dashed line), on the bottom wall (open symbols, blue dotted line) referenced to the left vertical axis, and $u_{y}^{\prime }$ on the top wall (filled symbols, black solid line) referenced to the inverted right vertical axis at locations II (▵), III (▿) and IV (◃) in figure 1(a) when the wall is made with shear modulus 0.75 kPa (a) and 2.19 kPa (b) for a channel with height about 0.6 mm in the undeformed state. The Reynolds number for the soft-wall and wall-flutter transitions are shown using the labels SW and WF respectively.

3.2 Wall-flutter transition

3.2.1 Wall dynamics

There is a second transition as the Reynolds number is increased beyond about 550 when the wall is made of shear modulus 0.75 kPa, and 760 when the wall is made of shear modulus 2.19 kPa, as shown in figure 7. In the experiments, flutter of the top wall is observed in the soft section. A sharp increase is observed in $u_{x}^{\prime }$ and $u_{y}^{\prime }$ in figure 7. Figure 7 also shows that the displacement fluctuations are asymmetric – while there is a sharp increase in the displacement fluctuations of the top wall, there is very little increase in the displacement fluctuations on the bottom wall. The magnitudes of $u_{x}^{\prime }$ and $u_{y}^{\prime }$ are comparable when there is wall flutter. The amplitude of these downstream travelling waves decreases with distance travelled – the amplitude is largest at location II (figure 1 a) where the deformation is largest, and it decreases at locations III and IV. The amplitude of the waves first increases as the Reynolds number is increased above 545, reaches a maximum at a Reynolds number of approximately 750 and then appears to decrease again.

The frequency spectrum of the tangential displacement fluctuations shows a distinct maximum in the range $100{-}300~\text{rad}~\text{s}^{-1}$ after the onset of wall flutter, in contrast to the broad low-frequency spectrum for the soft-wall turbulence, as shown in figure 8. This frequency range is shown to correspond to that expected from the wall thickness and the shear wave speed in the discussion in § 6.2.

Figure 8. The frequency spectrum for $\tilde{u} _{x}$ (2.2) at different Reynolds numbers when the shear modulus of the channel wall is 0.75 kPa (a) and 2.19 kPa (b) for a channel with height approximately 0.6 mm in the undeformed state.

3.2.2 Flow characteristics

The motion of the top wall is also reflected in the fluid velocity field. The mean velocity profile, shown in figure 9(a), is symmetric for $Re=545$ , but develops a distinct asymmetry at $Re=599$ . It is interesting that there is a distinct shift in the maximum towards the upper wall. The formation of waves on the top wall cannot be modelled as just static roughness elements which would decrease the mean velocity, but these waves actually increase the mean velocity near the upper wall. This transition is also evident in the profiles of $v_{x}^{\prime }$ , $v_{y}^{\prime }$ and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ in figure 9(b–d), where the maximum is near the upper wall at $Re=599$ . This asymmetry further increases at $Re=741$ , and then decreases when the Reynolds number is further increased to 860 and 923. The velocity $v_{y}^{\prime }$ appears to increase monotonically with Reynolds number, in contrast to $v_{x}^{\prime }$ which appears to first increase and then decrease. A similar feature is observed in the shear stress profiles in figure 9(e), where there is a distinct increase in the asymmetry when the Reynolds number increases from approximately 500 to approximately 860; the stress profile then becomes more symmetric when the Reynolds number increases to approximately 923. While the logarithmic profile is a good fit for $Re=545$ , the fit is poor when the Reynolds number increases to 599 after the wall-flutter transition, as shown in figure 9(f). Thus, figure 9(f) shows that the velocity profile close to the walls can no longer be fitted by a logarithmic profile after the wall-flutter transition, at either the top or bottom walls.

