3.1 Velocity response

In figure 2 we show the normalized driving and response of the mid-gap flow velocity $u_{\unicode[STIX]{x1D703}}(\tilde{r}=0.5,t)$ for three different modulation periods. The radius is non-dimensionalized as $\tilde{r}=(r-r_{i})/d$ , so that $\tilde{r}=0$ corresponds to the inner cylinder and $\tilde{r}=1$ to the outer one. We non-dimensionalize both velocities by their time-averaged value, so both lines meander around $1$ . For all oscillation periods, the mid-gap flow velocity oscillates with the same period $T$ as the driving. The amplitude and phase delay of the response depend on the driving period. For the larger modulation periods $T$ , $u_{\unicode[STIX]{x1D703}}$ responds nearly in phase with the same amplitude as the driving. For smaller modulation periods, the response amplitude decreases and a phase delay is observed, just as in prior studies (Cadot et al. Reference Cadot, Titon and Bonn2003; von der Heydt et al. Reference von der Heydt, Grossmann and Lohse2003a ,Reference von der Heydt, Grossmann and Lohse b ; Kuczaj et al. Reference Kuczaj, Geurts, Lohse and van de Water2008; Hamlington & Dahm Reference Hamlington and Dahm2009).

Figure 4. Azimuthal velocity profiles as a function of dimensionless radius $\tilde{r}$ . All data are phase-averaged and normalized. (a–c) Here $u_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6F7})$ is normalized with the time-averaged inner cylinder velocity $\langle u_{i}\rangle _{t}=6.3~\text{m}~\text{s}^{-1}$ , i.e. the same constant value for all lines. A collapse of all lines indicates that the response amplitude is small, as is observed for large $Wo$ (see panel (a)). Furthermore, we show the response of laminar flow to the modulation, calculated numerically (see § 2). (d–f) Here $u_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6F7})$ is normalized by the instantaneous inner cylinder velocity at phase $\unicode[STIX]{x1D6F7}$ , i.e.  $u_{i}(\unicode[STIX]{x1D6F7})$ (a value between $u_{i}(0.5\unicode[STIX]{x03C0})=6.9~\text{m}~\text{s}^{-1}$ and $u_{i}(1.5\unicode[STIX]{x03C0})=5.7~\text{m}~\text{s}^{-1}$ ). A collapse of all lines indicates that the behaviour of the system is quasi-stationary, as can be seen for small $Wo$ in panel (f). The solid grey lines show the azimuthal velocity profile for $Re_{i}=5\times 10^{5}$ for the non-modulated, stationary case (data from Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013b )). (g) The azimuthal velocity $u_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D6F7})$ is shown for a series of phases of the modulation; here we show data for phases between $0.5\unicode[STIX]{x03C0}\leqslant \unicode[STIX]{x1D6F7}\leqslant 1.5\unicode[STIX]{x03C0}$ , i.e. half of a modulation cycle, as shown in this inset. See also figure 2 for the definition of phase $\unicode[STIX]{x1D6F7}$ .

A different representation of a modulation cycle is depicted in figure 3. Here we plot the data from figure 2 parametrically as a function of $\unicode[STIX]{x1D6F7}$ . A fully quasi-stationary cycle completely follows the grey line, in which $u_{\unicode[STIX]{x1D703}}/\langle u_{\unicode[STIX]{x1D703}}\rangle _{t}=u_{i}/\langle u_{i}\rangle _{t}$ . The $Wo=15$ measurement is close to this line. The deviation from this line, which indicates a phase delay, increases for smaller modulation periods.

