As reported in a number of recent studies, turbulence in pipe flow is transient for Re<2000 and the flow eventually always returns to the laminar state. Generally, the lifetime of turbulence has been observed to increase rapidly with Reynolds number but there is currently no accord on the exact scaling behaviour. In particular, it is not clear whether a critical point exists where turbulence becomes sustained or if it remains transient. We here aim to clarify if these conflicting results may have been caused by the different experimental and numerical protocols used to trigger turbulence in these studies.
Even 125 years after the pioneering study of Reynolds (1883) the transition between the laminar and the turbulent state in a cylindrical pipe is not clearly understood. As reported by Reynolds, pipe flow is laminar when inertia is small or more precisely when the Reynolds number Re≲2000.1 As the Reynolds number increases, however, flows will typically be turbulent. However, there is no clear transition point and if experiments are carried out with great care laminar flow can be achieved at much larger Reynolds numbers (Reynolds 1883; Draad et al. 1998). It is now generally believed that the laminar state is stable for all Reynolds numbers (Drazin & Reid 1981) and, hence, a perturbation with finite amplitude is required to trigger turbulence. At low Reynolds numbers perturbations decay almost immediately and, only when the Reynolds number is above 1600 or so, so-called turbulent puffs (Wygnanski & Champagne 1973) develop. These localized regions of disordered motion are typically 10–20 diameters long and travel downstream at about the mean velocity of the flow. Various observations have shown that these turbulent patches can decay back to laminar flow and that their lifetime increases with Re. The decay typically takes place suddenly without any precursors (Faisst & Eckhardt 2004). Equally lifetime measurements have shown that the probability for a turbulent puff to decay is independent of its age and that probability distributions in the asymptotic limit scale are exponentially with(1.1)where P is the probability for the flow to still be turbulent after a time t, τ is the characteristic lifetime and t0 is an initial period required for the system to reach the turbulent state. A theoretical model describing this type of behaviour is that of a chaotic saddle. An issue of current debate is the scaling of the characteristic lifetime τ with Reynolds number. A number of studies (Faisst & Eckhardt 2004; Peixinho & Mullin 2006; Willis & Kerswell 2007) have reported a divergence of lifetimes to infinity at a critical point when the chaotic saddle is believed to turn into a strange attractor and turbulence becomes sustained. Other studies have reported non-diverging lifetime scalings that when extrapolated to larger Reynolds numbers suggest that turbulence remains a transient (Hof et al. 2006), a feature also observed in superfluid turbulence (Schoepe 2004).
The above studies, however, differ in the way turbulence was created and it is not clear in how far the perturbation procedures used may be responsible for the different results observed. While in the experiments of Peixinho & Mullin (2006) and the simulations of Willis & Kerswell (2007) turbulence was triggered at a larger Reynolds number that was then reduced to the value where the lifetime study was carried out, in other studies triggering of the turbulent state and the subsequent lifetime measurement were carried out at the same Reynolds number (Faisst & Eckhardt 2004; Hof et al. 2006). Indeed, in some experimental studies, considerable changes in the lifetime scaling were observed for differing perturbation schemes (Mullin & Peixinho 2006a,b). In numerical studies, on the other hand (Schneider & Eckhardt 2008), only in the short time limit a different decay rate was observed while the asymptotic decay rate was found to be independent of the initial disturbance used.
One additional difficulty arising in the interpretation of previous results is the influence of the formation time t0 on the lifetime scaling. t0 typically depends on the perturbation used. In many studies, especially highly resolved numerical investigations, observation times are much shorter than in experiments and consequently they are more strongly affected by uncertainties in t0. The formation time t0 as well as possible deviations in the initial decay rate from the asymptotic are likely to depend on the initial trigger as well as on the Reynolds number. In order to clarify whether these issues are indeed important for lifetime studies in experiments where observation times are typically much larger than in simulations, we measured the decay rate of turbulence for four different perturbation schemes and two different pipe lengths. Finally, we qualitatively compare velocity fields of turbulent puffs triggered by different disturbances.
