Experiments show that transitional pipe flows are characterized by different flow structures usually referred to as “puffs” and “slugs” [28, 44, 45, 76, 103]. Early experimental investigation of turbulent slugs and streaks in transitional pipe flow could be found in Lindgren (1969) [88]. Due to restrictions with technology at the time, pressure drop measurements were adopted to investigate the flow. For the pressure measurements, the initial disturbance may lead to the signal flash, also referred to as a spike. The spike may or may not trigger a burst of turbulence forming a ‘turbulent spot’, characterized by intensive small scale flow fluctuation, travelling with the flow along the wall. As the turbulent spots are transported along the walls by the flow, they may decay, increase or maintain in size according to the Reynolds number. When Re< 2000, the flash will travel some distance down the pipe and then decay. With the increasing Reynolds number, the frequency of spikes and flashes increase. When 2000 < Re < 2400, the turbulent flashes appear unchanged in length and they do not appear to decay. Some apparent flow structures at the wall continuously regenerate eddies that diffuse into the more central part of the pipe. The downstream velocity of the flashes decrease with increasing Reynolds number and when Re ≈ 2400, the flash velocity downstream is about 0.9Ub [172, 173]. It
indicates laminar flow must therefore enter the rear of the slug and become turbulent, while at the front end, turbulent flow must leave the flash and become laminar again. Lindgren states that it seems logical that the higher kinetic energy dissipation of the turbulent flow within the flash should be supplied by the kinetic energy of the laminar flow entering the flash at its rear. When 2400 < Re < 2800, a flash may split in two or more flashes within this Re range. As the Reynolds number increases, the turbulent flashes elongate and follow each other more closely. The front velocity on the streak is larger than the rear velocity and increases as Reynolds number is increased. For Re > 2800, the flow disturbances at the entrance of the pipe is comparatively large and the disturbance leads to the formation of turbulent flashes which never return to the laminar state (a flash never splits in two). When the Re increases further, the turbulence regions become more and more frequent until the flow goes into the fully turbulent regime.
Wygnanski et al. (1973,1975) [172, 173] investigated transition from conditionally sampled hot-wire measurements. Based on the Reynolds number, flow perturbations evolve into two different turbulent states depending on the Reynolds number, i.e. puff (Re ≈ 2700, caused by the instability of the boundary layer to small disturbances) and slug (Re ≈ 3000, generated by large disturbances at the inlet). The distinction between the two structures is given by the behaviours of the velocity near the leading edge, with a puff showing a gradual change of the velocity in the leading edge while a slug has a more abrupt velocity gradient. The velocity measurement from a puff or a slug at a fixed point in the centreline of the pipe can be seen in Figure 3.1.
Figure 3.1: Schematic trace of the velocity of a puff (a) and slug (b) as a function of time during their passage past a fixed probe point on the centreline of pipe by hot-wire velocity measure- ment. The velocity fluctuation indicates inside the puff or slug, the flow state is essentially turbulent while out of them, the flow state is laminar. Taken from Durst et al. (2006) [45].
A puff is furthermore approximately of constant length and is convected downstream at a velocity slightly smaller than the bulk velocity. Slugs are characterized by a sharp laminar- turbulent interface at both the downstream end (leading edge) and the upstream end (trailing edge) of the turbulent region. Puffs were observed in the range of 2000 < Re < 2700, and they were generated by a very strong disturbance at the inlet of the pipe. At the downstream end of a puff the centreline velocity is found to decrease gradually, and therefore it is often said that the leading edge is not so well defined. In contrast, the trailing edge is well defined, but only in the central region of the pipe [7, 149].
3.1. PIPE FLOW OF NEWTONIAN FLUIDS
The velocity of the trailing edge is smaller than the bulk velocity, while the leading edge travels faster than the bulk velocity. This means that the length of a slug increases as it moves downstream [31, 88, 138]. Farther downstream different slugs will merge to form one large region of continuous turbulence [160]. Wygnanski et al. (1973) also studied the mean flow field, the turbulent statistics of 3 velocity components, the Reynolds stress, the dissipation and kinetic energy inside a slug and on the interface. The result shows that the flow inside the slug, about 20D from the leading edge to the trailing edge, is identical to fully developed turbulence. Wygnanski et al. (1975) further investigated the puff and showed that the flow structure of a puff does not depend on the form of disturbance by which it is created but depends on the Reynolds number only. The development of the puff was also studied. The puff remains unchanged over a some range of Reynolds number, which was named equilibrium puff and increasing the Reynolds number will lead to the puff splitting and merging. An explanation which not strictly proved, is that the sharp trailing edge could be explained by the quick transition from the laminar flow, whereas at the leading edge the flow relaminarizes and is governed by the relatively slow dissipation of the turbulent fluctuations that results in a more gradual change of the velocity at the leading edge. Nishi et al. (2008) [115] investigated the behaviours of puffs/slugs in transition. The information was obtained by measuring the time variation of the streamwise velocity at the centreline of the pipe by hot-wire measurements. The results show
(a) puff splits. Re = 2495 (b) puffs merge. Re = 2865
Figure 3.2: Puff development with increasing Reynolds number of different pipe length for (a) puff splits. (b) puffs merge. Taken from Nishi et al. (2008) [115]
how slugs and puffs develop, i.e. merge or split, along the pipe. The Reynolds number plays an important role on the flow structure of puffs and slugs. At a lower Reynolds number, puffs generate and remain unchanged along the pipe. However, when the Reynolds number increases, the puff will split into two or more puffs and for sufficiently high Reynolds number, the puffs merge into slugs and begin to expand as slugs in the pipe, see Figure 3.2. When Reynolds number continues to increase, the sequences of slugs will merge and finally form the state of fully turbulent flow.
Figure 3.3: Experimental flow visualization of puff (bottom) compared with axial vorticity for a numerically calculated puff (top) at the same Re=1800. The working fluid is water and the computational domain is 50 pipe diameter. Taken from Willis [170].
The flow structure visualized in Figure 3.3 (bottom) is an experimentally observed puff. These structures greatly determine the random appearance of the flow, and they could be vi- sualized by the iso-contours of the axial vorticity, see Figure 3.3 (top), which is taken from simulations performed by Willis et al. [168, 170] in the domain of 50 pipe diameters. The puff characterized by spatio-temporal intermittency appears stochastically in pipe, indicating the on- set of laminar turbulent transition. Some recent studies also focused on the threshold to trigger a transition. These studies investigated two questions, i.e. what amplitude of perturbation is re- quired to trigger turbulence, and what magnitude of Re is required to sustain turbulence. In the experiment, the transition to turbulence could be observed when a threshold have been crossed, i.e. the bulk velocity (Reynolds number) should be large enough or the perturbation has to be sufficiently strong [47]. Hof et al. (2004) [67] reported an experimental investigation to un- cover the scaling of the turbulence transition threshold in a pipe, which indicated the amplitude of disturbances (defined by the injection volume flux) required to trigger transition scales as O
3.1. PIPE FLOW OF NEWTONIAN FLUIDS
(Re−1). Peixinho and Mullin (2007) [120] changed the disturbance pattern where just one jet as
a perturbation is kept. By varying the angle of the jet the threshold changes from the scaling O (Re−1.3) to O (Re−1.5). Perturbations with amplitudes below the threshold decay as they propa-
gate downstream and no transition occurs. On the other hand, disturbances with the amplitude above this amplitudes will lead to a sustained chaotic flow downstream which had the shape of a localized puff or slug.