To study whether the flow responds similarly over the gap width, we extend the analysis from figure 2 to the entire radius (see figure 4). In figure 4(a–c), the data are normalized by $\langle u_{i}\rangle _{t}=2\unicode[STIX]{x03C0}\langle f_{i}\rangle _{t}r_{i}=6.3~\text{m}~\text{s}^{-1}$ , i.e. the same constant for all measurements. The better all lines collapse, the smaller the response amplitude is. For the bottom row, we chose to normalize with $u_{i}(\unicode[STIX]{x1D6F7})=2\unicode[STIX]{x03C0}r_{i}\langle f_{i}\rangle _{t}[1+e\sin (\unicode[STIX]{x1D6F7})]$ , i.e. the inner cylinder velocity at the corresponding phase in the modulation. Here, when all lines collapse, the modulation is slow enough for the flow to react to the modulation, i.e. the system is in a quasi-stationary state. For comparison, the azimuthal velocity profile for the non-modulated case is shown as a grey line (Huisman et al. Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013b ). Figures 4(a) and 4(f) depict the most extreme cases. Furthermore, we show the laminar flow response in the top row. In figure 4(a), the azimuthal velocity of the flow is almost constant over a modulation cycle, and therefore $u_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D6F7})$ is close to the non-modulated statistically stationary solution for $f_{i}=5$  Hz; the flow cannot adapt to the quick changes of the inner cylinder. For larger Womersley numbers, the opposite is the case (see figure 4 f). Here, for every phase $\unicode[STIX]{x1D6F7}$ , the azimuthal velocity profile is identical to the statistically stationary solution for $f_{i}(\unicode[STIX]{x1D6F7})$ . This behaviour is surprisingly constant over the entire radius. We note that it might appear as if the correct boundary conditions are not met. However, as shown in Huisman et al. (Reference Huisman, Scharnowski, Cierpka, Kähler, Lohse and Sun2013b ), the boundary layer at the studied Reynolds number is too thin to resolve from the current measurements.

The laminar flow response is completely different as compared to the measured turbulent case. First, the response in the flow is restricted to a thin layer close to the inner cylinder wall. Calculating the thickness of the Stokes boundary layer, although slightly off due to the presence of the outer cylinder and a cylindrical coordinate system, gives a similar result, i.e.  $\tilde{\unicode[STIX]{x1D6FF}}(Wo=88)=0.10$ , $\tilde{\unicode[STIX]{x1D6FF}}(Wo=36)=0.24$ and $\tilde{\unicode[STIX]{x1D6FF}}(Wo=15)=0.60$ (see table 1). Second, the response is radius-dependent, as is also known from Stokes oscillating plate theory, as the response decays exponentially with increasing distance from the oscillating wall. These observations highlight how turbulent mixing enhances the transport of the modulation over the entire radius.

Figure 5. The delay between the driving modulation and the fluid velocity response as a function of Womersley number $Wo$ . The delay $\unicode[STIX]{x1D6F7}_{delay}$ is normalized with $2\unicode[STIX]{x03C0}$ of the modulation. The phase delay is calculated for a number of radii, not showing much difference. The same data are shown in linear scale (a) and logarithmic scale (b). The results are radially binned within $\tilde{r}\pm 0.025$ . The inset in (a) shows how the phase delay $\unicode[STIX]{x1D6F7}_{delay}$ is defined: $\unicode[STIX]{x1D6F7}_{delay}$ is calculated by cross-correlating both signals. We included the scaling of the response for laminar flow, which equals $\unicode[STIX]{x1D6F7}_{delay}\propto Wo$ .

3.2 Phase delay

Up to now the conclusions drawn from figures 2–4 were only qualitative. Here, we quantify the phase shift and amplitude response for the turbulent case. We extract the phase delay $\unicode[STIX]{x1D6F7}_{delay}$ between the modulation and the response by cross-correlating $u_{i}(t)$ and $u_{\unicode[STIX]{x1D703}}(t)$ . We detect the first peak in $u_{i}\star u_{\unicode[STIX]{x1D703}}(\unicode[STIX]{x1D70F})$ , and obtain the phase delay by fitting a Gaussian function through this peak and its two neighbouring points, to obtain the peak with increased accuracy. As visible in figure 5, at large modulation periods, the phase delay is small, as we already qualitatively concluded from figure 2. As the Womersley number increases, the bulk flow cannot follow the changing boundary conditions any more and it responds with an increasing delay. Within this approximation, von der Heydt et al. (Reference von der Heydt, Grossmann and Lohse2003a ) calculated, and Cadot et al. (Reference Cadot, Titon and Bonn2003) experimentally found, that the phase delay has a linear dependence on the modulation frequency, i.e.  $\unicode[STIX]{x1D6F7}_{delay}\propto Wo^{2}$ . We do not observe a similar behaviour, however. The results in the aforementioned studies, which both study homogeneous and isotropic turbulence (HIT), are significantly different from what we observe in our TC set-up, which cannot be regarded as HIT (Huisman, Lohse & Sun Reference Huisman, Lohse and Sun2013a ).

