New Experiments Set the Scale for the Onset of Turbulence in Pipe Flow
Measurements of the stability of laminar flow
bring us closer to answering one of the biggest outstanding questions
in fluid mechanics.
In 1883, Osborne Reynolds published his landmark paper on the
transition from smooth, laminar flow to turbulent flow in cylindrical
pipes. Drawing water through a horizontal glass pipe, Reynolds injected
a narrow stream of dye and looked for the onset of eddies as he varied
the flow velocity and the water viscosity (dependent on water
temperature). He found that the transition to turbulence was very
sensitive to disturbances and typically occurred above a critical value
of about 2000 for the ratio of UD/ν, where U is the average (or bulk) velocity, D is the pipe diameter, and ν is the kinematic viscosity.^{1} This ratio, which parameterizes the relative strengths of inertial and viscous forces, is now known as the Reynolds number Re.
Understanding the nature of the transition to turbulence has been an
ongoing quest ever since Reynolds's first experiments (and was the
subject of Werner Heisenberg's PhD thesis in 1923). For pipe flow, the
underlying NavierStokes equations, which describe the fluid dynamics
of a system, have a laminar solution that's been found numerically to
be stable for all Reynolds numbers. Indeed, in exquisitely controlled
experiments, laminar flow at Reynolds numbers up to 100 000 has been
observed.
And yet in practice, most pipe flowsat least for Re
above about 2000, a typical value for a moderate flow of water from a
faucetare turbulent. Because laminar flow is linearly stablethat
is, stable against infinitesimal perturbationsa finiteamplitude
perturbation must be required to kick pipe flow out of that state and
into a turbulent mode.
The nature of that transition is of more than academic
interest. For a given pressure drop along a pipe, turbulent flow will
result in a flow rate an order of magnitude smaller than laminar flow.
To avoid large pressure and flow fluctuations associated with the
turbulence transition, oil and gas pipelines are usually operated in
the lessefficient turbulent regime. The ability to predictand
perhaps eventually controlthe transition to turbulence could be a
real boon.
In recent experiments,^{2} Björn
Hof, Anne Juel, and Tom Mullin of the University of Manchester have
measured how the threshold amplitude of turbulenceproducing
perturbations in pipe flow scales with Re. "They have
unambiguously determined the thresholds in controlled experiments for
the first time," comments Dan Henningson, a fluid dynamicist at
Sweden's Royal Institute of Technology (KTH).
Of pipes and planes
Pipe flow is just one case of important flow scenarios in fluid
dynamics. The other two canonical shear flows are plane Couette flow
and plane Poiseuille flow. In plane Couette flow, the fluid channel is
defined by parallel walls that are moving relative to each other; fluid
adjacent to each wall has the same velocity as the wall, and in laminar
flow there is a linear velocity profile from one wall to the other. (In
a variant geometry, TaylorCouette flow, the fluid is confined between
differentially rotating coaxial cylinders.)
Plane Poiseuille flow is confined by stationary parallel walls
and is typically driven by a pressure gradient along the channel. Fluid
next to the walls is stationary, which leads to a laminar parabolic
velocity distribution across the channel. Pipe flow, also called
HagenPoiseuille flow, is similar to plane Poiseuille flow in that the
walls are stationary and the flow is again typically driven by a
pressure gradient. As a consequence of the threedimensional
cylindrical geometry, theoretical treatments of pipe flow tend to be
more difficult than that of its planar cousins.
Like pipe flow, Couette flow is linearly stable at all Reynolds
numbers; plane Poiseuille flow, in contrast, develops an instability at
a finite Re.
Bringing modern mathematical ideas to bear on the turbulence transition
in all these flows has become prevalent over the past 10 years.^{3} A key question that's been looked at is how the threshold perturbation amplitude ε scales with Re: If ε~ Re^{γ},
as is generally assumed, what is the value of γ? Earlier studies placed
γ between 1 and 7/4. Looking at the asymptotic behavior of the
NavierStokes equation, for example, Jon Chapman of the University of
Oxford has predicted that γ is 1 for Couette flow and 3/2 for plane
Poiseuille flow below its instability.^{4}
A big syringe
To set about looking experimentally for the threshold scaling in pipe
flow, Mullin and colleagues built a special pipe, illustrated in figure 1a,
that is 15.7 meters long, 785 times its 20mm diameter. One cannot buy
such a long straight pipe, so the Manchester group fabricated theirs
out of 105 machined sections of acrylic, each 150 mm long. Having a
long pipe is important because it takes time to develop a steady flow,
particularly at larger Re (corresponding to faster flow rates).
After aligning the sections with a laser, the team succeeded in
generating laminar flow through their pipe at a very high Re,
24 000 (corresponding to a flow rate of 3 m/s). Such rapid laminar flow
was a testament to the quality of the assembled pipe, which took four
years to create, align, test, and calibrate.