Figure 9. The cross-stream variation of $\bar{v}_{x}$ (a), $v_{x}^{\prime }$ (b), $v_{y}^{\prime }$ (c) and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ (d), the stress $\unicode[STIX]{x1D70F}_{xy}$ (solid line) and viscous stress $\unicode[STIX]{x1D70F}_{xy}^{v}$ (red dashed line) (e); and the variation of $(\bar{v}_{x}/v_{\ast })$ with $(yv_{\ast }/\unicode[STIX]{x1D708})$ (f) at the location III (figure 1 a) when the wall is made of shear modulus 0.75 kPa at Reynolds number $545$ (○), $599$ (▵), $741$ (▿), $860$ (◃) and $923$ (▹) for a channel with height approximately 0.6 mm in the undeformed state. The vertical dashed lines show the location of the top wall. In panel (f), the dotted curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ , and the dashed line is $(\bar{v}_{x}/v_{\ast })=3.45(yv_{\ast }/\unicode[STIX]{x1D708})-1.8$ .

Figure 10. The variation with Reynolds number of $u_{x}^{\prime }$ on the top wall (open symbols, red dashed line), on the bottom wall (open symbols, blue dotted line) referenced to the left vertical axis, and $u_{y}^{\prime }$ on the top wall (filled symbols, solid line) referenced to the inverted right vertical axis at the downstream locations II (○), III (▵) and IV (▿) in figure 1(a), when the wall is made with shear modulus 0.75 kPa (a) and 2.19 kPa (b) for a channel with height approximately 1.8 mm in the undeformed state. The Reynolds number for the hard-wall laminar–turbulent transition is labelled HW, the soft-wall transition is labelled SW, and the wall flutter is labelled WF.

The wall flutter also causes a significant increase, by a factor of 2, in the maximum of $v_{x}^{\prime }$ shown in figure 5(b), and a smaller but still significant increase in the maximum of $v_{y}^{\prime }$ in figure 5(c). The increase in the fluid velocity fluctuations is not monotonic, and there is an increase in the amplitude of the fluctuations up to a Reynolds number of about 860 and a decrease when the Reynolds number is further increased, in correlation with the wall displacement fluctuations. The fluid velocity fluctuations are also higher at the upstream location II (figure 1 a), and they decrease with downstream position.

4.1 Wall dynamics

The root mean square of the displacement fluctuations for this case, shown in figure 10, do exhibit signatures of the soft-wall and wall-flutter transitions. At the laminar–turbulent transition at a Reynolds number of approximately 1000, there is no evidence of wall fluctuations in either the tangential or normal directions. However, there is evidence of a soft-wall transition at a Reynolds number of approximately 1400 when the wall is made of shear modulus 0.75 kPa and about 1850 when the wall is made of shear modulus 2.19 kPa. After the soft-wall transition, there is a significant increase in the wall displacement fluctuations tangential to the surface, but no visible increase in the fluctuations perpendicular to the surface. The magnitudes of the displacement fluctuations are approximately equal on the top and bottom walls, and are invariant with downstream distance.

As the Reynolds number is increased, the wall-flutter transition takes place at a Reynolds number of approximately 1700 for the wall with shear modulus 0.75 kPa, and approximately 2350 for the wall with shear modulus 2.19 kPa. The displacement fluctuations tangential to the surface at the top wall are observed to increase sharply, and the amplitude decreases with downstream distance. The root mean square displacement perpendicular to the top wall also shows a discontinuous increase after the wall-flutter transition. However, the magnitude of the tangential fluctuations at the bottom wall shows no discontinuity, and there is no visible fluctuation perpendicular to the bottom wall, indicating that the increase in fluctuations is confined to the top wall.

Figure 11. The frequency spectra $\tilde{u} _{x}$ (2.2) of the tangential wall oscillations at different Reynolds numbers after the soft-wall transition (a) and wall-flutter transition (b) for a channel with height 1.8 mm when the walls are made of gel with shear modulus 0.75 kPa at location II in figure 1(a).

