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 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 , 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 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 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

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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. The data show that rZ; a y-‘, in agreement with turbulence. similar measurements in Ref. 4.

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Downloaded 12 Dec 2001 to 128.122.80.28. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp i --: 0 0.06 i == .- ;e. -- 5 ‘- : - - ; -. -. 1 - - : -- 7 -1

FIG. 4. Resealed frequency of slugs, f/J,,, plotted versus the intermit- tency factor y. The curve is obtained from the model described in Sec. IV, using the parameters N= 150, Uc= 1, cc= 1.1, rr,,=O.38, and c1=0.6. FIG. 6. Inverse cumulative distributions P(tY) of laminar times obtained at different values of the intermittency factor y. From right to left: In Fig. 6 we show the inverse cumulative distributions y=O.O5,0.075,0.11, and 0.35. Insei: Decay rate (Tvs y. The curve is from P(tY) for different values of y close to the onset of inter- the model described in Sec. IV with the parameters in Fig. 4. mittency. The data show that the distributions are expo- nentially decaying, P( t-p-) - exp ( -of oty) with a decay rate cr that is proportional to y (cf. the inset in Fig. 6). At larger values of 7, deviations from an exponential decay are observed.

- IV. MODEL f&- In order to describe the experimental observations, we introduce a model, where the state of the fluid is described by an array, q(t) (i=l,...,N). This defines a length unit A.= L/N as the distance between neighboring sites. In our c simulations, N= 150. The state of the fluid, si(t>, at site i 0.0I-A -! can take only two values, 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (4 7 0, “laminar”; si(t>= 1 (1) I 3 “turbulent.” The evolution of the state of the fluid is described by the following transformations: (i) Downstream shift. The fluid moves down the pipe with bulk velocity U(t),

W+Af) =si-,W, (2) where At denotes the time step, determined from the in- stantaneous value of U, At=il/U(t). Here U is (without loss of generality) normalized to unity at the onset of in- termittency, Uo= 1, i.e., Re=Re,, U. The relation between pressure gradient and Reynolds number is assumed to be

(b) 7 G=(Nya~Reb~+N~a~Reb~)/N, N=Ny+Ny, (3) FIG. 5. (a) Reacaled mean turbulent time feg, plotted as a function of where NY(t) is the number of laminar sites and N-y(t) is the intermittency factor ‘y. (b) Double-logarithmic plot of the resealed the number of turbulent sites at time t. The parameters ay mean laminar time feg vs y. The curves are obtained from the model described in Sec. IV, using the parameters given in Fig. 4. The slope of the and bZ for the fully laminar flow (NY = N) are estimated curve in (b) is - 1 at low y values. from the experiment [Fig. 2(b)] to be aY=0.006 and

