Transition to turbulence when the Tollmien-Schlichting and bypass routes coexist
Stefan Zammert Bruno Eckhardt


Strong couplings and form factors of charmed mesons in holographic QCD
    Carlisson Miller
    
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Plane Poiseuille flow, the pressure driven flow between parallel plates, 
shows a route to turbulence connected with a linear instability
to Tollmien-Schlichting (TS) waves, and another one, the bypass transition, 
that is triggered with finite amplitude perturbation. 
We use direct numerical simulations to explore the arrangement of the different 
routes to turbulence among the set of initial conditions. 
For plates that are a distance 2H apart and in a domain of width 2π H and length 2π H
the subcritical instability to TS waves sets in at Re_c=5815 
that extends down to Re_TS≈4884. The bypass route becomes available above 
Re_E=459 with the appearance of three-dimensional finite-amplitude traveling waves.
The bypass transition covers a large set of finite amplitude perturbations.
Below Re_c, TS appear for a tiny set of initial conditions that grows with increasing 
Reynolds number. Above Re_c the previously
stable region becomes unstable via TS waves, but a sharp transition to the bypass route can
still be identified. Both routes lead to the same turbulent in the final stage of the transition, 
but on different time scales. Similar phenomena can be expected in other flows where two or more routes to turbulence
compete.








§ INTRODUCTION

The application of ideas from dynamical systems theory to the turbulence transition in flows without linear instability of the
laminar profile, such as pipe flow or plane Couette flow have provided a framework in which many of the
observed phenomena can be rationalized. This includes the sensitive dependence on initial conditions
<cit.>, the appearance of exact coherent states around which the turbulent state
can form <cit.>, 
the transience of the turbulent state <cit.>, 
or the complex spatio-temporal dynamics in large 
systems <cit.>. Methods to identify
the critical thresholds that have to be crossed before the turbulent state can be reached have been developed <cit.> 
and the bifurcation and manifold structures that
explains this behavior in the state space of the system have been identified
<cit.>. 
Extensions to open external
flows, like asymptotic suction boundary layers <cit.> 
and developing boundary layers 
<cit.> have been proposed.  

Plane Poiseuille flow (PPF), the pressure driven flow between parallel plates, 
shows a transition to turbulence near a Reynolds number of about 1000 <cit.>.
In the subcritical range the flow shows much of the transition phenomenology observed in other
subcritical flows, such as plane Couette flow or pipe flow, but it also has a linear instability of 
the laminar profile at a Reynolds number of 5772 <cit.>. 
This raises the question about the relation between the transition via an instability
to the formation of Tollmien-Schlichting (TS) waves and the transition triggered by large amplitude
perturbations that bypass the linear instability 
(henceforth referred to as the "bypass" transition)<cit.>.
For instance, one could imagine that the exact coherent structures related to the bypass transition
are connected to the TS waves in some kind of subcritical bifurcation. However, the flow structures
are very different, with the exact coherent structures being dominated by downstream vortices 
<cit.>,
and the TS waves dominated by spanwise vortices. 

In order to explore the arrangement of the different transition pathways we will use direct
numerical simulations to map out the regions of initial conditions that follow one or the other path.
Such explorations of the state space of a flow have been useful in the identification of the sensitive dependence on initial conditions for the transition
<cit.>, and in the exploration of the bifurcations <cit.>. 

We start with a description of the system and the bifurcations of the relevant
coherent states in section <ref>. Afterwards, in section <ref> we describe the exploration of the state space of the
system. Conclusions are summarized in section <ref>.

 


§ PLANE POISEUILLE FLOW AND ITS COHERENT STRUCTURES



To fix the geometry, let 
 x, y, and z be the downstream, normal and spanwise directions, 
 and let the flow be bounded by parallel plates at y=± H. The flow is driven by
 a pressure gradient, giving a parabolic profile for the laminar flow.
Dimensionless units are formed with the height H
and the center line velocity U_0 so that the unit of time is H/U_0 and the
Reynolds number becomes Re=U_0 H / ν, with ν the fluid viscosity. In these
units the laminar profile becomes u⃗_0=(1-y^2) e⃗_x. 
The equations of motion, the incompressible Navier-Stokes equations,
are solved using Channelflow <cit.>, with a spatial resolution 
of N_x=N_z=32 and N_y=65 for a domain
of length 2 π and width 2π and at fixed mass flux.  
The chosen resolution is sufficient to resolve the exact solutions and the transition process
but underresolved in the turbulent case.

