3.1. Fluid flow

Results for the mean and root mean square fluctuating velocities are plotted for three different parameter sets, based on the wall shear modulus and the transition Reynolds number. The transition Reynolds number has already been determined using the dye-stream method (Verma & Kumaran Reference Verma and Kumaran2013) for different downstream locations and for different shear moduli of the soft wall. In experiments, transition was first observed in the downstream converging section of the channel, and not in the upstream diverging section where the slope of the wall is largest. In the theoretical linear stability analysis, it was found that the modification of the mean velocity profile due to channel deformation had a significant effect on the flow stability, and the flow in the downstream converging section is more unstable than that in the upstream converging section. For a soft wall with shear modulus 18 kPa, the transition is observed at a lower Reynolds number at the downstream section C in comparison to the upstream sections A and B – the flow is laminar up to a Reynolds number of 222, and the non-laminar flow (described in § 3) is first observed at a Reynolds number of 250 in zone C, while it is first observed at a $Re=277$ in zones A and B. For a soft wall with shear modulus 26 kPa, the flow is laminar up to a Reynolds number of 332, and the non-laminar flow is first observed in all three zones at a Reynolds number of 388. Therefore, results are discussed for two different Reynolds numbers when the wall is made of shear modulus 18 kPa, $Re=277$ just above transition and $Re=416$ which was the maximum achievable Reynolds number in the experiments. For a soft wall made of shear modulus 26 kPa, the profiles are shown only for $Re=416$ . For comparison, the velocity statistics for a microchannel made with four hard walls of shear modulus 0.55 MPa are also shown using the ♢ symbol. These statistics were evaluated at a distance 0.1 cm upstream of the exit of the microchannel, though we have verified that the statistics do not vary appreciably from a distance of approximately 1.5 cm upstream of the exit.

It was mentioned in § 2.5 that the Reynolds number $\mathit{Re}$ (2.1), based on the flow rate and the width of the channel, is used to characterise the flow velocity. In order to provide a reference, the relation between the Reynolds number used here and other Reynolds numbers commonly used to characterise the flow is provided in figure 9. The Reynolds number based on the maximum velocity along the central plane in the spanwise direction and one-half of the channel width, referred to as the local Reynolds number $\mathit{Re}_{l}$ , is defined as

where ${\overline{v}_{x}}^{max}$ is the maximum velocity. For a two-dimensional flow, the maximum velocity is ( $3/2$ ) times the average velocity, and so $\mathit{Re}_{l}=(3\mathit{Re}/4)$ , where $\mathit{Re}$ is based on the average velocity and the channel height. The local Reynolds number, shown in figure 9(a), is somewhat larger than $(3\mathit{Re}/4)$ for the laminar flow for two reasons – the height along the central plane is larger than the undeformed height of the channel and the maximum velocity along the central plane is larger than that expected for a two-dimensional flow for the same pressure gradient due to the slowing down of the velocity near the sidewalls. The Reynolds number based on the friction velocity is defined as

where the friction velocity $v_{\ast }=\sqrt{{\it\tau}_{w}/{\it\rho}}$ and ${\it\tau}_{w}$ is the wall shear stress. For a laminar flow, ${\it\tau}_{w}=(4{\it\eta}{\overline{v}_{x}}^{max}/h)$ and the friction Reynolds number $\mathit{Re}_{\ast }=\sqrt{6\mathit{Re}}$ . Figure 9(b) shows that the Reynolds number based on the friction velocity is a little larger than $\sqrt{6\mathit{Re}}$ in the laminar regime, but it increases when there is a transition due to the increased wall shear stress.

Figure 9.

The local Reynolds number $\mathit{Re}_{l}=({\it\rho}{\overline{v}_{x}}^{max}h/2{\it\eta})$ (a) and the friction Reynolds number $\mathit{Re}_{\ast }=({\it\rho}v_{\ast }h/{\it\eta})$ (b) as a function of the Reynolds number based on the flow rate and channel width $\mathit{Re}=({\it\rho}Q/{\it\eta}W)$ in the zones A (○), B (▵) and C (▿) when the soft wall is made of shear modulus 18 kPa (solid lines) and 26 kPa (dashed lines). The dotted lines show the values for a two-dimensional laminar flow.

Figure 10.