2. Experimental methods
The pipe was made of glass and consisted of eight sections, each 1.2 m in length and with an inner diameter of D=4.0±0.01 mm. A sketch of the experimental setup is presented in figure 1. Using gravity as the driving force, water entered the pipe from a reservoir of adjustable height. Before entering the pipe, the water passed a 15 m long tube with an inner diameter of 6 mm. Since the flow rate is constant, the larger diameter of this tube ensured that the flow here was always laminar. In essence, this tube provided a constant resistance to the flow in series with the actual pipe and increased the pressure drop required to reach a certain Reynolds number in the pipe. In this way, fluctuations of the flow rate during transition, which are caused by the increased pressure drop across the turbulent section relative to the total pressure drop, were minimized. After triggering a single turbulent puff at Re=2000 the drop in the flow rate was measured to be smaller than 0.1 per cent. Hence, in this Reynolds number regime the mass flux is constant for all practical purposes. In order to guarantee that the applied pressure difference is constant for long times, the fluid level in the reservoir had to be kept constant. This was achieved by continuously pumping water from a larger container into the reservoir that consequently was permanently overflowing. Fluctuations of the fluid level in the reservoir were lower than 1 mm. The height difference between the water in the reservoir and the pipe exit, required to reach Re∼2000, was approximately (depending on the water temperature) 1.1 m. Hence, pressure fluctuations are smaller than one part in a thousand. In order to perform good statistics over many experiments, the Reynolds number has to be kept constant to within approximately 0.1 per cent over several hours and therefore temperature variations have to be avoided. A temperature control was implemented by immersing the 15 m long tube connecting the reservoir and the pipe in a temperature-controlled bath that hence acted as a heat exchanger. The temperature of the liquid was monitored at the outlet of the pipe. Using this information, a computer program adjusts the temperature of the heat exchanger accordingly (figure 1). With this procedure the water temperature at the outlet is kept constant with a precision of ±0.05 K. Additional measurements showed that the temperatures at the inlet and outlet of the pipe differ by less than 0.01 K. This procedure allows the Reynolds number to be kept constant with a precision of ±0.15 per cent over a period sufficient to carry out more than 500 measurements.
The experimental procedure then was as follows. First, a perturbation was applied at a fixed position in the pipe. This perturbation generates a turbulent puff of fixed length that travels downstream the pipe at a velocity similar to the mean velocity of the flow. Four different types of perturbations were used as discussed below. The survival or decay of the puff after a time t was observed at the pipe outlet by monitoring the outflow angle. Since the velocity profile of the turbulent state is different from the laminar one, a change of the outflow angle is clearly observable when the puff exits the pipe (Rotta 1956; Hof et al. 2006). For each Reynolds number and perturbation, between 50 and 500 puffs were generated to determine the puff survival rate P(t, Re).
In our studies of lifetimes, four different types of perturbations were used to trigger the turbulent puff.
Injection of liquid through a hole in the pipe. This method is identical to that used in the previous lifetime study of Hof et al. (2006). The perturbation amplitude was chosen large enough to trigger the turbulent state and the duration of the perturbation was set to between 10 and 20D/U.
Simultaneous injection and extraction of equal quantities of liquid through two opposite holes in the pipe. This procedure is less intrusive because the pressure and mean flow in the pipe are unaffected. The injection/extraction was achieved by using two coupled syringes and the volume of fluid to be extracted/injected was 0.1 ml.
Reduction in Re: a puff was created at a higher Reynolds number (2000<Re<2100). After the puff had propagated for a distance of about L/D=800 Re was reduced by ΔRe=150 to a prescribed value. This perturbation is similar to those used by Peixinho & Mullin (2006) and by Willis & Kerswell (2007). In our case, the Reynolds number quenching was achieved in the following way: a long tube of diameter D=5 mm and a short tube of diameter D=10 mm were placed in front of the pipe entrance in an equivalent way to parallel resistances in an electric circuit (see switching valve in figure 1). With this construction, most of the liquid flows through the tube with the larger diameter, which presents a smaller resistance. The D=10 mm tube could be blocked with the aid of a solenoid valve so that the liquid had to flow through the 5 mm tube only, increasing the pressure drop across this segment. As a consequence, the pressure drop across the actual pipe decreases, resulting in a reduction of the Reynolds number by 150. The perturbation was triggered at the higher Re and, after a programmed time corresponding to L/D=800, the valve was closed abruptly reducing the Reynolds number. The solenoid valve closes within 0.1 s corresponding to 12D/U.
