Stochastic transition intermittency in pipe flows: Experiment and model Jun Zhang, Dimitris Stassinopoulos, Preben Alstra)m, and Mogens T. Levinsen The Niels Bohr Institute, 0rsted Laboratoty, Universitetsparken 5, DK-2100 Copenhagen, Denmark (Received 16 September 1992; accepted 18 November 1993) New experimental results at the onset of turbulence in a gravity-driven pipe flow are presented, and a simple phenomenological model is introduced to describe the intermittent behavior observed. In this model slugs are stochastically produced at the pipe inlet, and the decrease in velocity due to turbulent friction is taken into account. The present approach shows that stochastic arguments account well for several experimental observations at low intermittency factors. In particular, it is shown that special intermittency routes to chaos are not needed to explain the exponentially decaying inverse cumulative distribution of laminar times. 1. INTRODUCTION horizontally and connected via plastic tubes, 4 cm in di- ameter, to reservoirs containing deionized water. The pipe The intermittency transition to turbulence in pipe and the plastic tubes were connected by sections, welded to flows is one of the oldest known transitions from laminar the pipe by a smooth contraction (Fig. 1) . Throughout the flow to turbulence.’ The most striking feature of the phe- contraction regions the angle between the glass wall and nomenon is the sudden and seemingly random appearance the axial direction is less than 9”. The inlet section was of turbulent slugs,53 as the flow becomes unstable to oc- fitted with tightly packed straws, 9 cm long and 0.5 cm in curring disturbances. A slug is being convected by the flow diameter. A fly-screen (aperture size = 2.0 mm) was placed downstream, while at the same time it spreads-laminar 5 cm downstream from the straws and 30.5 cm upstream fluid at both ends of the slug becomes turbulent. In pipe from the pipe entrance. A difference in height level, and flows driven by gravity (constant pressure difference), an consequently in pressure, between the two reservoirs was additional complication arises because the velocity con- maintained by means of a pump. The value of the pressure stantly alternates. The reason for this is that the friction is drop was controlled by adjusting the pumping rate. larger for a turbulent flow than for a laminar flow, there- The water was seeded with few drops of homogenized fore the velocity decreases as a slug spreads. Near the onset milk (resulting concentration ratio - 10m6), and laser of turbulence the formation of one slug thus prevents the Doppler velocimetry was used to measure the axial velocity formation of new slugs until the slug has left the pipe and in the center of the pipe, 7 cm from the outlet. Two parallel the flow again is laminar. linearly polarized laser beams from a 15 mW He-Ne laser In this paper we present new experimental results for a were focused into a measuring volume of diameter - 0.1 gravity-driven pipe flow. Furthermore, we introduce a mm; to avoid undesirable refraction from the pipe wall a model with stochastic rules for the generation and growth rectangular cell filled with water was placed around the of slugs, which takes into account the change in bulk ve- pipe where the velocity was measured. One beam was locity due to the turbulent friction. The purpose is to iden- shifted by about 40 MHz with respect to the other by use tify the aspects of intermittency that are nonspecific to the of a Bragg cell. The scattered light was detected by a pho- details of the generation and growth of slugs, and to show tomultiplier, and the output was passed to a DANTEC how average flow properties can be understood by simple 58N20 flow velocity analyzer. This was operated so that stochastic arguments. Our approach is different from that Doppler signals with less than 60 cycles were rejected. The of Sreenivasan and Ramshankar,4 where a description of typical sampling rate was about 100 Hz. The exact fre- experimental observations was based on a connection be- quency shift between the laser beams was set to optimize tween transition intermittency and the intermittency route the resolution. to chaos.5 The layout of the paper is as follows. In Sec. II we describe the experimental setup, and in Sec. III we present III. EXPERiMENTAL RESULTS our experimental results. In Sec. IV we define our model and the stochastic rules employed. In Sec. V we relate the Increasing the pressure drop, the Reynolds