The mean velocity profiles in the zones A (○), B (▵) and C (▿) in figure 5 for a microchannel with soft wall made with shear modulus 18 kPa and Reynolds number 277 (a), shear modulus 18 kPa and Reynolds number 416 (b), shear modulus 26 kPa and Reynolds number 416 (c), and (d) the mean square of the difference in the mean velocities from experiments and ANSYS FLUENT simulations, (3.3), as a function of the Reynolds number for a microchannel with soft wall made of shear modulus 18 kPa (solid lines) and 26 kPa (dashed lines) in zones A (○), B (▵) and C (▿). In (a–c), the lines show the results obtained using the ANSYS FLUENT simulations.

The mean velocity profiles in the three different zones A, B and C (figure 5) are shown as a function of the cross-stream distance in figure 10. Also shown by the filled symbols are the results of the ANSYS FLUENT simulations for the laminar flow at the same Reynolds number. It is clearly observed that the experimental velocity profiles are flatter at the centre and steeper near the walls in comparison to the simulation results. The difference is relatively small but discernible at a Reynolds number of 277 when the soft wall is made with shear modulus 18 kPa, but the difference is much larger at a Reynolds number of 416. The mean velocity profiles do not show much variation with downstream location when the soft wall is made with shear modulus 26 kPa (possibly because there is very little variation in the height in the three different zones, as shown in figure 8 b), indicating that the flow is nearly fully developed. There is a significant change in the velocity profiles with downstream location when the soft wall is made with shear modulus 18 kPa, due to the larger height variation shown in figure 8(a). In all cases, the experimental velocity profile is significantly different from the laminar velocity profile for the same geometry when the Reynolds number exceeds the transition Reynolds number. This is in contrast to the excellent agreement between the experimental and simulation velocity profiles when the Reynolds number is below the transition Reynolds number, as shown in figure 7. A quantitative estimate of the difference between the experimental and simulation velocity profiles can be obtained by defining the scaled mean square of the difference between the experimental and simulation velocity profiles,

where ${\overline{v}_{x}}^{\!\!\ast }(y)$ is the mean velocity at the location $y$ scaled by the maximum velocity from the experiments, ${\overline{v}_{x}}_{sim}^{\!\!\ast }$ is the mean velocity predicted by the ANSYS FLUENT simulation, scaled by the maximum velocity, at the same location for a laminar flow. This measure, shown as a function of the Reynolds number in figure 10(d), is close to zero when the flow is laminar, but increases sharply after transition at a Reynolds number of 250 (in zone C) and 277 (in zones A and B) for a soft wall with shear modulus 18 kPa, and at 388 for a soft wall with shear modulus 26 kPa.

The root mean square of the velocity fluctuations is shown as a function of the cross-stream distance in figure 11(a–c). The streamwise root mean square of the fluctuating velocity, $v_{x}^{\prime }$ , displays a maximum close to the wall and decreases towards the centre. This near-wall maximum in $v_{x}^{\prime }$ is clearly visible even at a Reynolds number of 277, just above the transition Reynolds number, when the soft wall is made of shear modulus 18 kPa. This near-wall maximum is a characteristic of the turbulence in a rigid channel and is attributed to the energy production due to turbulent bursts close to the wall. The $v_{x}^{\prime }$ profiles are clearly not symmetrical, and the value of $v_{x}^{\prime }$ close to the soft wall is higher, by a factor of 2–3, in comparison to that close to the hard wall. This is very different from the flow in a rigid channel (Appendix, figure 20), where the $v_{x}^{\prime }$ profiles are symmetric about the centreline, and suggests the role of the soft wall in the production of the turbulent energy. The root mean square of the cross-stream velocity fluctuations, $v_{y}^{\prime }$ , is much smaller in magnitude than $v_{x}^{\prime }$ , but it is also not symmetrical, and the magnitude is significantly larger near the soft wall. Figure 11(d) does indicate a shallow maximum in $v_{y}^{\prime }$ close to the wall, at a distance comparable to the distance of the maximum of $v_{x}^{\prime }$ from the wall. This is, again, very different form the flow in a rigid channel where the maximum in $v_{y}^{\prime }$ is further away from the wall in comparison to $v_{x}^{\prime }$ , as shown in the Appendix in figure 20. In figure 11(a–c), we have not extrapolated the data to $v_{x}^{\prime }=0$ and $v_{y}^{\prime }=0$ at the soft wall, as is conventionally done in the profiles of the fluctuating velocities, since there is the possibility of velocity fluctuations at the soft wall. In order to examine the near-wall behaviour, higher-resolution images were analysed close to the wall, though we were not able to accurately resolve the region within approximately $10~{\rm\mu}\text{m}$ of the wall due reflections off the wall. The results of the high-resolution measurements within approximately $100~{\rm\mu}\text{m}$ of the wall are shown in figure 11(d). The value of $v_{x}^{\prime }$ at the wall cannot be conclusively established from the results in figure 11(d); the data could plausibly be extrapolated to $v_{x}^{\prime }=0$ at the wall, but the data also support a non-zero $v_{x}^{\prime }$ at the wall. In the case of the wall-normal velocity $v_{y}^{\prime }$ , the data do not suggest a decrease to zero at the wall. This is in contrast to the flow in a rigid channel (Appendix, figure 20), where $v_{x}^{\prime }$ and $v_{y}^{\prime }$ are zero at the wall due to the no-slip condition.