1724 Phys. Fluids, Vol. 6, No. 5, May 1994 Zhang et al.

Downloaded 12 Dec 2001 to 128.122.80.28. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp by -2, assuming that Re/Reosr V/V, ,6*gFor the fully tur- JG=Y7 (84 bulent flow ( N.T==N), we use the parameters ay=0.32 and by=; found from Ref. 3. f (Tp+F) = 1. (8b) (ii) Generation of slugs. It is generally accepted that Thus, knowing one, e.g., f(r), the other two are known as the flow in an infinite cylindrical pipe remains stable to a function of y as well. In this section we derive approxi- infinitesimal perturbations. Experiments in pipe flows mate functional forms near the onset of intermittency of show, however, that for high enough Reynolds numbers the measured quantities from our model. This enables us to the flow eventually becomes turbulent. Both the departure relate the measured physical quantities with the model pa- from the parabolic profile and instabilities due to finite-size rameters, and eventually to test to what extent the model is disturbances play a role in the interpretation of this fact. consistent with experiment. We introduce a scheme in which the generation of slugs is For the theoretical part, we neglect the fluctuations in a stochastic event occurring only at the pipe entrance (site velocity. Then, the frequency of slugs at the inlet f I can be 1). In the update time interval, At, a slug is generated with written in terms of p( U), and the mean turbulent time a probability p( U), which close to the onset of intermit- c can be written in terms of a( U), tency is assumed to be linear, fl=pC W/At, (9) 10, if Ul. I F=mu* (10) Hence, the state of the first site at time t undergoes a transformation s1 -+ s; , where Equation (10) follows from the fact that F is the time difference between the time L/( 1 -rr) U, at which the 1; with probability p(U); trailing edge of a slug reaches the measuring point (site (5) “= I0, otherwise, N), and the time L/( 1 +r> U, at which the leading edge of the slug reaches the same point. and U is the velocity at time t, determined from Eq. (3 ). In the U range, where no merging of slugs are taking The N dependence in Eq. (4) reflects that p ( U) a/z (from place, fI can be replaced by J: To first order in U- 1, Eq. coarse-graining arguments). (9) and Eq. (10) yield (iii) Growth of slugs. Once slugs are generated in the pipe, they grow. This growth is characterized by the veloc- f/fo==co(U-l), (11) ity of the leading edge, U, , and the velocity of the trailing edge, U_, both depending on the Reynolds number. A f&T&l+fi cl(l+d; -1 (U-l), (12) simplification is achieved by assuming that U-U- and ( ~o(l--ml)) ) U, - U are equal in the intermittency range, i.e., U,/U= 1 =t’i~( U). In terms of our discrete model, rr( U) is where p=2nb/( 1 -d) is identified with the value of the the probability for a neighbor site to a slug to become (resealed) mean turbulent time at U= Uo. This value is turbulent. Close to U= U,=l, n(U) is to first order a experimentally found to be fl,O.9 [Fig. 5 (a)],” corre- linear function, sponding to the value ~~~-0.38. By Eq. (8a), the intermittency factor y is from Eq. ?T(U)=7re+ci(U-l). (6) ( 11) and Eq. (12) found, to first order, The transformation sf -, s; that governs the downstream y&2()( u- 1). (13) growth (at time t) is For U right above U,, U/U, is close to the laminar branch 1, with probability rr( U)si-i; of V/V,, [Fig. 2(b)]. From the experimental data in Fig. s;= (74 St, otherwise, 2(b) and Fig. 3, y is for U=: U. found to follow the linear form ( 13) with &,a 1. This fixes the parameter co to while the transformation sf + si that governs the upstream cocr. 1.1. growth is According to Eq. (8a), the leading contribution to the 1, with probability V( U)Si+l; frequency of slugs f(y) (Fig. 4) is determined by the s;= (7b) value /3 of the (resealed) mean turbulent time foG at Sf9 otherwise. u= u,, Again, U is given by Eq. (3). Notice that Eqs. (7a) and f/fo-Y@. (14) (7b) imply that growth can only take place at laminar sites that are adjacent to turbulent ones. In agreement with theory, the experimental slope in Fig. 4 is found to be l@=l.l at U-,U,. Also, the expression (12) for the mean turbulent time V. MODEL PARAMETERS AND RESULTS can be recast in terms of the intermittency factor y [cf. Eq. In Sec. III, our experimental data for the frequency of (13>1, slugs f and for the mean laminar and turbulent times, G cdl+& and GY were presented independently. They are, however, foF-P+cl$ -1 y. (15) related, ( ro(l -d) 1

Phys. Fluids, Vol. 6, No. 5, May 1994 Zhang et a/. 1725

Downloaded 12 Dec 2001 to 128.122.80.28. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp The first-order term [the slope in Fig. 5(a) near the onset order of magnitude in Reynolds number. We understand of intermittency] depends on all three parameters co, ro, this as coming from an increase in p and a corresponding and ct. From a best fit to our experimental data, we find decrease in co at increasing Re. The value of /3 increases because the spreading velocity of a slug increases at higher cl -0.6. - For the mean laminar time ty (y), we note that G Re; the value of co is a measure of the noise in the system s G near the onset of intermittency. From Eq. (8b) we that is larger at lower values of Re for the onset of turbu- then have t, = l/f, and by ( 14)) lence.

