The splitting of a turbulent puﬀ in pipe ﬂow
Masaki Shimizu,1 Paul Manneville,2 Yohann Duguet,3 Genta Kawahara1
1Osaka University, Graduate School of Engineering Science, Toyonaka 560-8531, Japan

2

Ecole Polytechnique, Hydrodynamics Laboratory, F-91128 Palaiseau, France

3

LIMSI CNRS, UPR 3251, F-91403 Orsay, France
We consider this splitting process and show how it develops.

1 Introduction
In 1883, Osborne Reynolds experimentally discoveredthat the state of the ﬂow in a pipe depends on a single
parameter, now called Reynolds number Re, and that itchanges qualitatively from laminar to turbulent at somecritical Re number, Rec  2000, under suﬃciently large
disturbance. Steady laminar ﬂow being linearly stable
for all Reynolds numbers, the transition to turbulenceis caused by ﬁnite amplitude perturbations. This nonlinearity makes it diﬃcult to locate Rec . Around Rec ,axially localized turbulent states, called ”puﬀs”, can beobserved.[1] Because the length of a puﬀ seems statistically constant at moderate Re number, it was thought
to be an equilibrium state. However recent many experimental and numerical studies have revealed that a turbulent puﬀ decays within a ﬁnite time and the statisticalFig. 2: Mean lifetime of a puﬀ.[2] Dashed and solid
curves represent decaying and splitting time respectively.

2 Equations

probability to persist can be understood as a stochas-

We consider the ﬂow of an incompressible viscous

tic process. Furthermore some studies have shown thatﬂuid driven by an external force and imposed time inde-

the decay time does not diverge at any ﬁnite Re num-

pendent mass ﬂux U in a straight circular pipe of radiusber. These facts make the determination of Rec more

a. A cylindrical polar coordinate system (r, θ, z) is in-complicated.

troduced to describe the velocity ﬁeld u with the z axis
taken on the pipe centerline. We apply no-slip conditions at the pipe wall and periodic boundary conditionsin the z direction with period aL. Then the governing equations of the ﬂow are given by the continuity
equation and Navier-Stokes equations,Fig. 1: Turbulent puﬀ in pipe ﬂow.[1] Gray scale rep-∇ · u = 0,(1)resents perturbation velocity, uz − uz θz , in a plane1 2∂u+ (u · ∇)u = −∇p +∇ u∂tRe(2)through the pipe’s centerline.

together with the no-slip boundary conditions
On the other hand, a puﬀ can not only decay but
also split. More recently, Avila et al. (2011) made abreak-through in this problem. They have also shownu(1, θ, z) = 0(3)

and the periodic boundary conditions

that the splitting time can be explained by a stochas-u(r, θ, z + L) = u(r, θ, z).

tic process, and they deﬁned Rec as the intersection ofHere, the Reynolds number is deﬁned as Re ≡two curves for decaying and splitting times above which

2aU/ν and, all the physical quantities have been non-[2]dimensionalized by a for length, 2U for velocity and ν

turbulence is sustainable in the thermodynamic limit.

(4)

being the kinematic viscosity. The reduced pressure p
includes an external force which is consistent with theconstant mass ﬂux condition.

3 Numerical Scheme
The solenoidal ﬁeld u can be expressed asu = ∇ × (ψ ẑ) + ∇ × (∇ × (φẑ)) .

(5)Re Using a spectral method, we solve numerically the setof evolution equations for these scalar functions which
are equivalent to above equations for u. We approximate the scalar functions by a ﬁnite series of expansionas ψ
φ=K


M


N

 mk 
ψnk=−K m=−M n=|m|n+m=evenφmkni[mθ+(2π/L)kz]
Φmn (r)eRe Re 

(6)
where K, M and N (≥ M ) are positive integers
and Φmn (r) are Zernike circular polynomials. We take(N, M, K) = (40, 21, 1535).

Fig. 3: Spreading of turbulence.
CV

4 Splitting
The time variation of the streamwise velocity on thecenterline is plotted in Fig.

3 that shows spread-ing of puﬀs at Re =2100,2150 and 2200.

DV

Dark re-gions represent disturbed ﬂow. Each simulation starts
from the same initial condition, a single puﬀ state at
Re = 2000. We take the streamwise period L very long,E

L = 400, to weaken the eﬀect of the boundary condi-Vtions. At Re = 2100 a puﬀ splits irregularly, whereas at
Re = 2200 the turbulent region seems to be expand continuously while containing laminar spots at places. An
example of puﬀ splitting at Re = 2100 is represented inFVFig. 4. A puﬀ sometimes throws a disturbance ahead,
which is associated with a low-speed streak. At lower
Re, most of these disturbances dissipate (Fig.4 (a)(b)),
while the others lead to the creation of a new puﬀ aheadof the original one (Fig.4 (c)(d)). In case of a successful event, thrown-ahead disturbance spreads around in
the spanwise (θ) direction after suﬃcient downstream
motion. The frequency of these events and the successratio increase as Re increases and the spreading gets
more continuous at higher Re number.

Fig. 4: Puﬀ splitting. Streamwise velocity is plottedin the θz plane at r = 0.8.
[References][1] Shimizu, M. and Kida, S. FDR., 41, 045501 (2009)[2] Avila, K. at al., Science,333, pp.192 (2011)