Figure 11.

The root mean square of the fluctuating velocities in the flow direction $v_{x}^{\prime }$ (solid lines and open symbols) and in the cross-stream direction $v_{y}^{\prime }$ (dashed lines and filled symbols) in the zones A (○), B (▵) and C (▿) in figure 5 for a microchannel with soft wall made with shear modulus 18 kPa and Reynolds number 277 (a), shear modulus 26 kPa and Reynolds number 416 (b), shear modulus 26 kPa and Reynolds number 416 (c), and a magnified view close to the soft wall of the root mean square fluctuating velocities for the soft wall made with shear modulus 18 kPa and Reynolds number 416 (d). The vertical dotted lines show, from right to left, the locations of the top wall in zones A, B and C respectively.

The cross-correlation $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ is of significance because it is proportional to the Reynolds stress, or the rate of transport of momentum due to the velocity fluctuations. As in the case of $v_{x}^{\prime }$ and $v_{y}^{\prime }$ , the profile of $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ is not symmetrical, and the zero is not at the centre of the channel, as shown in figure 12. Near the top rigid wall, the qualitative features of $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ are similar to those for the flow in a rigid channel shown in figure 20 in the Appendix for the flow in a rigid channel, and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ does tend to zero at the top wall. However, based on the data and error in figure 12, the graph of $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ cannot plausibly be extrapolated to zero at the soft wall. This is significant because it implies a non-zero Reynolds stress at the soft wall and the presence of wall motion if no-slip boundary conditions are applied at the soft wall. As we show in § 3.2, wall motion is indeed detected after transition. Figure 12 shows that $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ has a maximum at the soft wall or close to the soft wall. In the following discussion, the value of $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ at the soft wall is determined by a cubic spline extrapolation using the four data points closest to the soft wall.

Figure 12.

The root mean square of the fluctuating velocities in the flow direction $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ in the zones A (○), B (▵) and C (▿) in figure 5 for a microchannel with soft wall made with shear modulus 18 kPa and Reynolds number 277 (a), shear modulus 18 kPa and Reynolds number 416 (b), and shear modulus 26 kPa and Reynolds number 416 (c). The vertical dotted lines show, from right to left, the locations of the top wall in zones A, B and C respectively.

Figure 13.

The maximum values of $v_{x}^{\prime }$ (a), $v_{y}^{\prime }$ (b) and $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ (c) across the channel, scaled by suitable powers of the maximum mean velocity $\bar{v}_{xm}$ , as a function of the Reynolds number for soft walls made with shear modulus 18 kPa (solid line) and 26 kPa (dashed line) in zone A (○), B (▵) and C (▿) in figure 5. For comparison, the symbol ♢ shows the results at the downstream location $x=2.9~\text{cm}$ when the bottom wall is hard (shear modulus 0.55 MPa). The * symbol, referenced to the top $x$ axis, shows the fluctuation intensities scaled by suitable powers of the maximum velocity from DNS (Moser, Kim & Mansour Reference Moser, Kim and Mansour1999; del Alamo et al. Reference del Alamo, Jimenez, Zandonade and Moser2003; Hoya & Jimenez Reference Hoya and Jimenez2006) at much higher Reynolds number.

