Periodically driven Taylor-Couette turbulence

We study periodically driven Taylor-Couette turbulence, i.e. the flow confined between two concentric, independently rotating cylinders. Here, the inner cylinder is driven sinusoidally while the outer cylinder is kept at rest (time-averaged Reynolds number is $Re_i = 5 \times 10^5$). Using particle image velocimetry (PIV), we measure the velocity over a wide range of modulation periods, corresponding to a change in Womersley number in the range $15 \leq Wo \leq 114$. To understand how the flow responds to a given modulation, we calculate the phase delay and amplitude response of the azimuthal velocity. In agreement with earlier theoretical and numerical work, we find that for large modulation periods the system follows the given modulation of the driving, i.e. the system behaves quasi-stationary. For smaller modulation periods, the flow cannot follow the modulation, and the flow velocity responds with a phase delay and a smaller amplitude response to the given modulation. If we compare our results with numerical and theoretical results for the laminar case, we find that the scalings of the phase delay and the amplitude response are similar. However, the local response in the bulk of the flow is independent of the distance to the modulated boundary. Apparently, the turbulent mixing is strong enough to prevent the flow from having radius-dependent responses to the given modulation.


Introduction
Periodically driven turbulent flows are omnipresent. Well-known examples include blood flow driven by the beating heart, the flow in internal combustion engines, the earth's atmosphere which is periodically heated by the sun, and tidal currents caused by periodic changes in the gravitational attraction of both the moon and sun. Despite their ubiquity, periodically driven turbulent flows received relatively little scientific attention.
In the last few decades, only some efforts were made to explore the field of periodically driven turbulence, and most of them only on the theoretical or numerical side. Analytical studies focussed on the global response of the system, i.e. the response amplitude and the phase shift of the quantities such as a global Reynolds number (Lohse 2000), or the total energy in the system (von der Heydt et al. 2003a). Most numerical studies in addition only used simplified models, such as the GOY shell model or the reduced wave vector set approximation (REWA) (Hooghoudt et al. 2001;von der Heydt et al. 2003b;Bos et al. 2007;Hamlington & Dahm 2009). Only a limited number of DNS studies have been performed in this field, because of the computational costs needed to achieve both fully developed turbulence and sufficient statistical convergence with temporal dependence (Yu & Girimaji 2006;Kuczaj et al. 2006Kuczaj et al. , 2008. Experimentally, periodically driven turbulence has been studied in a number of well-known systems, such as Rayleigh-Bénard convection (Jin & Xia 2008;Sterl et al. 2016), von Kármán flow (Cadot et al. 2003) and wind tunnels (Cekli et al. 2010). In these systems the forcing was periodically varied over time, with the variations being of O(10%) of either the average forcing or the energy input.
Two different ways to periodically drive the turbulence have been studied, namely either a kicked energy input, followed by energy decay (Lohse 1994(Lohse , 2000Hooghoudt et al. 2001;Jin & Xia 2008), or a sinusoidal driving of the flow, in which either the energy input or the forcing is modulated (von der Heydt et al. 2003a,b;Cadot et al. 2003;Kuczaj et al. 2006;Chien et al. 2013). In kicked turbulence, the Reynolds number of the flow saturates depending on the kicking strength and kicking frequency (Lohse 2000;Jin & Xia 2008). The main observations made in the studies on sinusoidal driven turbulence were similar regarding the global response of the system. In the limit of extremely small modulation frequencies of the boundary conditions (BCs), the flow can fully respond to these changes, meaning that the flow behaves quasi-stationary. In this regime, no phase delay Φ delay between the response and the modulation is observed, and the response amplitude is identical to the modulation amplitude. As the modulation frequency is increased, the fluid system cannot follow the changing BC: the response amplitude decreases and a phase delay between input and response is observed. In the extreme case of an infinite modulation frequency, the response amplitude vanishes and a phase delay can no longer be defined. In this regime, the system behaves as if it were constantly driven. Several studies observe resonances in the response, i.e. certain modulation frequencies for which the amplitude of the response is strongly enhanced (von der Heydt et al. 2003a;Cadot et al. 2003;Cekli et al. 2010;Sterl et al. 2016). These resonance peaks were absent in others, presumably because the peaks are "washed out" by turbulent fluctuations (von der Heydt et al. 2003b).
In this manuscript, we study the physics of periodically driven turbulence in a Taylor-Couette (TC) apparatus, employing a sinusoidally driven inner cylinder. TC flow, i.e. the flow of a fluid confined in the gap between two concentric cylinders, is one of the canonical systems in which the physics of fluids is studied, see e.g. the recent reviews by Fardin et al. (2014) and Grossmann et al. (2016). It has the advantage of being a closed system with an exact global energy balance (Eckhardt, Grossmann & Lohse 2007), and due to its simple geometry TC systems can be accessed experimentally with high precision. Apart from several recent studies which focussed on the decay of turbulent TC flow (Ostilla-Mónico et al. 2014;Verschoof et al. 2016;Ostilla-Mónico et al. 2017), or time-dependent driving close to the low Reynolds number Taylor-vortex regime (Ahlers 1987;Walsh & Donnelly 1988;Barenghi & Jones 1989;Ganske et al. 1994a,b;Borrero-Echeverry et al. 2010), to our knowledge no work has been conducted so far on TC turbulence with time-dependent driving.
The outline of this article is as follows. We start by explaining the experimental method in §2. The results, in which we present the response of the flow, are shown in §3. Finally, we conclude this paper in §5. camera laser z r Figure 1. Schematic of the vertical cross-section of the T 3 C facility. The laser illuminates a horizontal plane (r, θ) at midheight (z = l/2) for all PIV measurements. The flow is imaged from the bottom with a high resolution sCMOS camera to obtain the velocity components u θ and ur in the (r, θ) plane.

