The results of the previous two sections clearly demonstrate the stochastic and memoryless nature of the splitting process beyond the formation time, and thus the scaling of the mean lifetime of an isolated puff must be calculated for each process. The numerical results of
section 5.2 are confined to a narrow range of Re due to the large computational requirements at low Re, and thus to determine an accurate scaling we must obtain data points as close to the suspected critical Reynolds number as possible. Since the data from table 5.1 clearly demonstrates the quick growth of as Re decreases, it is not computationally feasible to
obtain these numerically.
To this end, as part of the Warwick-Göttingen collaboration we combine the numerical data obtained from DNS1 and DNS2 with the experimental data of Kerstin Avila, Alberto de Lozar and Björn Hof in a very long pipe of lengthLD3;750D. The setup of the experiment
is detailed thoroughly in de Lozar & Hof (2009) and Avilaet al. (2011), and allows for up to 60,000 samples to be obtained at a single Re if required. Splitting events are detected using the pressure signal at the pipe wall, and thus the numerical results presented here therefore provide an immediate validation for the experimental results and vice versa.
In figure 5.10 (which is adapted from Avilaet al., 2011), we visualise the characteristic puff lifetime as a function of Re for both decaying (left) and spreading (right) turbulence, using
all of the available experimental (coloured points) and numerical data (solid black triangles). Decay lifetimes are presented using the experimental and numerical data from Hofet al. (2008), Kuiket al. (2010) and Avilaet al. (2010).
For both decay and splitting processes, super-exponential curves of the form
.Re/DexpŒexp.aReCb/
provide a very good fit to the data, where for decay we obtain.a; b/D.0:005556; 8:499/
numerical data, whilst limited in scope to a small range of Re, provides excellent agreement with the experimental data, with the super-exponential fit lying within the 95% confidence intervals at each of the measured data points.
One important consequence of the data presented here is that there is no particular Re beyond which spatial proliferation abruptly sets in, as suggested in the results of the previous chapter and other studies (e.g. Wygnanski & Champagne, 1973, Nishiet al., 2008). Indeed, like decay, it appears that the tendency to split is intrinsic to all puffs, regardless of the Reynolds number. Therefore ‘equilibrium’ puffs, described by Wygnanskiet al. (1975) as those which do not grow in length and advect at a constant speed, do not exist: after a time governed by a random variable, any localised puff will either decay or split.
This point is especially emphasised in the experimental samples, for which much larger data sets can be obtained and thus far lower Re examined. Here, the measured samples extend as low as ReD2;032. At this Reynolds number, splits are extremely infrequent, but still occur:
out of 57,823 initial conditions only 7 were observed, so that 2:7107. Further details
of the experimental data can be found in the supplementary materials of Avilaet al. (2011). The intersection of the super-exponential fits at Re2;040therefore marks a tipping point
between the two competing processes of splitting and decay. As Re increases, pipe flow undergoes a statistical phase transition so that the probability of splitting outweigh that of decay.
A typical trait of many systems which undergo stochastic phase transitions is that the tipping point between competing processes often under- or over-estimates the true critical point of the system. A classical example of this is in the standard contact process, in which the infection rate must outweight the rate of recovery by a factor of just over three (see Hinrichsen, 2000, Harris, 1974) before the absorbing state yields to intermittent or active states which are fully active throughout the domain.
However, for pipe flow, the super-exponential scalings ofmean that the intersection point
must be almost indistinguishable from the critical point. At Re D 2;050, 10 above the
intersection point, the splitting rate already outweighs the decay rate by a factor of four. Indeed, reduced models of pipe flow which exhibit the same stochastic properties seen here indicate that the actual critical point is within around 0.3% of the value stated here (Barkley, 2011). Therefore we may surmise that the actual critical point lies within the
1800 1900 2000 2100 2200 2300 2400 Re 102 103 104 105 106 107 108 Spreading turbulence Experiment DNS1 DNS2 Decaying turbulence Injection (LD3380D) Hof et al. (2008) Kuik et al. (2010) Avila et al. (2010)
Figure 5.10:Characteristic lifetime as a function of Reynolds number for an isolated puff.
Coloured points represent experimental data (courtesy of Kerstin Avila, Alberto de Lozar and Björn Hof), and solid black triangles those obtained fromSemtex(DNS1) and a hybrid
spectral finite-difference code (DNS2). The dashed line denotes the super-exponentially increasing fit for mean decay times, and the solid line the decreasing fit for mean splitting times. The crossover point at Rec D2;040˙10, determines the transition between transient
and sustained turbulence in pipe flow.
interval Rec D2;040˙10, and this marks the onset of sustained turbulence in pipe flow.
One further point requiring clarification is the definition of ‘sustained turbulence’. In the original experiments of Reynolds (1883a,b) Rec is defined to be the number above
which turbulence persists indefinitely, and the results presented here clearly indicate an exponentially small probability of decay above Rec. However in this context, a more natural
definition of ‘sustained turbulence’ is instead to consider the thermodynamic limit (i.e. an average over space, time and ensemble of initial conditions, where space and time tend to infinity) of the turbulence fraction or intermittency factor. Below Rec, D0as decay is