Models for Turbulent Pipe FlowI have
proposed modeling transitional pipe flow as an excitable and
bistable medium^{1}
^{2}.
The models are presented in two variables, turbulence intensity
and mean shear, that evolve according to established properties
of transitional turbulence described below. A continuous model
captures the essence of the puffslug transition as a change
from excitability to bistability. A discrete model and a
stochastic PDE model additionally incorporate turbulence that
is locally transient. These models reproduce almost all
largescale features of transitional pipe flow. In particular
they captures metastable localized puffs, puff splitting,
slugs, localized edge states, a continuous transition to
sustained turbulence via spatiotemporal intermittency (directed
percolation), and a subsequent increase in turbulence fraction
towards uniform, featureless turbulence. 
Two VariablesShown below are the instantaneous turbulent fluctuations and the mean shear profiles from direct numerical simulation (DNS) of a turbulent puff at Re=2000. (The turbulent fluctuations are here simply the magnitude of transverse velocity. The mean is shear profiles are instantaneous azimuthal averages of streamwise velocity.) 
Two scalar variables, q and u, depending only on streamwise coordinate x are obtained by sampling the above fields on the centerline. The axial velocity on the centerline provides a convenient proxy for the mean shear. 
Physical FeaturesModels are based on the following established properties of transitional turbulence. Most of these features are seen in the figure below. The first four properties are contained in Wygnanski et al.^{3} ^{4},Sreenivasan et al.^{5}, and other “classic” works. The last two properties, and especially the crucial final property, have been understood only more recently^{6} ^{7}. 



PuffSlug TransitionThe following PDE model capture all of the above physical properties, except the last. The parameter r plays the role of Reynolds number. The most essential point is that these equations capture the difference between puffs and slugs as a difference between excitability and bistability. The transition occurs at r = r_{c}, where r_{c} is easily computable. Further details of these equations can be found in papers and talks here. It is probably useful to understand basic properties of excitable media^{10} ^{11}. Puff Regime – excitability There is a reverse transition on the downstream side of model puffs. This gives puffs a finite, selected streamwise width. They propagate at a selected speed determined by the upstream front. Slug Regime – bistability There is no reverse transition on the downstream side of model slugs. The system is locally bistable. The upstream and downstream fronts have different selected speeds and the slugs can expand to arbitrary streamwise length. 

Locally Transient ChaosThe above PDE model captures the essence of the puffslug transition, but misses the crucial property of locally transient chaos (chaotic saddle). Here two approaches have been used to incorporate this while keeping with a twovariable description.


Map Model (Discrete) Space and
time are discrete. The uequation is obtained an Euler
time step of the PDE model. The qequation uses a tent
map (given by the function F) to generate locally transient
chaotic dynamics. The dynamics is deterministic. Noise Model (SPDE) Multiplicative
noise is added to the qequation. The dynamics is
random. These models capture nearly all largescale features of transitional pipe flow. Below are illustrations of puffs, puffsplitting, and slugs in the two models. Further comparisons can be seen here.


Map Model 
Noise
Model 