In figure 13, the maximum values of $v_{x}^{\prime }$ , $v_{y}^{\prime }$ and $|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |$ across the channel, scaled by suitable powers of the maximum of $\bar{v}_{xm}$ , are shown as a function of the Reynolds number. The scaled velocities $(v_{x}^{\prime }/\bar{v}_{xm})$ and $(v_{y}^{\prime }/\bar{v}_{xm})$ increase threefold from approximately 8 % to approximately 25 % and from approximately 2 % to approximately 6 % respectively at transition. The maximum of the ratio $(|\langle v_{x}^{\prime }v_{y}^{\prime }\rangle |/\bar{v}_{xm}^{2})$ also shows a threefold increase at transition. It is evident that the levels of the fluctuation intensities, scaled by the maximum velocity, vary little with Reynolds number before and after transition, but there is a near discontinuous change in the levels of the fluctuation intensities when there is a transition. Also shown in figure 13, referenced to the top $x$ axis, are the maximum fluctuation intensities scaled by suitable powers of the maximum velocity for a rigid channel at much higher Reynolds numbers. The scaled maximum fluctuation intensity in a soft-walled microchannel at a Reynolds number in the range 250–400 is comparable to that in a rigid-walled channel at a Reynolds number in the range 5000–20 000. The root mean square velocities scaled by the friction velocity at the soft wall have also been analysed, though the data are not shown here for conciseness. The ratio $v_{x}^{\prime }/v_{\ast }$ for the flow after transition in a soft-walled microchannel is approximately 20–30 % smaller, while $v_{y}^{\prime }/v_{\ast }$ is approximately one-half of that in a rigid-walled microchannel for Reynolds number in the range 5000–20 000. Similarly, the ratio $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle /v_{\ast }^{2}$ for the flow after transition in a soft-walled microchannel for Reynolds number in the range 250–400 is approximately one-half of that in a rigid-walled microchannel for Reynolds number in the range 5000–20 000. Thus, the fluctuation intensities in a soft-walled channel at a Reynolds number in the range 250–400 are comparable to those for a turbulent flow in a rigid channel at a Reynolds number one to two orders of magnitude higher.

The total shear stress, ${\it\tau}_{xy}$ , is defined as

where the velocity gradient is determined from the slope of the mean velocity profile using a cubic spline fit for the mean velocity. The total stress and the viscous stress ${\it\eta}(\text{d}\overline{v}_{x}/\text{d}y)$ are shown as a function of the $y$ coordinate in figure 14. The error bars are not shown in this figure for clarity. The error bars for the viscous stress are small, because the mean velocity is determined with good accuracy, as shown in figure 10. The error bars in the Reynolds stress are higher, by a factor of 1000 (density of water in SI units), in comparison to the error bars in the correlation $\langle v_{x}^{\prime }v_{y}^{\prime }\rangle$ in figure 12. The Reynolds stress decreases to zero at the top rigid wall, as expected from the no-slip boundary conditions at a rigid wall. However, the Reynolds stress does not decrease to zero at the bottom soft wall, and it constitutes approximately 30 % of the total stress at the soft wall. This implies that the force exerted by the fluid on the wall contains a substantial component due to the fluid velocity fluctuations, and it is not entirely due to the viscous friction at the wall. An important feature to note is the curvature of the stress profiles. For a fully developed unidirectional flow in a channel with a constant pressure gradient, the total stress is a linear function of height. In the present experiments, the flow is not fully developed, since there is a variation in the channel height in the streamwise direction. However, the slope of the wall is numerically small, typically approximately 2 %, in the downstream sections where the velocities are measured. Therefore, the total stress is expected to be close to a linear function of the height. Figure 14 indicates that the slope of the viscous stress is certainly not a constant. However, the slope of the total stress profile is nearly a constant. In order to obtain a nearly constant slope, it is necessary to incorporate the Reynolds stress which is non-zero at the soft wall. This serves as a consistency check for the Reynolds stress which was measured independently from the velocity fluctuations; if the Reynolds stress is calculated from the pressure gradient, the values are close to those evaluated from the velocity fluctuations. The slope of the stress, which is the pressure gradient, does vary significantly with downstream position when the soft wall is made of shear modulus 18 kPa. This is due to the relatively large variation in the height of the microchannel with downstream position; this variation in the pressure gradient was observed even for a laminar flow before transition in Verma & Kumaran (Reference Verma and Kumaran2013). However, when the soft wall is made of shear modulus 26 kPa, there is a relatively small variation of approximately 20 % in the height of the microchannel and a corresponding small variation in the pressure gradient.