Experimental method
In this study, we restrict ourselves to the case of inner cylinder rotation, while keeping the outer cylinder at rest. The inner cylinder rotation is set to is the rotation rate of the inner cylinder at time t and T is the period of the modulation. The time t is related to the phase Φ by Φ = 2πt/T . The modulation amplitude is set to e = 0.10 throughout this work, so that the mean flow is one order of magnitude larger than the induced modulation. The time-averaged rotation rate f i t is set to f i t = 5 Hz, resulting in a time-averaged Reynolds number of Re i t = u i t d/ν = 2π f i t r i d/ν = 5 × 10 5 . In this equation, u i = 2πf i r i equals the velocity of the inner cylinder with radius r i , ν is the kinematic viscosity and d is the gap width between the cylinders. Here, we are in the so-called 'ultimate turbulence' regime, in which both the bulk flow and boundary layers are fully turbulent (Kraichnan 1962;Chavanne et al. 1997;Grossmann & Lohse 2011;Huisman et al. 2012). We varied the modulation period T from 180 s down to 3 s. The modulation frequency was limited by the power of the motor needed to accelerate and decelerate the mass of the inner cylinder (80 kg). We then simultaneously measured the rotational speed of the inner cylinder f i (t) and the fluid velocity by using non-intrusive Particle Image Velocimetry (PIV). The experiments were performed in the Twente Turbulent Taylor-Couette (T 3 C) facility (van Gils et al. 2011), as shown schematically in figure 1. The apparatus has an inner cylinder with a radius of r i = 200 mm and a transparent outer cylinder with a radius of r o = 279 mm, resulting in a radius ratio of η = r i /r o = 0.716, a gap width d = r o − r i = 79 mm. The height of the setup is l = 927 mm, giving an aspect ratio of Γ = l/d = 11.7. As working fluid we use water with a temperature of T = 20 • C, which is kept constant within 0.2 K by active cooling through the end-plates of the setup. More experimental details of this facility can be found in van Gils et al. (2011).
The PIV measurements were performed in the r−θ plane at mid-height (z = l/2) using a high-resolution camera operating at 15 fps (pco.edge camera, double frame sCMOS, 2560×2160 pixel resolution). We illuminate the flow from the side with a horizontal laser sheet, as shown in figure 1. The used laser is a pulsed dual-cavity 532 nm Quantel Evergreen 145 Nd:YAG laser. We seeded the water with 1-20 µm fluorescent polyamide particles. We calculate the Stokes number which equals St = τ p /τ η = 0.0019 ≪ 1. Furthermore, the mean particle radius is roughly 6 times smaller than our Kolmogorov  length scale, thus we can be sure that the particles faithfully follow the flow. The images are processed with interrogation windows of 32 × 32 pixel with 50% overlap, resulting in u θ (r, θ, t) and u r (r, θ, t).

Velocity response
In figure 2 we show the normalized driving and response of the mid-gap flow velocity u θ (r = 0.5, t) for three different modulation periods. The radius is non-dimensionalized asr = (r − r i )/d, so thatr = 0 corresponds to the inner cylinder andr = 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 Bottom row (d-f ) u θ (Φ) is normalized by the instantaneous inner cylinder velocity at phase Φ, i.e. ui(Φ) (a value between ui(0.5π) = 6.9 m/s and ui(1.5π) = 5.7 m/s). A collapse of all lines indicates that the system behaves quasi-stationary, as can be seen for large T in figure (f). The solid black lines show the azimuthal velocity profile for Rei = 5×10 5 for the non-modulated, stationary case (data from Huisman et al. (2013)). Bottom right (g) The azimuthal velocity u θ (Φ) is shown for a series of phases of the modulation; here we show data for phases between 0.5π Φ 1.5π, i.e. half of a modulation cycle, as shown in this inset. See also figure 2 for the definition of phase Φ.
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 θ 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 (von der Heydt et al. 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 Φ. A fully quasi-stationary cycle completely follows the grey line, in which u θ / u θ t = u i / u i t . The T = 180 s modulation period 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 the top row, the data is normalized by u i t = 2π f i t r i = 6.3 m/s, i.e. the same constant for all driving periods T . The better all lines collapse, the smaller the response amplitude is. For the bottom row, we chose to normalize with u i (Φ) = 2πr i f i t [1 + e sin(Φ)], 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. 2013). Figure 4(a) and (f) depict the most extreme cases. In figure 4(a), the azimuthal velocity of the flow is almost constant over a modulation cycle, and therefore u θ (r, Φ) 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. When T = 180 s, the opposite is the case, see figure 4(f). Here, for every phase Φ, the azimuthal velocity profile is identical to the statistically stationary solution for f i (Φ). This behaviour is surprisingly constant over the entire radius.