As visible in figure 5(b), in this work the dependence of $\unicode[STIX]{x1D6F7}_{delay}$ is better described by an effective power law over a range of larger values of $Wo$ , with $\unicode[STIX]{x1D6F7}_{delay}\propto Wo^{1.1}$ . For the laminar case, the phase lag in the Stokes boundary layer problem is calculated as $\unicode[STIX]{x1D6F7}_{delay}=\sqrt{2}\tilde{r}Wo$ . The exponent $1.1$ is close to the value of the laminar flow response, in which there is a linear dependence between the Womersley number and the phase delay. The phase lag saturates at around $\unicode[STIX]{x1D6F7}_{delay}=\unicode[STIX]{x03C0}/2$ , similar to what is known in pulsating pipe flow (Womersley Reference Womersley1955; Shemer et al. Reference Shemer, Wygnanski and Kit1985) and in periodically forced harmonic oscillators, for example.

We now come to the spatial dependence of the response. Intuitively, one expects an increasing phase delay further away from the modulated wall. Surprisingly, this is not the case. Apparently, the turbulent mixing of this highly turbulent flow prevents the system from having a range of phase delays over the radius, given the fact that the modulation has been ‘passed on’ from the boundary layer to the bulk flow. This can be explained by calculating a characteristic time scale $\unicode[STIX]{x1D70F}_{bulk}$ for the movement from the inner to the outer cylinder, using the Reynolds wind number $Re_{w}=\unicode[STIX]{x1D70E}(u_{r})d/\unicode[STIX]{x1D708}$ , in which $\unicode[STIX]{x1D70E}(u_{r})$ is the standard deviation of the radial velocity. We estimate $\unicode[STIX]{x1D70F}_{bulk}=d/\unicode[STIX]{x1D70E}(u_{r})=d^{2}/Re_{w}\unicode[STIX]{x1D708}$ . For the corresponding $\langle Re_{i}\rangle _{t}=5\times 10^{5}$ , the $Re_{w}$ value is known from Huisman et al. (Reference Huisman, van Gils, Grossmann, Sun and Lohse2012), resulting in a $\unicode[STIX]{x1D70F}_{bulk}=0.27$  s. As long as $\unicode[STIX]{x1D70F}_{bulk}\ll T$ , the radial dependence of the phase delay and amplitude should be negligible, in agreement with our observations. Such small periods $T$ are unfortunately not accessible experimentally due to the moment of inertia of the cylinders.

3.3 Amplitude response

We calculate the amplitude $A$ of the response for both the velocity and kinetic energy, which is defined as $E=(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{u})/2\approx u_{\unicode[STIX]{x1D703}}^{2}/2$ . Following the approach of von der Heydt et al. (Reference von der Heydt, Grossmann and Lohse2003a ), the local oscillating responses of the velocity and energy are calculated as

(3.1a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{u}(t)=\frac{u_{\unicode[STIX]{x1D703}}(t)}{\langle u_{\unicode[STIX]{x1D703}}\rangle _{t}}-1\quad \text{and}\quad \unicode[STIX]{x1D6E5}_{E}(t)=\frac{E(t)}{\langle E\rangle _{t}}-1.\end{eqnarray}$$