Figure 14.

The shear stress ${\it\tau}_{xy}$ (solid line) and the viscous component of the shear stress $({\it\eta}\text{d}\overline{v}_{x}/\text{d}y)$ (dashed line) as a function of the cross-stream distance $y$ for a microchannel with soft wall made with shear modulus 18 kPa and Reynolds number 277 (a), shear modulus 18 kPa and Reynolds number 416 (b), and shear modulus 26 kPa and Reynolds number 416 (c) in zone A (○), B (▵) and C (▿).

The rate of fluctuating energy production per unit mass, $-\langle v_{x}^{\prime }v_{y}^{\prime }\rangle (\text{d}\overline{v}_{x}/\text{d}y)$ , calculated from the Reynolds stress and the mean velocity gradient, is shown in figure 15. In contrast to the near-wall maximum expected for the flow near a rigid wall, it is observed that the fluctuating energy production rate is a maximum at the soft wall. This is due to the non-zero Reynolds stress and the maximum in the velocity gradient at the wall. The fluctuating energy production rate does decrease to zero at the hard upper wall and there is a maximum close to the upper wall. However, the maximum fluctuating energy production rate near the top wall is smaller, by a factor of 4, in comparison to that at the soft wall. The maximum of the fluctuating energy production rate also increases rapidly with a decrease in the channel height, both due to a significant increase in the fluid strain rate at the wall and the Reynolds stress. The rate of energy production at the wall increases sharply with an increase in the Reynolds number after transition in the downstream location C where the microchannel height is a minimum, as shown in figure 15(d), though the increase is not as large at the upstream locations A and B; this is because the strain rate at the wall is smaller at the upstream locations due to the channel expansion. The shape of the fluctuating energy production curve intriguingly suggests that the mechanism that generates velocity fluctuations is not the near-wall turbulence bursting responsible for turbulence production in the flow past a hard surface. Further work is required to establish the mechanism of near-wall fluctuating energy production near the soft wall, though the mechanism is certainly qualitatively different from that in a rigid channel.

Figure 15.

The fluctuating energy production rate per unit mass, $-\langle v_{x}^{\prime }v_{y}^{\prime }\rangle (\text{d}\overline{v}_{x}/\text{d}y)$ , as a function of the cross-stream distance $y$ for a microchannel with soft wall made with shear modulus 18 kPa and Reynolds number 277 (a), shear modulus 18 kPa and Reynolds number 416 (b), and shear modulus 26 kPa and Reynolds number 416 (c) in zone A (○), B (▵) and C (▿), and (d) the maximum rate of fluctuating energy production per unit mass at the wall as a function of the Reynolds number for a microchannel with soft wall made of shear modulus 18 kPa (solid line) and 26 kPa (dashed line) in zone A (○), B (▵) and C (▿).

Figure 16.

The reduced velocity $(\overline{v}_{x}/v_{\ast })$ as a function of the reduced distance $(yv_{\ast }/{\it\nu})$ in the downstream zone C (figure 5) for a microchannel when the soft wall is made of shear modulus 18 kPa (a) and 26 kPa (b), and the Reynolds number is 110 (○), 222 (▵), 277 (▿), 332 (◃), 388 (▹) and 416 (♢), and for a rigid channel of height 1.52 mm at a Reynolds number of 3500 (●). Also shown are the curves for $(\overline{v}_{x}/v_{\ast })=(yv_{\ast }/{\it\nu})$ (dotted line), $(\overline{v}_{x}/v_{\ast })=2.5\log (yv_{\ast }/{\it\nu})+5$ (dashed line), $(\overline{v}_{x}/v_{\ast })=3\log (yv_{\ast }/{\it\nu})-2.3$ (solid line in (a)) and $(\overline{v}_{x}/v_{\ast })=2.8\log (yv_{\ast }/{\it\nu})-2.3$ (solid line in (b)). The data collapse can be compared with the dimensional velocity profiles shown in figure 10.