Phase delay
Up to now the conclusions drawn from figures 2, 3, and 4 were only qualitative. Here, we quantify the phase shift and amplitude response. We extract the phase delay Φ delay between the modulation and the response by cross-correlating u i (t) and u θ (t). We detect the first peak in u i ⋆ u θ (τ ), 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 modulation period T decreases, the bulk flow cannot follow the changing BCs anymore and it responds with an increasing delay. Within this approximation, von der Heydt et al. (2003a) calculated, and Cadot et al. (2003) experimentally found, that the phase delay has a linear dependence on the modulation frequency. We do not observe a similar behaviour, however. As visible in figure 5b, in this work the dependence of Φ delay is better described by an effective power law over a range of larger values of measured T , with Φ delay ∝ T −0.55 . The exponent −0.55 is to be seen as an effective exponent, describing the experimentally observed results, and not as a theoretical result.
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 timescale τ bulk for the movement from the inner to the outer cylinder, using the Reynolds wind number Re w = σ(u r )d/ν, in which σ(u r ) is the standard deviation of the radial velocity. We estimate τ bulk = d/σ(u r ) = d 2 /Re w ν. Re w for the corresponding Re i t = 5 × 10 5 is known from Huisman et al. (2012), resulting in a τ bulk = 0.27 s. As long as τ bulk ≪ T , the radial dependence of the phase delay and amplitude should be negligible, in agreement with our observations. Such periods T are unfortunately not accessible experimentally due to the moment of inertia of the cylinders.

Amplitude response
We calculate the amplitude A of the response for both the velocity and kinetic energy, which is defined as E = 1 2 u · u ≈ 1 2 u 2 θ . Following the approach of von der Heydt et al. (2003a), the local oscillating response of the velocity and energy is calculated as (3.1) We average ∆ u (t) and ∆ E (t) radially and azimuthally, and make the ansatz that ∆ u,E (t) can be described as: with A(T ) as sole fitting parameter. Φ delay is not a fitting parameter, as it is calculated using cross-correlation, see figure 5. In the case of slow, quasi-stationairy modulations, the amplitude response of the azimuthal velocity can be calculated from equations (3.1), namely A u = (1+e) 1 − 1 /e = 1. Strictly speaking is it 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 straight-forward, as the response amplitude varies over the sine wave. We calculate A max E = (1 + e) 2 − 1 /e = 2.1 and A min E = (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. T → 0 implies that A u,E → 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 by von der Heydt et al. (2003a) and Cadot et al. (2003)), although the overall trend is similar as in those papers.
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 modulated wall. Because of the no-slip condition at the wall, the values of A and Φ delay directly at the wall are fixed, i.e. A u (r i ) = 1 and Φ delay (r i ) = 0. At the outer cylinder, A u (r o ) = 0, hence Φ 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.

Summary and conclusions
To conclude, we studied periodically driven Taylor-Couette turbulence. We drove the inner cylinder sinusoidally, and measured the local velocity using PIV. Consistent with earlier studies and theoretical expectations, we observe a phase delay and declining velocity response as we decrease the modulation period. Most surprisingly, we did not observe a radial dependence of the phase delay in the bulk of the flow, nor of the amplitude response, in contrast to the expectation one might have that there could be a larger influence of the modulation on the flow close to the modulated wall. Apparently, a radial dependence of A and Φ delay is prevented by the strong mixing in this turbulent flow. Even though we did not measure directly in the boundary layers, their vital importance in transferring modulations to the bulk flow is evident.
To further study this interesting phenomenon, direct numerical simulations are necessary to cover the extremely small modulation period range, which is inaccessible in experiments. Using such data, it will be possible to study the interplay between the modulated cylinder, the boundary layers and the bulk in more detail, as the entire flowfield will then be available. Another domain of "terra incognita" is the study of modulations with larger amplitude. Here, we limited ourselves to a modulation amplitude of e = 0.1. Larger values induce non-linear effects, and linear response type assumptions such as those made in equations (3.1) and (3.2) will then not be valid anymore.