We average $\unicode[STIX]{x1D6E5}_{u}(t)$ and $\unicode[STIX]{x1D6E5}_{E}(t)$ radially and azimuthally, and make the ansatz that $\unicode[STIX]{x1D6E5}_{u,E}(t)$ can be described as

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{fit}(t)=eA(T)\sin (2\unicode[STIX]{x03C0}t/T+\unicode[STIX]{x1D6F7}_{delay}).\end{eqnarray}$$

Then $\unicode[STIX]{x1D6E5}_{fit}(t)$ is fitted to $\unicode[STIX]{x0394}(t)$ with $A(T)$ as sole fitting parameter; $\unicode[STIX]{x1D6F7}_{delay}$ is not a fitting parameter, as it is calculated using cross-correlation (see figure 5). In the case of slow, quasi-stationary modulations, the amplitude response of the azimuthal velocity can be calculated from (3.1), namely $A_{u}=((1+e)/1-1)/e=1$ . Strictly speaking it is impossible to describe the kinetic energy with a sinusoidal function, as it has a squared dependence on the velocity, but, as $e$ is small, a sine wave can be used within the assumption of a linear response. However, the calculation of $A_{E}$ in the quasi-stationary case is less straightforward, as the response amplitude varies over the sine wave. We calculate $A_{E}^{max}=((1+e)^{2}-1)/e=2.1$ and $A_{E}^{min}=((1-e)^{2}-1)/(-e)=1.9$ as the two extremes, leading to a phase-averaged value of $A_{E}=2.0$ . Both response amplitudes will vanish in the limit of infinitely fast modulations, i.e.  $Wo\rightarrow \infty$ implies that $A_{u,E}\rightarrow 0$ .

As figure 6 clearly shows, the fluid completely follows the imposed modulation at larger modulation periods, i.e. amplitude responses of $A_{u}=1$ and $A_{E}=2$ are observed, which corresponds to our expectations. For smaller modulation periods, the response amplitude decreases. We do not observe clean power laws, as predicted assuming HIT by von der Heydt et al. (Reference von der Heydt, Grossmann and Lohse2003a ) and Cadot et al. (Reference Cadot, Titon and Bonn2003) shown as dashed lines. The response of the flow can better be described by an exponential function, as indicated by the solid black line. This is in line with the laminar flow response, in which the amplitude of the response also is an exponential function of the Womersley number and the distance to the modulated wall. Note that, in contrast to the turbulent case, the amplitude response of the laminar case depends on the radius.

Figure 6. Amplitude response as a function of the Womersley number $Wo$ for various dimensionless radii. The coloured lines represent our measurements, and the solid grey lines are numerically calculated laminar flow responses. (a) The response amplitude of the velocity $A_{u}$ and (b) the response amplitude of the energy $A_{E}$ . The experimental results are radially binned between $\tilde{r}\pm 0.025$ . The dashed grey lines show the scalings of $A$ as predicted by von der Heydt et al. (Reference von der Heydt, Grossmann and Lohse2003a ). We included the laminar responses, shown in solid grey lines. A number of radii are included, to highlight the dependence on the radius, which does not exist in the well-mixed turbulent case. The effective slope of the measurements $A\propto \text{e}^{-0.025Wo}$ is shown in dashed black. This would correspond to the slope of the laminar flow response at $\tilde{r}\approx 0.035$ .

Similar to the phase delay between modulation and response, also in the response amplitude we do not observe any trend over the radius. Here, one could expect a decreasing $A$ at higher radii, i.e. further away from the modulated wall. Because of the no-slip condition at the wall, the values of $A$ and $\unicode[STIX]{x1D6F7}_{delay}$ directly at the wall are fixed, i.e. $A_{u}(r_{i})=1$ and $\unicode[STIX]{x1D6F7}_{delay}(r_{i})=0$ . At the outer cylinder, $A_{u}(r_{o})=0$ , hence $\unicode[STIX]{x1D6F7}_{delay}(r_{o})$ cannot be defined. Clearly, the boundary layers play a pivotal role in transferring perturbations and modulations to the bulk of the flow.