The two distinct transitions are also evident in the frequency spectra of the tangential wall oscillations, shown in figure 11. After the soft-wall transition at a Reynolds number of approximately 1400 and before the wall-flutter transition at a Reynolds number of approximately 1700 in figure 11(a), there is a sharp peak in the frequency spectrum at a relatively high frequency of approximately $395~\text{rad}~\text{s}^{-1}$ when the wall is made of shear modulus 0.75 kPa. Similarly, the data for the wall with shear modulus 2.19 kPa (not shown here for conciseness) indicate that after the soft-wall transition ( $Re=1850$ ) and before the wall-flutter transition ( $Re=2350$ ), there is one distinct maximum in the frequency spectrum at a frequency of approximately $210~\text{rad}~\text{s}^{-1}$ . The frequency spectra after the wall-flutter transition, (figure 11 b) indicate that the magnitude of the fluctuations increases significantly (as indicated by the a comparison of the scales on the vertical axes in figure 11 a,b). The maximum at $395~\text{rad}~\text{s}^{-1}$ after the soft-wall transition is no longer observed in figure 11(b) after the wall-flutter transition. Instead, there is a distinct maximum at a lower frequency of approximately $195~\text{rad}~\text{s}^{-1}$ , which appears at a Reynolds number of approximately 1900 and then grows rapidly as the Reynolds number is increased. In a similar manner, when the soft wall is made of shear modulus 2.19 kPa, the spectrum (not shown here) exhibits a maximum at a lower frequency of approximately $155~\text{rad}~\text{s}^{-1}$ after the wall-flutter transition.

The wall dynamics for the soft-wall and wall-flutter transitions in a turbulent flow is very similar to that in a laminar flow, except in one respect. In the laminar flow, a specific frequency for the wall oscillations was not detected within the range of $0{-}500~\text{rad}~\text{s}^{-1}$ accessible in our experiments; this is consistent with the expectation of a frequency of approximately $10^{3}{-}10^{4}~\text{Hz}$ predicted by the linear stability analysis (Verma & Kumaran Reference Verma and Kumaran2013, Reference Verma and Kumaran2015). In contrast, in the soft-wall transition in a turbulent flow, there was a distinct maximum in the frequency spectrum at a frequency of approximately $395~\text{rad}~\text{s}^{-1}$ for the wall made of shear modulus 0.75 kPa, and approximately $210~\text{rad}~\text{s}^{-1}$ for the wall made of shear modulus 2.19 kPa.

4.2 Flow characteristics

The evolution of the fluid velocity field is summarised in figure 12 at the location II in figure 1(a); the results for the other locations are qualitatively similar. The hard-wall laminar–turbulent transition is first observed when Reynolds number is increased from 768 to 1071. The mean velocity profile (figure 12 a) is parabolic at $Re=768$ , but is clearly non-parabolic, with a lower curvature at the centre and higher gradient at the wall, at $Re=1071$ . The hard-wall transition is also accompanied an increase in the amplitude of the velocity fluctuations, as shown in figure 12(b–d). The effects of the soft-wall transition are visible when the Reynolds number increases from 1332 to 1515. There is a distinct change in the shape of the mean velocity profile accompanying this transition in figure 12(a). The magnitude of the streamwise velocity fluctuations increases and the prominence of the near-wall maxima decreases in figure 12(b). Figure 12(d) shows that the Reynolds stress increases significantly after the soft-wall transition. In the von Kármán plot in figure 12(f), there is a clear shift in the near-wall velocity profile when the Reynolds number increases from 1332 to 1551. While there is no clear logarithmic region when the Reynolds number is 1332 or less, there velocity profiles for Reynolds number 1551 and 1734 are clearly very well fitted by a logarithmic law.

The wall-flutter transition at a Reynolds number of approximately 1850 results in a distinct asymmetry in the mean velocity profile, as shown in figure 12(a) at a Reynolds number of 1973. The near-wall velocity profile in the von Kármán plot in figure 12(f) shows a clear shift from a logarithmic law for Reynolds number between 1515 to 1734, to a different profile when the Reynolds number increases to 1973. There is a significant increase in the magnitude, and a distinct asymmetry, in the streamwise root mean square velocity when the Reynolds number is increased from 1734 to 1973, as shown in figure 12(b). The Reynolds stress also increases significantly after the wall-flutter transition, as shown in figure 12(d). Thus, the qualitative features for flow after the soft-wall transition and wall-flutter transition are common to transitions from a laminar and turbulent flow.