The viscous sublayer and the logarithmic layer close to the wall are universal features of turbulent flows near rigid surfaces. The viscous sublayer with a linear profile extends up to $(yv_{\ast }/{\it\nu})\sim 10$ and the velocity is given by $(\overline{v}_{x}/v_{\ast })=(yv_{\ast }/{\it\nu})$ , and the logarithmic layer extends from $30<(yv_{\ast }/{\it\nu})<200$ where the velocity profile satisfies the von Karman law $(\overline{v}_{x}/v_{\ast })=A\log (yv_{\ast }/{\it\nu})+B$ , where $A$ and $B$ are the von Karman constants. Here, $v_{\ast }=\sqrt{{\it\tau}_{w}/{\it\rho}}$ is the friction velocity, and ${\it\tau}_{w}$ is the wall shear stress at the soft wall shown in figure 14. Here, an attempt has been made to examine the velocity profile close to the wall, to check whether there is a discernible viscous sublayer and logarithmic layer in the flow past a soft surface. The results for the variation of the reduced velocity $(\overline{v}_{x}/v_{\ast })$ with the reduced distance $(yv_{\ast }/{\it\nu})$ are shown in figure 16. Also shown, for comparison, is the velocity profile in a rigid channel at a Reynolds number of 3500 from the experiments discussed in the Appendix. The velocity profile in the rigid channel shows evidence of a viscous sublayer with the scaling $(\overline{v}_{x}/v_{\ast })=(yv_{\ast }/{\it\nu})$ for $(yv_{\ast }/{\it\nu})<5$ . After this, there is a transition to a profile that seems to agree well with the logarithmic law $(\overline{v}_{x}/v_{\ast })=2.5\log (yv_{\ast }/{\it\nu})+5.0$ , even though the Reynolds number of 3500 for the rigid channel is a little low. In the case of the flow through the microchannel, there is a data collapse when we plot $(\overline{v}_{x}/v_{\ast })$ as a function of $(yv_{\ast }/{\it\nu})$ only after transition, that is, for Reynolds numbers higher than 277 when the soft wall is made of shear modulus 18 kPa and higher than 332 when the soft wall is made of shear modulus 26 kPa. There is no evidence of the linear profile $(\overline{v}_{x}/v_{\ast })=(yv_{\ast }/{\it\nu})$ expected for a viscous sublayer even for $(yv_{\ast }/{\it\nu})$ as low as 2, though a viscous sublayer for smaller values of $(yv_{\ast }/{\it\nu})$ cannot be ruled out based on our data which extend only up to $(yv_{\ast }/{\it\nu})\sim 2$ . The logarithmic layer seems to extend from $(yv_{\ast }/{\it\nu})=4$ to approximately $(yv_{\ast }/{\it\nu})=30$ . This is in contrast to the logarithmic layer in a pipe flow which extends from $(yv_{\ast }/{\it\nu})=30$ to approximately $(yv_{\ast }/{\it\nu})=200$ . In the logarithmic layer, the velocity profile seems to follow the equation $(\overline{v}_{x}/v_{\ast })=A\log (yv_{\ast }/{\it\nu})+B$ , where best fits are obtained when $A$ assumes slightly different values of 3.0 and 2.8 for soft walls made with shear modulus 18 kPa and 26 kPa, and $B$ has a value of approximately $-2.3$ in both cases.

In contrast to the universal logarithmic law for the mean velocity when scaled by the friction velocity, it has been known for some time now that the fluctuating velocity, scaled by the friction velocity, does not exhibit universal ‘wall scaling’ when expressed as a function of $(yv_{\ast }/{\it\nu})$ . This has been interpreted as an indication that the large-scale fluctuations in the outer flow far from the wall affect the turbulent bursting near the wall (Rao, Narasimha & Badri Narayanan Reference Rao, Narasimha and Badri Narayanan1971; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010) which generates the near-wall velocity fluctuations. It is of interest to examine whether this lack of ‘wall scaling’ is also observed for the flow at small dimensions and low Reynolds number. The root mean square of the streamwise fluctuating velocity, scaled by the friction velocity, is shown as a function of the scaled distance $(yv_{\ast }/{\it\nu})$ in figure 17. The data for the scaled fluctuating velocity $v_{x}^{\prime }$ do not collapse onto a single curve, in contrast to those for the mean velocity in figure 16. This is similar to turbulence in a channel with rigid walls, where also it is observed that the streamwise root mean square velocity does not exhibit scaling when expressed in terms of inner variables. However, there is an important difference from rigid-wall-bounded turbulence. The location of the maximum in figure 17, approximately $(yv_{\ast }/{\it\nu})=8$ , is smaller than the value of approximately 15 usually reported for the flow past a rigid surface at much higher Reynolds number. In figure 17, the maximum value of $(v_{x}^{\prime }/v_{\ast })$ is in the range 1.8–2.3, which is not much smaller than the maximum of 2.8–3 for the flow past a rigid surface at Reynolds numbers approximately two to three orders of magnitude higher. It should be noted that $(v_{x}^{\prime }/v_{\ast })$ is large only near the soft wall; it is smaller by a factor of at least two near the hard wall.