j-&am (16) VI. CONCLUSIONS which is also verified experimentally [Fig. 5(b)]. Based on new experimental studies of pipe flows, we Finally, consider the set of laminar times obtained have shown that a simple phenomenological model, where from a particular time series. The inverse cumulative dis- slugs are generated stochastically at the pipe entrance, and tribution P(ty) is the fraction of laminar times that are where the decrease in velocity due to turbulent friction is larger than t2. For foty,p, we can neglect the down- taken into account, is sufficient to describe the intermittent stream reduction in ty due to slug spreading, and P(ty) behavior at the onset of turbulence, given that the inter- equals the probability not to have a slug generated at the mittency factor is not too high. More specifically, we have pipe entrance in ty/At subsequent steps (still neglecting measured the frequency of slugs, the mean laminar and fluctuations in U), turbulent times, and the distribution of laminar times. In P(tJ+[l--p(U)]t4”@ (17) all cases the behavior at the onset of intermittency is con- sistent with our stochastic model. Clearly, our model is too In the “continuous” limit N, 1, i.e., p(U) & 1, we have, simple to reproduce all the experimental results. It pro- according to Eq. (9 ), vides, however, a new picture for understanding the as- log WY) = -fItye (18) pects of intermittency that stem from the stochastic dy- namics and not from the specific details of the slug In the range where f = fI (no merging), the inverse cu- generation or the slug growth. mulative distribution in terms of y thus has the form (1%) ACKNOWLEDGMENT with the decay rate This work was supported by the Novo-Nordisk Foun- dation and the EC Science plan. 17czy/p (19b) We also find experimentally that the decay rate (T is linear ‘0. Reynolds, “On the dynamical theory of incompressible viscous fluids in y (the inset of Fig. 6). The proportionality constant is and the determination of the criterion,” Philos. Trans. R. Sot. London Ser. A 186, 123 (1895). again given by l/p= 1.1. “I. J. Wygnanski and F. H. Champagne, “On transition in a pipe, part I: Using the values of the three model parameters co, ro, The origin of pulfs and slugs, and the flow in a turbulent slug,” J. Fluid and cl relevant for our experiment, we have carried out Mech. 59, 281 (1973). simulations for our model. The results are shown as curves ‘D. J. T&ton, PhysicuZFZuid Dynamics (Oxford Science, Oxford, 1988). ‘K. R. Sreenivasan and R. Ramshankar, “Transition intermittency in in Figs. 4-6. Surprisingly, the model works well beyond the open flows, and intermittency routes to chaos,” Physica D 23, 246 linear regime, indicating that stochastic effects are impor- (1986). tant for the main features of the quantities discussed. It has ‘Y. Pomeau and P. Manneville, “Intermittent transition to turbulence in dissipative dynamical systems,” Comments Math Phys. 74, 189 (1980). been a purpose of this work to introduce the stochastic “At the onset of turbulence, the ratio between the measured central concept in terms of p( V), and see how far such a model is velocity V, and the bulk velocity U, is estimated to be 1.5 for the useful. Indeed, it breaks down later than naively expected. laminar flow [derived from the velocity profile given by E. M. Sparrow, There are some comments to add. S. H. Lin, and T. S. Lundgren, Phys. Fluids 7, 338 (1964). These profiles are in good agreement with those we find experimentally]. (i) The linear terms of U- 1 in p( U) and rr( U) were 7 V, and I’, were found as peak values in the velocity distribution. The chosen to make the model simple. They seem to be exper- scatter in V, was l%, in v, it was 15%. imentally justified. We stress, however, that the model is ‘Our experimental setup does not allow a direct determination of the not linear; it has a nonlinear relationship between the pres- bulk velocity, i.e., the Reynolds number. As an alternative measure of the degree of turbulence we therefore use the intermittency factor y. sure gradient and the velocity. 90ur laminar flow data lie much closer to the curve obtained for high (ii) The stochastic model explains the exponential de- (laminar) Re [ay=d/2L=0.0033, bip=2) than to the curve obtained cay of the inverse cumulative distribution of laminar times. for low Re ((~;r = 32, by = 1) . This is not surprising, since the Reynolds No special types of intermittency are needed to understand number always must lie below both curves, which cross each other at Re=64L/d=9600. this decay. t ’ “From Fig. 5(a), it is seen that the resealed mean turbulent time at the (iii) Comparing various experiments, Sreenivasan and onset of intermittency is less than the parameter p. This is because the Ramshanka? found that experimental data for y vs bulk velocity right above the onset is less than U, when the flow is ( U- Uo)/Uo was falling on approximately the same curve, turbulent. “The fact that a simple mechanism as a Poisson process can be respon- with an initial slope of about 1. In regard to our model, it sible for the exponential variation in Fig. 6 was mentioned (but not means that fico=: 1 for a set of experiments that varies an shown) in Ref. 4.

1726 Phys. Fluids, Vol. 6, No. 5, May 1994 Zhang et a/.

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