Figure 12. The cross-stream variation of $\bar{v}_{x}$ (a); $v_{x}^{\prime }$ (b); $v_{y}^{\prime }$ (c); $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ (d); the stress $\unicode[STIX]{x1D70F}_{xy}$ (solid line) and viscous stress $\unicode[STIX]{x1D70F}_{xy}^{v}$ (red dashed line) (e); and the variation of $(\bar{v}_{x}/v_{\ast })$ with $(y_{w}v_{\ast }/\unicode[STIX]{x1D708})$ (f), at the streamwise location II in figure 1(a) when the wall is made of shear modulus 0.75 kPa at Reynolds number 768 (○), 1071 (▵), 1332 (▿), 1515 (◃), 1734 (▹) and 1973 (♢) for a channel with height approximately 1.8 mm in the undeformed state. The vertical dashed lines show the location of the top wall, and the dashed curves in panel (a) show parabolic velocity profiles with equal average velocity. In panel (f), the dotted curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ the dashed line is $(\bar{v}_{x}/v_{\ast })=4.20(yv_{\ast }/\unicode[STIX]{x1D708})-3.12$ .

The von Kármán plots for the mean velocity are shown in figure 13 for the channels with height 1.8 mm in the undeformed state, only for the range of Reynolds numbers where soft-wall turbulence is observed. The scaled velocity profiles do follow a logarithmic law only for the range of Reynolds numbers where soft-wall turbulence is observed, but not after the wall-flutter transition. In figure 13, a viscous sublayer is not visible to within experimental resolution even for $(yv_{\ast }/\unicode[STIX]{x1D708})$ as low as $3$ , but there is a logarithmic layer for $3\leqslant (yv_{\ast }/\unicode[STIX]{x1D708})\leqslant 70$ where $y$ is the distance from the wall, although it must be cautioned that there is a difference of approximately 10–20 % between the data and the log law for the lowest point. Figure 13 also shows that the best fits for the von Kármán constants do change with the shear modulus, and the constants are also very different from those in the turbulent flow past a rigid surface.

Figure 13. The near-wall velocity profiles when the wall is made of shear modulus 0.75 kPa (a) and 2.19 kPa (b) for a channel with height approximately 1.8 mm in the undeformed state. In (a), ○ location II, $Re=1515$ ; ▵ location II, $Re=1621$ ; ▿ location III, $Re=1515$ ; ◃ location III, $Re=1621$ ; ▹ location IV, $Re=1580$ ; ♢ location IV, $Re=1621$ ; the dashed curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ ; and the dashed line is $(\bar{v}_{x}/v_{\ast })=4.20\log (yv_{\ast }/\unicode[STIX]{x1D708})-3.12$ . In (b), ○ location II, $Re=1948$ ; ▵ location II, $Re=2290$ ; ▿ location III, $Re=1900$ ; ◃ location III, $Re=2187$ ; ▹ location IV, $Re=1936$ ; ♢ location IV, $Re=2151$ ; the dotted curve is $(\bar{v}_{x}/v_{\ast })=(yv_{\ast }/\unicode[STIX]{x1D708})$ ; and the dashed line is $(\bar{v}_{x}/v_{\ast })=4.54\log (yv_{\ast }/\unicode[STIX]{x1D708})-4.38$ .

