Puff–spot transition in rectangular-duct flow
Keisuke Takeishi 1, Genta Kawahara 1, Markus Uhlmann 2, Alfred Pinelli 3 and Susumu Goto1
121

Graduate School of Engineering Science, Osaka University, JapanTurbulent Flow Group, Institute for Hydromechanics, University of Karlsruhe, 76128
Karlsruhe, Germany
3
Modeling and Numerical Simulation Unit, CIEMAT, 28040 Madrid, Spain

Introduction

Generally, the motion of fluids is smooth and laminar at low
speeds, but it becomes highly disordered and turbulent as thevelocity increases. The onset of turbulence in the flow through along circular straight pipe has intrigued scientists since
Reynolds’ experimental investigations1 at the end of the 19thcentury. Despite more than a century of research, the observed
turbulent motions remain largely mysterious. The principle issue
is that the flow is linearly stable2,3, i.e. it remains laminar for all
Reynolds number. However most practical pipe flows areturbulent. Hence there is a direct conflict between the theory andexperiments.
In the transitional region, turbulence coexists with laminar
flow, and the localized turbulent structures survive. Fig.1 shows
the localized turbulent structures at the transitional Reynolds
numbers. Notice that, in the pipe flow, turbulence occurs in theform of localized patches called puffs, which are embedded in
the surrounding laminar flow and are localized in thestreamwise direction. In the plane channel flow, on the otherhand, turbulence occurs in the form of localized patches calledspots, which are localized in the streamwise direction and in the
spanwise direction.

numerical simulations of the rectangular-duct flow to investigate
the transition dependence on the aspect ratio.
We also study on the marginal Reynolds numbers in therectangular-duct flow. For the plane channel flow, the marginal
Re number is found by a numerical experiment4. Recently themarginal Re number in the pipe flow has been revealed by adirect numerical simulation and an experiment5. However, it is
not known how the marginal Reynolds numbers depend on theaspect ratio in the rectangular duct.

Lx  40  hyxz

h0h h Asmeanflow0 h  AsFig. 2. Computational model of the rectangular-duct flow.
2(a) The turbulent puff structure in a pipe flow.

(b)The turbulent spot structure in a plane channel flow.
Fig. 1. The localized turbulent structures at the transitional
Reynolds numbers.
When the aspect ratio is small, the rectangular duct flow isclose to the pipe flow, so the localized turbulent structure
becomes a turbulent puff. When the aspect ratio is large, therectangular duct flow is close to the plane channel flow, so thelocalized turbulent structure becomes a turbulent spot. However,
it is not known at what aspect ratio the transition from laminar
to turbulent flow takes place. In the present study, conduct direct

Numerical Methods

Fig. 2 shows the coordinate system: the x-axis is in thestreamwise direction, the y-axis is in the duct-height direction
and the z-axis is in the spanwise direction. We fix the grid
numbers of the duct-height direction and of the streamwise
direction, and change the grid numbers of the spanwise direction.
We have numerically integrate the Navier-Stokes equations ofan incompressible fluid by the spectral method.
We decrease the Reynolds number by 5 from an initialcondition, which is a fully turbulent state at Re=1500 (hereRe=Uh/ ν , where U is the mean velocity, h is the duct halfheight and ν is the kinematic viscosity of the fluid). If thelocalized turbulent structure survives for the duration of 400+
after decreasing the Reynolds number, we conclude that theturbulent structure survives (here the symbol + denotes the unit
normalized by the wall friction velocity and the kinematic
viscosity of the fluid). If the localized turbulent structuredisappears before 400+, we conclude that the flow becomesfully laminar. The marginal Re number is evaluated as theminimum Re number, in which the localized turbulent structure
survives.

3Result

 x2 As h 2 ( y / h  1) dzdx
2 A  x  As h x
 1
 x2 h 2

21


x

 ( z / h   As) dydx
2 A x h x
narrow
 1
 x2

 1wide  

In the streamwise direction we integrate these indices over theinternal

whereTurbulent stripe structureStreamwise
directionis satisfied. HereFig. 5 The turbulent stripe structure.

is the cross-sectional mean vorticity strength,x'  1 A

A

Streamwise
direction

Spanwise direction

Spanwise direction

Fig. 3 shows a result of visualization of vortical structuresfrom the positive direction of the y-axis near the sidewall aroundthe marginal Reynolds numbers. Here, to visualize the vortex
structure is used positive isosurfaces of the second invariant ofvelocity gradient tensor Q (Q+=2.0 × 10-3). To distinguishturbulent puffs from turbulent spots, we introduce a new index
to evaluate the strength of the turbulence in the vicinity of thesidewalls (z/h=±As , here As is the aspect ratio.);Fig. 5 shows that the localized turbulent structures obliqueand become turbulent stripe structure for As ≧ 4.0.

x2dydz

Fig. 4 shows that the vortical structures near the sidewall
disappear in the vicinity of As = 4.0. That is, the localized
turbulent structures change from turbulent puffs to turbulent
spots around As = 4.0.

Fig. 6 shows the dependence of marginal Reynolds numbers
on As of rectangular duct. Note that As = 1 corresponds to thesquare duct, whereas As → ∞ corresponds to the plane channelflow. It is observed in this figure that, for As ≧ 4.0, themarginal Reynolds number in the rectangular duct flow
coincides with that in the plane channel flow.

The marginal Reynolds number

DNS to the rectangular-duct flow

Carlson et al.

planechannelflow

Aspect ratio

Fig. 6. The marginal Reynolds numbers as the function of theaspect ratio.
4Fig. 3 Three-dimensional representation of vorticalstructures visualized by the isosurface of Q around themarginal Reynolds number.
PipeSquare duct

Plane channel
flowwidth

LaminarConclusion

We have conducted the direct numerical simulations ofturbulence in a rectangular duct at transitional Reynolds
numbers (between turbulent and laminar flows), and revealed
the dependence of the localized turbulent structures on the
aspect ratio As of the cross-section of the duct. When As < 4, thestructure is puff similarly to the flow in a square duct; whereas
when As > 4 it becomes spot like in the plane channel turbulence.
The dependence of the marginal Reynolds number on As is also
shown to be consistent with these observations.
References[1] O. Reynolds, Proc. R. Soc, A35, 88(1883)heightTurbulentspot[2] P. G. Drazin and W. H. Reid, Hydrodynamic Stability,Cambridge University Press (1980)[3] Y. Pomeau, Physica D 23, 3 (1986).

Aspect ratio
Fig. 4. The magnitude of the vorticity near the sidewall as thefunction of the aspect ratio.

[4] D. Carlson, S. Widnall & M. Peeters, J .Fluid
Mech.,121(1982), 487-505[5] K. Avila, D. Moxey, A. Lozar, M. Avila, D. Barkley, B.
Hof, Science, 2011, vol.333