In the studied domain, the linear 
instability occurs at Re_c=5815, slightly higher than the value 
found by <cit.> on account of the slightly different
domain size.

The full velocity field U⃗=u⃗_0+u⃗ can be written as a sum of
the laminar flow u⃗_0 and deviations u⃗=(u,v,w).  In the following
we always mean u⃗ when we refer to the velocity field. 
Tollmien-Schlichting (TS) waves are travelling waves formed by spanwise vortices. 

They appear in a subcritical bifurcation that extends down to Re≈ 2610 
for a streamwise wavenumber of 1.36.  
The TS wave is independent of spanwise position z and consists of two spanwise vortices, as
shown in figure <ref>(a).



The Reynolds number range over which the transition to TS waves is subcritical depends on the 
domain size. For our domain (streamwise wave number of 1.0)
the turning point is at Re≈ 4685. 
A bifurcation diagram of this exact solution, referred to as TW_TS 
in the remainder of the  paper, is shown in figure <ref>(a). 
The ordinate in the bifurcation diagram is the amplitude of the flow field

    a(u⃗)=||u⃗||=√(1/L_xL_yL_z∫u⃗^2 dx  dy  dz).

A study of the stability of the state in the full three-dimensional space shows that this lower
branch state has only one unstable direction in the used computational domain for 5727<Re<5815=Re_c.
Thus, for these Reynolds numbers the state is an edge state whose stable
manifold can divide the state space in two parts <cit.>.
For lower Re, there are secondary bifurcations  that add more unstable directions to the
state. Specifically, near the turning point at Re=4690, the lower branch has acquired 
about 350 unstable directions.
Because of the high critical Reynolds numbers this state cannot explain the transition to 
turbulence observed in experiments at  Reynolds numbers around 1000 
<cit.> or even lower <cit.>.



The states that are relevant to the bypass transition can be found using the method of edge tracking 
<cit.>.
Starting from an arbitrary turbulent initial condition, trajectories in the laminar-turbulent-boundary that 
are followed with the edge-tracking algorithm converge to a travelling wave <cit.> which we referred to as TW_E in the following. 
The visualization in figure <ref>(b) shows that this state has a strong narrow upstream streak, a weaker but more extended downstream streak and streamwise vortices. 
Moreover, TW_E  has a wall-normal reflection symmetry

    s_y: [u,v,w](x,y,z)=[u,-v,w](x,-y,z),

a shift-and-reflect symmetry

    s_zτ_x: [u,v,w](x,y,z)=[u,v,-w](x+0.5 · L_x,y,-z),

and exists for a wide range in Reynolds numbers. 
It is created in a saddle-node bifurcation near Re ≈ 459 
(see the bifurcation diagram in figure <ref>(a));
for other combinations of spanwise and streamwise wavelengths
the state appears at a even lower Reynolds numbers of 319 <cit.>.
 The corresponding lower branch state can be continued to
Reynolds numbers far above 3· 10^5, and its amplitude decreases 
with increasing Reynolds number as shown in 
figure <ref>(b). A fit to the amplitude for large Reynolds numbers gives a scaling like
Re^-0.52, similar to that of the solution embedded in the edge of plane Couette flow <cit.>.
A stability analysis of the lower branch of TW_E  shows that the travelling wave has 
one unstable eigenvalue for 510<Re<5850. Therefore, TW_E is a second travelling wave with a stable 
manifold that can  divide the state space into two disconnected parts. How the two edge states interact
and divide up the state space will be discussed in section <ref>.