The measures $\bar{v}_{diff}$ (3.2), $(\max (v_{x}^{\prime })/\bar{v})$ , $(\max (v_{y}^{\prime })/\bar{v})$ and $(\max (|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |)/\bar{v}^{2})$ are shown in figure 14. The results for a rigid channel are shown using the ♢ symbol. The laminar–turbulent transition is clearly visible at a Reynolds number of approximately 1000 at all downstream locations. As the Reynolds number is increased, there is a sharp increase in all the measures at the soft-wall transition Reynolds number of approximately 1400 for the wall made with shear modulus 0.75 kPa, and about 1850 for the wall made with shear modulus 2.19 kPa. The increase is rather sharp in the measure $\bar{v}_{diff}$ , indicating a significant change in the form of the mean velocity profile in comparison to that for the hard-wall turbulence. The increase is not as evident in the measure $(v_{x}^{\prime }/\bar{v})$ for the streamwise root mean square velocity, because the fluctuation intensity after the hard-wall transition is rather large. However, figure 14 does not capture the significant change in the profile of $v_{x}^{\prime }$ shown in figure 12, where the low central minimum observed after the hard-wall transition transforms into a much higher minimum, without a significant increase in the near-wall peaks, after the soft-wall transition. There is an increase by a factor of 1.5 in $(\max (v_{y}^{\prime })/\bar{v})$ , and a larger increase by a factor of 2–3 in $(\max (|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |)/\bar{v})^{2}$ , at the soft-wall transition.

There is a further increase in the all measures after the wall-flutter transition at a Reynolds number of approximately 1700 for the wall made with shear modulus 0.75 kPa, and approximately 2350 for the wall made with shear modulus 2.19 kPa, but the values do depend on downstream position. The maximum increase is observed at the upstream location II in figure 1(a), and the magnitudes of all measures decrease progressively at the downstream locations III and IV. The fluctuation intensities also do not increase monotonically with Reynolds number. The magnitudes of $(\max (v_{x}^{\prime })/\bar{v})$ and $(\max (|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |)/\bar{v}^{2})$ are almost always higher than those for the hard channel, but the cross-stream fluctuating velocity appears, at the downstream locations, to be smaller than that for a rigid channel at the same Reynolds number.

Figure 14. The measures $\bar{v}_{diff}$ (a), $(\max (v_{x}^{\prime })/\bar{v})$ (b), $(\max (v_{y}^{\prime })/\bar{v})$ (c) and $(\max (|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |)/\bar{v}^{2})$ (d), as a function of the Reynolds number at different downstream locations II (○), III (▵), and IV (▿) in figure 1(a), when the channel is made of gel of shear modulus 0.75 kPa (solid line) and 2.19 kPa (dashed line) for a channel with height approximately 1.8 mm in the undeformed state. The results for a hard-wall channel are shown by the ♢ symbol. The Reynolds number for the hard-wall laminar–turbulent transition is labelled HW, the soft-wall transition is labelled SW, and the wall flutter is labelled WF.

An interesting feature observed in figure 10, at Reynolds numbers greater than approximately 3300 for the wall made with shear modulus 0.75 kPa, and approximately 3800 for the wall with shear modulus 2.19 kPa, is a sharp increase in the root mean square of the fluctuations on the bottom wall. At this point, we also visually observe normal oscillations on the bottom wall, not shown in figure 10 to enhance clarity. This has not been explored in detail in the present analysis because the range of Reynolds numbers is too small to observe the evolution of the fluctuation amplitude and frequency. However, these oscillations could be analogous to the hydroelastic instabilities observed by Hansen & Hunston (Reference Hansen and Hunston1974), Hansen & Hunston (Reference Hansen and Hunston1983) and Gad-el Hak et al. (Reference Gad-el Hak, Blackwelder and Riley1985) in open flows, where waves were observed on a surface that is fixed to a bottom substrate. This is a subject that requires further study.

Figure 15. The transition Reynolds number for the soft-wall transition from the present study (●), the experimental study of Verma & Kumaran (Reference Verma and Kumaran2012) (○), the experimental study of Kumaran & Bandaru (Reference Kumaran and Bandaru2016) ( $\oplus$ ) and the transition Reynolds number for the wall flutter from the present study (▵) as a function of the parameter $\unicode[STIX]{x1D6F4}$ . The lower dashed line shows the relation $Re=0.11\unicode[STIX]{x1D6F4}^{5/8}$ , while the upper dashed line indicates a slope of $(1/2)$ on a log–log graph.