number in- model parameters with the average flow properties at low creases. Here, the Reynolds number is defined as Re= Ud/ intermittency factors. These parameters are then used in Y, where U is the bulk velocity and v is the viscosity of simulations of the model, which are compared with the water. The onset of turbulence is found at relatively high experimental observations. The paper is concluded in Sec. Reynolds numbers, Re=Rec-26 000. At this Reynolds VI. number the velocity profile has a flat part6 Above Re,,, a typical intermittent flow is observed with a measured axial velocity V that alternates between a laminar ( V,) and a II. EXPERIMENTAL SETUP turbulent (ye) value [Fig. 2(a>J7 The sharp transition A cylindrical glass pipe, d= 1 cm in inner diameter and between the two values suggests a well-defined interface L = 150 cm in length, was used. The pipe was positioned between the laminar and the turbulent flow, and allows a 1722 Phys. Fluids 6 (5), May 1994 1070-6631/94/6(5)/1722/5/ $6.00 0 1994 American Institute of Physics Downloaded 12 Dec 2001 to 128.122.80.28. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp 0.6 0.6 0.4 ro,llow “dorily onoryzer andprrmnol compu,er Y Pumping 0.3 0.2 FIG. 1. Schematic illustration of the experimental setup. S: straws; F: fly screen; C: smooth contractions; L: He-Ne laser; R: rectangular cell; and 0.1 PM: photomultiplier. 0.0 clear distinction between the two states. The laminar and 1.0 1.1 1.2 1.3 the turbulent velocities are in Fig. 2(b), plotted versus the G/Go normalized pressure gradient G=gd’ Ah/dL, where Ah is the height difference between the water levels of the two FIG. 3. Intermittency factor y versus the resealed pressure gradient reservoirs and g is the gravitational acceleration. G/G,, . For larger pressure drops the flow spends more and more time in the turbulent state. A measure of how much time the flow spends in each of the two states is given by the intermittency factor y, which is defined as the fraction of time the flow is turbulent (at the measuring point). Figure 3 shows the intermittency factor as a function of the J normalized pressure gradient G. At sufficiently high values Vc of Ah, above the range considered here, the flow becomes fully turbulent with y= 1. V I44 v, We have already mentioned that the two essential mechanisms for interpreting the above observations are the generation of slugs and the growth process that takes place 2 I I 0 5 10 15 20 as the slugs are convected by the flow. Consider the fre- quency f of slugs observed at the outlet. For low Re, the frequency f is low due to sparse generation of slugs. As Re (4 increases, f also increases. However, this increase is not indefinite: When two slugs are formed sufficiently close to 1.3 each other and grow enough they will merge, forming a single slug. For large y the merging eventually causesf to decrease. This trend is seen in Fig. 4, where the frequency 1.2 f is shown as a function of y.’ The frequency is resealed by fo= UdL, where U. is the bulk velocity at the onset of intermittency.6 We find that for low y values the frequency 1.1 G/Go of slugs is a linear function of y. Further quantitative characterization of the transition 1.0 intermittency can be obtained through a statistical analysis of the turbulent and laminar times. The turbulent time tr is the time it takes for a slug to pass the measuring point, 0.9 and the laminar time ty is the time elapsing between the end of a slug (the trailing edge) and the beginning of the 0.8 next (the leading edge). Here we consider the mean tur- (b) 0.7 0.8 0.9 1.0 1.1 1.2 bulent time g, the mean laminar time g, and the inverse cumulative distribution of laminar times P( ty ), defined as 17/vo the fraction of laminar times that are larger than ty . Fig- FIG. 2. (a) Axial velocity V versus time f, measured at the center and near the exit of the pipe (G=4.75 X 106). A sharp transition between a ure 5(a) shows the y dependence of the mean turbulent laminar state with V=Vy=4.24 m/s and a turbulent state with time (resealed by fo> . Notice that at the onset of intermit- V=Vrm3.34 m/s is observed. (b) Phase diagram. The laminar and tency the mean turbulent time remains finite. In Fig. 5(b) turbulent velocities, resealed by Vo=3.9 m/s, are shown versus the nor- a log-log plot of the resealed mean laminar time versus y is malized pressure gradient, rescakd by G0=4.00x106. Here, V, is the laminar velocity and Go is the normaked pressure gradient at the onset of shown.