At Re=510  the lower branch undergoes a supercritical 
pitchfork bifurcation that breaks the s_y symmetry and adds a second unstable eigenvalues for Re<510.
The upper branch of the travelling wave has three unstable eigenvalues for Re<1000.  
Investigation of different systems which show subcritical turbulence
revealed that bifurcations of exact solutions connected to the edge state of the system lead to the formation 
of a chaotic saddle that shows transient turbulence with exponential distributed lifetimes <cit.>.
In the present systems the formation of chaotic saddle 
cannot be studied in detail since it takes place in an unstable subspace.
However, previous investigations in a symmetry restricted system did show that the 
states follow such a sequence of bifurcations to the formation of a chaotic saddle  <cit.>, so that we expect that also the states in the unstable subspace
follow this phenomenology.

The two travelling waves described above are clearly related to the two different transition mechanism 
that exist in the flow. For Reynolds numbers below the onset of TS waves (here: Re_c=5815), 
initial conditions that start close to TW_E in the state space will either decay 
or become turbulent without showing any approach to a TS wave: 
they will follow the bypass transition to turbulence. 
Initial condition that start close to TW_TS can also either decay or swing up to turbulence, but
they will first form TS waves. Above Re_c all initial conditions will show a transition 
to turbulence, but it will still be possible to distinguish whether they follow the bypass or
TS route to turbulence, as we will see.



§ STATE SPACE STRUCTURE



In order to explore the arrangement of the different routes to turbulence in the space of initial
conditions we pick initial conditions and integrate them until the flow either becomes turbulent or 
until it returns to the laminar profile. The initial conditions are taken in a two-dimensional slice of the
high-dimensional space, spanned by two flow fields u⃗_1 and u⃗_2.
The choice of the flow fields allows to explore different cross sections of state space. For the most
part, we will use u⃗_1 and u⃗_2 to be the travelling waves  TW_E and TW_TS,
so that both states are part of the cross section.  The initial conditions are then 
parametrized by a mixing parameter α and an amplitude A, i.e.,

    u⃗(α,A)=
    A  (1-α) u⃗_1 + αu⃗_2/||(1-α) u⃗_1 + 
    αu⃗_2||.

For α=0 one explores the state space along velocity field u⃗_1 and for
α=1 along velocity field u⃗_2.
If the upper and lower branch of TW_E are used to create such a slice, one recognizes that
the turbulence in PPF appears in similar chaotic bubbles as in 
plane Couette <cit.>.


Lower branch states are relevant for the transition to turbulence, so begin by exploring the
slice spanned by the lower branches of TW_E and TW_TS. We assign to each 
initial condition the time it takes to become turbulent, with an upper cut-off for initial conditions
that either take longer or that never become turbulent because they return to the laminar profile.
Color coded transition-time plots are shown in figure <ref>(a) - (e) for
different Reynolds numbers below Re_c.
The boundary between initial conditions that relaminarize and those that become turbulent stands out
clearly. They are formed by the stable manifold of the states and their crossings with the cross section.
Parts of the stable manifold are indicated by the dashed white lines for better visibility. 
The part of the laminar-turbulent boundary connected with TW_E can be distinguished from that 
connected to TW_TS by the huge differences in transition times:
for TW_TS transition times are significantly longer and even exceed 2· 10^4 time units. 
The interaction between the two domains is rather intricate. 
For Reynolds number 5780, shown in  figure <ref>(d), it seems that the borders 
do not cross but rather wind around each other in a spiral shape down to very small scales.
Although the wave TW_TS has still only one unstable eigenvalues, 
the size of the structure that is directly connected to TW_TS shrinks 
with decreasing Re and is not visible in these kind of 
projection for Re<5727 where TW_TS has more than one unstable eigenvalue. 
 

In figure <ref> the evolution of the amplitude for different initial conditions marked in figure <ref>(d) 
is shown. The green and blue lines are typical representatives of the slow TS transition. Starting with
a three dimensional initial condition, their amplitude decays and the two-dimensional TS wave TW_TS, whose amplitude is marked by the black line, is approached.
Afterwards, they depart from TW_TS again, which is a slow process because of the small growth rate.
Ultimately, the transition is caused by secondary instabilities of the TS waves <cit.>.