Figure 17.

The root mean square of the streamwise fluctuating velocities, $(v_{x}^{\prime }/v_{\ast })$ , as a function of the reduced distance $(yv_{\ast }/{\it\nu})$ in the downstream zone C (figure 5) for a microchannel when the soft wall is made of shear modulus 18 kPa (a) and 26 kPa (b), and the Reynolds number is 110 (○), 222 (▵), 277 (▿), 332 (◃), 388 (▹) and 416 (♢).

3.2. Wall displacement

The average displacement of the soft wall in the streamwise direction, $\bar{u}_{x}$ , at the fluid–wall interface, as measured by the motion of the ink spot discussed in § 2.3, is shown in figure 18(a). There is a displacement of up to $10~{\rm\mu}\text{m}$ in the soft wall with shear modulus 18 kPa, and approximately $6~{\rm\mu}\text{m}$ for the soft wall with shear modulus 26 kPa. Also shown in figure 18(a), by the filled symbols, is the expected displacement from the fluid shear stress at the wall, shown in figure 14, in zone B (figure 5) where the dye spot is located. The expected displacement is calculated as the strain (ratio of the wall shear stress and the shear modulus) times the thickness (2 mm) of the bottom wall. The observed mean displacement is in good quantitative agreement with that expected from the shear stress exerted on the soft wall. It should be mentioned that there is a much larger displacement of the order of $100~{\rm\mu}\text{m}$ in the wall-normal direction as the Reynolds number increases, as is evident from figure 8. Despite this, the displacement tangential to the wall is accurately measured and correlates well with the shear stress exerted on the wall. The root mean squares of the displacement in the streamwise and spanwise directions, $u_{x}^{\prime }$ and $u_{z}^{\prime }$ , calculated from the displacement time series, are shown in figure 18(b). The root mean square displacements $u_{x}^{\prime }$ and $u_{y}^{\prime }$ are non-zero even at zero Reynolds number, due to lack of experimental resolution of distances below approximately $1~{\rm\mu}\text{m}$ . However, a sharp increase in the root mean square displacements in both the streamwise and spanwise directions occurs at the flow transition Reynolds number. The root mean squares of the wall displacement fluctuations are relatively small, of the order of $1{-}2~{\rm\mu}\text{m}$ , but these are clearly discernible in figure 18(b), as well as the time traces of the fluctuations in the position of the dye spot discussed next. This confirms the onset of wall motion coupled with the transition in the fluid.

Figure 18.

The steady displacement in the streamwise direction (a) and the root mean square of the fluctuations in the displacement in the streamwise (solid line) and spanwise (dashed line) directions (b) for the flow through a microchannel with soft wall made of shear modulus 18 kPa (○) and 26 kPa (▵). The filled symbols in (a) show the mean displacement calculated from the wall shear stress in the fluid and the shear modulus and thickness of the soft wall.

A more detailed picture of the wall fluctuations is obtained from the time variation of the displacement, measured as discussed in § 2.3. The time series of the streamwise displacement at different Reynolds numbers are shown in figure 19. High-frequency fluctuations are present even in the laminar regime, due to background noise and the noise in the CCD sensors of the camera. However, after transition (for Reynolds number greater than 250 for a soft wall with shear modulus 18 kPa in figure 19(a) and Reynolds number greater than 332 for a soft wall with shear modulus 26 kPa in figure 19(b)), variations at much lower frequency are visible in figure 19. The amplitudes of the fluctuations are relatively small, of the order of $1{-}2~{\rm\mu}\text{m}$ , but these fluctuations are clearly discernible in the time traces in figure 19(a,b). Based on the linear stability analysis (Verma & Kumaran Reference Verma and Kumaran2013), the frequency of the most unstable modes is expected to be $10^{3}{-}10^{4}~\text{Hz}$ . Such high frequencies cannot be measured with the imaging procedure used here, because the Nyquist frequency is 500 Hz for our imaging system with a framing rate of 1000 f.p.s.; considering an oversampling factor of 10, a framing rate of at least $10^{4}{-}10^{5}~\text{Hz}$ is required for capturing the frequencies predicted by the linear analysis. However, even with the present sampling rates, the appearance of relatively low-frequency structure after transition is also evident in the frequency spectra of the displacement shown in figure 19(c,d).