Most experiments studying pipe flow have driven the flow by
applying a pressure gradient along the length of the pipe, with either
an elevated source of fluid or, as Reynolds originally used, a lowered
drain at the end of the pipe. Mullin and coworkers took a different
tack: Like a big syringe, a computercontrolled piston pulled water
through the pipe at a specified rate from a large settling tank. That
approach has the advantage of allowing direct control of the mean fluid
velocity and hence Re. In contrast, for flows driven by pressure gradients, the velocityand Rewill vary during the transition to turbulence. For looking at how the threshold depends on Re, the ability to fix Re is important.
The perturbations to the flow were generated using an automobile fuel
injector connected to a fastswitching pump. That configuration allowed
the experimenters to inject tangentially a controlled flux of water for
a controlled duration through six holes evenly spaced around the pipe.
The injected flux was the experimental knob corresponding to the
perturbation amplitude; typical values were a few tenths of milliliters
per second, two to three orders of magnitude smaller than the total
flux through the pipe.
The thresholdmeasuring experiments were very time consuming.
The researchers took 40 measurements to determine each threshold for a
given Re
and injection duration. To ensure that the observed flow
characteristics were due solely to the controlled perturbation and not
to any spurious background influences such as temperature gradients in
the water tank, they waited for an hour before each run to give the
system time to settle. The flow was imaged by monitoring the reflection
off anisotropic platelets that had been dispersed in the water (see figure 1b).
Water injected into the pipe gets pulled along by the flow; how far an
injected pulse got carried along the length of the pipe turned out to
be a key parameter affecting the system's response. When the Manchester
group plotted their threshold data against the spatial length of the
perturbation, the results for different Re
collapsed onto a single curve. The threshold depended sharply on that
length for short pulses, for which larger amplitudes were needed;
otherwise the disturbances decayed as the injected pulse traveled down
the pipe. The threshold amplitude for perturbation lengths longer than
about six pipe diameters became independent of length.
For
comparison to theory, the natural means of describing the threshold
amplitude is the volume flux of the perturbation normalized by the flux
in the pipe. When plotted this way, the threshold amplitudes showed an
inversely proportional dependence on Rethat is, a scaling exponent γ of 1, as seen in figure 2. That scaling relationship extends over an order of magnitude in Re,
from 2000 to nearly 20 000. "What's interesting is that the scaling
looks so clean at relatively low Reynolds number," comments Rich
Kerswell of the University of Bristol. The scaling exponent of 1
agrees with calculations by Chapman, based on the growth of transient
fluctuations, but it's not yet clear whether that mechanism is what's
at work in the Manchester experiments.
The nature of the transition
Of course, determining the size of the kick needed to drive a flow
turbulent is only one part of understanding the turbulence transition.
The mechanism by which the turbulent state develops is also vital to a
full understanding.
The laminar flow state can be viewed as an island (or,
technically, a basin) of stability in the sea of phase space. The
Manchester work has measured how the island size decreases as Re increases. But what happens when a sufficiently strong perturbation knocks the system off the island?
A mathematical solution to that question may be emerging. Fabian
Waleffe of the University of WisconsinMadison has formulated a
socalled selfsustaining process that leads to nonlinear,
threedimensional travelingwave solutions of the Navier Stokes
equations for plane Couette and plane Poiseuille flow.^{5} Holger Faisst and Bruno Eckhardt of the University of Marburg^{6} and Hakan Wedin and Kerswell at Bristol^{7} have recently reported a class of similar travelingwave solutions in pipe flow.
Those solutions, one of which is shown in figure 3,
have a discrete number of fastermoving streaks of fluid near the wall
and slower streaks near the center, and vortices around which the fluid
spirals as it travels down the pipe. Hof, now with Frans Nieuwstadt at
the Delft University of Technology, has found some evidence for such
states in pipe flow. Although the travelingwave flow states are
unstable, they may represent the first indications of increasing
complexity in phase space, which ultimately harbors a turbulent
attractor as Re increases.
Richard Fitzgerald
References
1. O. Reynolds, Proc. R. Soc. London 35, 84 (1883).
2. B. Hof, A. Juel, T. Mullin, Phys. Rev. Lett. 91, 244502 (2003).
3. See, for example, the discussion and references in P. J. Schmid, D. S. Henningson, Stability and Transition in Shear Flows, SpringerVerlag, New York (2001).
4. S. J. Chapman, J. Fluid Mech. 451, 35 (2002).
5. F. Waleffe, Phys. Fluids 9, 883 (1997); 15, 1517 (2003).
6. H. Faisst, B. Eckhardt, Phys. Rev. Lett. 91, 224502 (2003).
2004 American Institute of Physics


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