The solid yellow line in figure <ref> is an initial condition that undergoes bypass transition. It
quickly swings up to higher amplitudes and does not approach the TS wave on its way to turbulence.
The dashed yellow line is of an intermediate type. It takes a long time to become turbulent but it does not
come very close to the TS wave. The relation between time-evolution, transient amplification,
and final state is complicated and non-intuitive. For instance, the dashed red and green trajectories 
share a transients increase near t≈4000, but differ in their final state: the red curve, with the higher
maximum, eventually returns to the laminar profile, but the green curve, with the smaller maximum,
approaches the TS level and eventually becomes turbulent following the TS route.
Similarly, the red, blue and green continuous lines start with high amplitude slightly below the 
threshold for the bypass route. They all decay, but while the red initial conditions ends up on 
the decaying side of the TS wave, the green and blue one eventually become turbulent
via the TS route.

For plane Couette flow it was found that a small chaotic saddle can appear inside of existing larger ones <cit.>. 
There, trajectories that escape from the inner saddle are still captured by the outer one. The appearance of TS transition
in PPF follows a comparable mechanism. With increasing Reynolds number the chaotic saddle of subcritical bypass turbulence is surrounded
by the stable manifold of the TS wave that above Re=5727 can separate two parts of the state space and therefore prevent trajectories in the interior from becoming laminar.

With increasing Reynolds number the number of initial conditions becoming turbulent increases. 
Finally, for Re>Re_c no initial conditions that return to the laminar state exist anymore. 
Nevertheless, also in this supercritical regime a sudden change in the type of transition can be identified:
when the amplitude increases and crosses the stable manifold of TW_E, the transition time
drops dramatically and turbulence is reached via the bypass route. 
In the state space visualization for Re=5855 that is shown in figure <ref>(c), these change of the transition type presents itself in the rapid drop of the transition time with increasing a for α values between 0 and 0.6.

In the supercritical range the stable manifold of the bypass edge state TW_E separates initial conditions undergoing the quick bypass
transition form initial conditions that become turbulent by TS transition.
The state space picture at a higher Reynolds number of 6000 looks qualitatively similar to 
the one shown in figure <ref>(c), including the switch from TS to bypass transition when
the  stable manifold of TW_E is crossed.



§ CONCLUSIONS 


We have explored the coexistence of two types of transition in subcritical plane Poiseuille flow 
connected with the existence of states dominated by streamwise and spanwise vortices
(bypass and TS transition). 
Probing the state space by scanning initial conditions in two-dimensional 
cross sections gave information on the sets of initial conditions that follow one or the other route to turbulence. 
The results show that the transition via TS waves initially occupies a tiny region of state space. As this
region expands it approaches the bypass-dominated regions, but a boundary between the two 
remains visible because of the very different times needed to reach turbulence. This extends to the
parameter range where the laminar profile is unstable to the formation of TS waves. 

The results shown here are obtained for small domains, where the extensive numerical computations
for very many initial conditions are feasible. For larger domains, the corresponding exact coherent
structures are localized, as shown by <cit.> and <cit.> for TS waves and 
by <cit.> for the bypass transition. Since the bifurcation diagrams for the 
localized states are similar to that of the extended states, we anticipate a similar phenomenology 
also for localized perturbations in spatially extended states. 

The methods presented here can also be used to explore the relation between bypass transition 
and TS waves in boundary layers <cit.>. More generally,
they can be applied to any kind of transition where two different paths compete: examples include
shear driven or convection driven instabilities in thermal convection <cit.>, 
the interaction between transitions driven by different symmetries <cit.>, 
or the interaction between the established subcritical scenario and the recently discovered
linear instability in Taylor-Couette flow with rotating outer cylinder <cit.>.



This work was supported in part by the German Research Foundation (DFG) within Forschergruppe 1182.



jfm


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