Figure 19.

Time series (a,b) and frequency spectra (c,d) of the displacement of a dye spot in the streamwise direction at different Reynolds numbers when the soft wall is made with shear modulus 18 kPa (a,c) and 26 kPa (b,d). The curves are displaced vertically with respect to each other to enhance clarity, and the dotted lines show the baseline for each curve.

The observation of a non-zero stress at the wall and wall displacement suggests that the mechanism of fluctuating energy production may be different from that in rigid-walled channels, where the production term is given by $-{\it\rho}\langle v_{x}^{\prime }v_{y}^{\prime }\rangle (\text{d}\bar{v}_{x}/\text{d}y)$ due to the interaction of the Reynolds stress and the mean velocity gradient. Linear stability studies (Verma & Kumaran Reference Verma and Kumaran2013) indicate that the mechanism of destabilisation of the laminar flow is the transfer of energy from the mean flow to the fluctuations due to the shear work done at the fluid–solid interface. The rate at which shear work is done, $\mathscr{S}$ , at the interface is the product of the shear stress and the tangential velocity integrated over the surface,

where $\text{d}S$ is the differential surface area and $\boldsymbol{n}$ is the outward unit normal to the surface. When there is a no-slip condition at the surface, the tangential velocity is zero and so the shear work is zero. When there is a steady flow, the tangential velocity at the surface is once again zero (though there is a non-zero tangential displacement), and the shear work done at the interface is once again zero. When there are fluctuations, it is convenient to separate the tangential velocity at the surface into its mean and fluctuating components, $v_{x}=\bar{v}_{x}+v_{x}^{\prime }$ . The rates of shear work done by the fluctuations on the fluid side $\mathscr{S}_{f}$ and the solid side $\mathscr{S}_{s}$ are

where the subscripts $f$ and $s$ refer to fluid and solid respectively, and the velocity fluctuation in the solid has been set equal to the substantial derivative of the surface displacement, $(\text{D}u_{x}^{\prime }/\text{D}t)$ (the time rate of change of the mean displacement is zero). There is a negative sign in the expression for the shear work on the fluid side, because the outward unit normal is directed in the $-y$ direction. When there is stress continuity, the shear stresses at the surface are equal on the fluid and solid sides. However, the velocity fluctuations on the two sides are not equal when the surface is displaced. For a surface displacement $u_{y}^{\prime }$ , the mean velocity at the displaced surface on the fluid side is $(\text{d}\bar{v}_{x}/\text{d}y)u_{y}^{\prime }$ , where $(\text{d}\bar{v}_{x}/\text{d}y)$ is the mean strain rate, while the mean velocity at the displaced surface on the solid side is zero, since there is no mean motion in the solid. Therefore, the no-slip condition at the displaced surface, linearised about the base surface configuration, is

Due to this additional term in the velocity condition at the interface, the rates of shear work on the fluid and solid sides do not sum to zero,

When the stress is separated into a mean and a fluctuating part, ${\it\tau}_{xy}=\bar{{\it\tau}}_{xy}+{\it\tau}_{xy}^{\prime }$ , the correlations between the stress fluctuations ${\it\tau}_{xy}^{\prime }$ and the displacement fluctuations $u_{y}^{\prime }$ can be non-zero, resulting a transfer of energy from the mean flow to the fluctuations which destabilises the laminar flow (Kumaran Reference Kumaran1995). While this mechanism is well established for the flow instability, it has not yet been demonstrated for energy production after transition. Although this provides a plausible mechanism different from the usual fluctuating energy production, a lot more theoretical work is required to conclusively establish the energy production mechanism in the present case.