An obstacle was placed in the pipe blocking half of its cross section. This configuration intermittently generates puffs and the frequency of puff occurrences increases with Re (Rotta 1956; Wygnanski & Champagne 1973). The passage of the puffs was observed by monitoring the pressure of the fluid at a fixed position (L/D=120) behind the obstacle. A sudden change in the pressure indicated the passage of a puff. The outflow, at a distance L=930D downstream of this point, was then monitored to determine the survival rate of the so-created turbulent patches.
The lifetime measurements employing the injection, the injection/extraction and the obstacle methods were carried out with a fixed distance between the perturbation and the outlet, L (in the obstacle method L is the distance between the position at which the pressure is monitored and the outlet). For the Reynolds number quenching, on the other hand, the observation time is fixed. Here for each Re the time between triggering the puff and reducing Re was adjusted accordingly to keep the time interval required for puffs to cover the remaining distance to the outlet constant. Differences between the experiments at constant length and those at constant observation time are discussed in §3.
(a) Influence of velocity variations on puff statistics
In the previous experimental study of lifetimes by Hof et al. (2006), the separation between the perturbation and the outlet in which the survival or decay or a puff was observed, L, was fixed. In these experiments, it was assumed that the puff moves with the mean velocity of the flow (Upuff=1), so that the observation time is the same for the points taken for all different Reynolds numbers (t=L/U). However, the puff propagation speed shows a small variation with Re that was neglected in this earlier study. In order to test the validity of this approximation, first an accurate measurement of the puff propagation speed with Re was necessary.
Puffs were created by applying the single jet perturbation and the time between the creation of the puff and its arrival at the pipe exit was recorded. This procedure was carried out for four different distances between the perturbation and the outlet L/D=328, 656, 927 and 1915. In figure 2, the measured times at each distance are presented for four different Reynolds numbers. Each point represents the mean value of 10 measurements. Our results show that the puff moves with a constant velocity for the pipe length we have considered, 270≤L≤2000. Indeed, the behaviour observed in figure 2 is almost perfectly linear.2 This behaviour is not surprising and confirms the classical picture of the so-called equilibrium puff (Wygnanski et al. 1975): after a time for its formation, the puff does not evolve anymore and therefore its velocity is independent of the position.
In figure 3, the puff velocities are plotted as a function of the Reynolds number. Our results are in very good agreement with the measurements of Hof et al. (2005; line in figure 3), the maximum difference between both datasets being 0.015. In this previous study, the speed was measured only for a single distance and it was hence not possible to determine whether the propagation speed is constant. As our data show, the propagation speed is indeed constant, and even a relatively short observation time of L/D of approximately 600 is sufficient to measure the advection speed accurately. Subsequently, the velocity data will be used to transform the data recorded for fixed distances into observation times.
(b) Decay of turbulent puffs
Here, we measured the probability of puffs created by the four different perturbations to survive for the constant observation time t=933D in the case of the Re reduction experiments or while travelling a constant length of L=933D for the other perturbation types. The injection and Re-reduction measurements were automated so that it was possible to take between 200 and 500 experiments for each datum point. For the other perturbations between 50 and 200 experiments were carried out for each point. In figure 4, the probability of the puff survival is shown as a function of the Reynolds number. Over 7500 experiments were necessary for this plot. The collapse of all datasets indicates that the lifetimes are independent of the type of perturbation. This observation disagrees with some earlier results (Mullin & Peixinho 2006a; Peixinho & Mullin 2006) but it is in accord with the general picture that trajectories on a chaotic saddle or repellor quickly lose all memory of the initial conditions. In the case of the injection and injection/extraction perturbation the probability of survival of puffs was also measured for a shorter pipe with L=269D, as shown in figure 5. As for the longer pipe, no differences between the datasets are observable. Recent numerical data (Schneider & Eckhardt 2008) have suggested that, depending on initial conditions, not only can the formation period of puffs t0 vary but also the initial puff decay rate can differ considerably from its asymptotic value. Our observations show that such dependences are not observed for the perturbations typically used in experiments. By contrast to these numerical studies, we here always applied perturbation amplitudes well above the lowest amplitude necessary to trigger turbulence (Hof et al. 2003). Our experiments strengthen the picture that the turbulent dynamics quickly erase any memory of the initial perturbation.
Further support of the independence of turbulent puffs from their creation can be obtained from detailed recordings of the flow field. In an upscaled version of the present pipe that had an inner diameter of 30 mm, we carried out stereoscopic particle image velocimetry measurements (using a Lavision high-speed PIV system). Here, a cross-sectional plane of the pipe is illuminated by a pulsed laser and two high-speed cameras record the positions of tracer particles, 13 μm in size, within the plane. The measurement method is identical to that used by Hof et al. (2004), van Doorne (2004) and van Doorne & Westerweel (2007). The stereoscopic view allowed us to determine all three velocity components within the plane and by applying Taylor's hypothesis an estimate of the three-dimension flow field can be obtained. In figure 6, iso-surfaces of the streamwise vorticity of turbulent puffs are plotted for a turbulent puff created with the injection perturbation at Re=1860 (figure 6b) and for a puff created at Re=2030 and recorded 200L/D after a sudden reduction in the flow rate to Re=1880 (quenching procedure; figure 6a). Both pictures look qualitatively very similar, indicating that the puffs created by different methods have similar length and structural composition. The detailed description of the puff structure can also be found in this voulme, given by van Doorne & Westerweel (2009). A better comparison can be obtained from the kinetic energy (w2+v2)/U2 averaged over the cross-sectional plane as a function of the downstream distance (shown in figure 7). The close similarity of the energies of the two single puffs further supports that the ‘puff state’ is well defined in the sense that it is independent of initial conditions.
With the aid of equation (1.1), the decay rates shown in figure 4 can be converted into lifetimes τ(Re) shown in figure 8. In these calculations, the observation times were corrected using the measured puff velocities . The so-calculated lifetimes differ up to 5 per cent from the calculations under the approximation Upuff=1. However, this difference is barely observable in the semilogarithmic plot and it was hence omitted from the plot. The value t0=70D was taken from Hof et al. (2008), where accurate measurements of t0 were performed for single jet perturbations using laser Doppler measurements. Although t0 might vary with the perturbation type, the observation time is long enough so that even an error in t0 of 30 per cent would cause a relative error in the lifetimes of only 2 per cent.
As in Hof et al. (2006) the lifetimes in figure 8 vary exponentially with the Reynolds number following τ=exp(a+bRe). The linear fit yields a=71.65 and b=−0.0419. Although, the functional form of τ is the same as in the cited paper, the parameters a and b differ by approximately 20 per cent.3 In particular, the exponential curve is shifted by ΔRe=50 to smaller Reynolds numbers. A possible reason for this shift is the relative large uncertainty in the absolute value of the Reynolds number in the earlier study by Hof et al. (2006) which was largely caused by uncertainties in the pipe diameter. The more accurate pipes used in the present study and an improved temperature calibration allow us to determine the absolute Reynolds number values more accurately than previously possible. The exact scaling of the lifetimes measured over a larger Reynolds number regime will be reported in another study (Hof et al. 2008). The determined functional dependence of τ(Re) can now be used to estimate the influence of the speed variation of turbulent puffs on the S-shapes in figure 4. The solid line in figure 4 gives the Reynolds number-dependent probability for puffs to decay after a fixed time t=933. Since this line differs only marginally from the fitted curve (dotted line in figure 4) the assumption that puffs travel at the mean velocity in this Reynolds number regime has no noticeable effect on the lifetime scaling.
In conclusion, our measurements show that turbulent puffs created by four different perturbation mechanisms all show the same decay statistics. The lifetime scaling of the turbulent puffs qualitatively agrees with that observed by Hof et al. (2006) and over the limited Reynolds number regime studied an exponential lifetime scaling is observed. For the four different perturbation schemes used to create turbulence the decay statistics are found to be identical. Unlike in previous lifetime studies applying Reynolds number reduction (Peixinho & Mullin 2006; Willis & Kerswell 2007) no divergence of lifetimes could be observed and the puffs remain transient well beyond the critical Reynolds numbers for sustained turbulence reported in these earlier studies.
The authors gratefully acknowledge the financial support provided by the Max Planck Society and by the EPSRC (grant EP/F017413/1).
One contribution of 10 to a Theme Issue ‘Turbulence transition in pipe flow: 125th anniversary of the publication of Reynolds' paper’.
↵In pipe flow Re=UD/ν, where U is the mean velocity, D is the diameter and ν is the kinematic viscosity.
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