1 Introduction

Deterministic and stochastic aspects of the transition to turbulence

Baofang Song and Björn Hof

IST Austria (Institute of Science and Technology Austria), 3400 Klosterneuburg, Austria

Max Planck Institute for Dynamics and Self-Organization, Bunsenstrasse 10, Göttingen, Germany


Abstract

The purpose of this contribution is to summarize and discuss recent advances regarding the onset of turbulence in shear flows. The absence of a clear cut instability mechanism, the spatio-temporal intermittent character and extremely long lived transients are some of the major difficulties encountered in these flows and have hindered progress towards understanding the transition process. We will show for the case of pipe flow that concepts from nonlinear dynamics and statistical physics can help to explain the onset of turbulence. In particular the turbulent structures (’puffs’) observed close to onset are spatially localized chaotic transients and their lifetimes increase super exponentially with Reynolds number. At the same time fluctuations of individual turbulent puffs can (although very rarely) lead to the nucleation of new puffs. The competition between these two stochastic processes gives rise to a non-equilibrium phase transition where turbulence changes from a super-transient to a sustained state.

1 Introduction

How turbulence first arises in simple shear flows has remained an open question for well over a century. Osborne Reynolds [1] was the first to observe that this transition depends on a dimensionless group, i.e. the Reynolds number, as well as on the amplitude of disturbances present in the system. Some of the leading theorists at the time (e.g. Lord Kelvin, Lord Rayleigh, Arnold Sommerfeld, Werner Heisenberg, Hendrik Antoon Lorentz [44]) attempted to probe the stability of pipe and related shear flows (i.e. channel and Couette flow) with essentially linear methods. After many unsuccessful attempts it has become clear (e.g. see [2]) that the occurrence of turbulence is unrelated to the stability of the laminar state, as Reynolds had already concluded from his experimental observations many years earlier. While pipe and Couette flow are believed to be stable for all Reynolds numbers, plane Poiseuille (i.e. channel) flow becomes unstable at Re of about 5800. However in the latter case turbulence is typically already observed at Reynolds numbers a little above 1000 and to hold the flow laminar up to the linear stability threshold actually requires considerable effort in experiments.
Transition in the type of flows discussed above is qualitatively different from the classical pictures for the transition to turbulence which goes back to Landau and to Ruelle and Takens [3]. In both scenarios turbulence arises following a sequence of instabilities of the base flow (which hence becomes linearly unstable). While the Ruelle Takens type transition has been observed in several closed flows (e.g. [4]) it is noteworthy that even in this case the transition sequence only explains the onset of comparably low dimensional chaotic motion which dynamically is still far from the full spatio-temporal complexity encountered in turbulent flows. In open shear flows such as pipes all experimental observations show that the transition in contrast is rather abrupt directly from laminar to turbulent. The latter type of transition has turned out to be far more difficult to understand.
In more recent years a new transition mechanism has been proposed based on dynamical systems concepts. Invariant solutions of the Navier Stokes equations such as periodic orbits or traveling waves are deemed to be ultimately responsible for the existence of the turbulent state. These new solutions 1 arise as the Reynolds number is increased and importantly are entirely disconnected form the laminar flow. The proposition is then that chaotic and ultimately turbulent motion will arise following instabilities of these disconnected solutions (independent of the laminar state). Over the last two decades or so many traveling waves and periodic orbits have been found in direct numerical simulations for shear flows (for reviews see [6, 7, 8]. While coherent structures resembling travelling waves have also been observed in turbulent flows in experiments [9, 10, 11] a main discrepancy remains: close to onset turbulence is always confined to structures localized in the streamwise direction and surrounded by laminar flow, called ‘puffs’ for the case of pipe flow. Invariant solutions in this range on the other hand are usually periodic in this direction. Only very recently the first streamwise localized invariant solutions were discovered for pipe flow[12]. In addition it could be shown that chaotic motion indeed originates from one such localized periodic orbit. Albeit in this study the dynamics were limited to a symmetry subspace it nevertheless illustrates how turbulence can arise from a simple invariant solution, unrelated to the laminar state, as proposed above. Also this study could explain the origin of another property of localized puffs in pipe flow which is their transient nature. How (and if) turbulence develops from a transient to a sustained state has been subject to much recent debate. As we will argue below spatial aspects are crucial in the transition mechanism and eventually lead to a non equilibrium phase transition giving rise to sustained turbulence. In the following we will discuss this transition in more detail and briefly review important recent results.

2 Discussion

In pipe flow turbulent puffs (Fig. 1(top)) are typically observed in a Reynolds number regime of approximately . They result from perturbations of finite amplitude and in experiments if no great care is taken they will often result from distortions the flow experiences directly at the inlet. If the inlet is designed carefully to avoid such disturbances and if the pipe is sufficiently straight and smooth, flows can be held laminar up to much higher ( the record in experiments is currently at =100000). For controlled studies of transition it is desirable to start with a laminar flow and to study its response to a well controlled disturbance which for example can be a jet of fluid injected for a brief period through a small hole in the pipe wall. If the amplitude of such a perturbation is large enough a turbulent puff is created. While directly after the flow has been perturbed the dynamics depend on the nature of the disturbance (the puff first has to develop), after 100 to 150 advective time units (measured in D/U, where D is the diameter and U the mean velocity) the resulting puff is independent of the perturbation that triggered it.
Curiously in this Reynolds number regime turbulence always remains localized. Even if an extended (axially) part of the pipe is disturbed the flow will always arrange itself in turbulent segments (i.e. puffs) which are about to long (the turbulent core excluding the leading edge) and interspersed by laminar fluid. Vorticity isosurfaces of a turbulent puff at are shown in Fig. 1(top). Fig. 1(bottom) shows energy levels in a space time plot (time from bottom to top) of a simulation of an initially (t=0) fully turbulent flow. Upon reduction of to 2200 laminar regions (blue) appear and turbulence becomes confined to localized regions, i.e. puffs (red vertical stripes in Fig. 1(bottom)). The direct numerical simulations were carried out with a spectral code [13]. Fourier modes in the axial and azimuthal direction and finite difference in the radial are employed and the resolution chosen here is in radial, axial, and azimuthal directions. The domain is in the axial direction ( being the diameter) with periodic boundary conditions.

Figure 1: (top) Isosurfaces of the streamwise vorticity of a puff at , taken from the segment denoted by a black horrizontal bar in the bottom figure. Flow is from left to right. (bottom) Space-time diagram for the reduction from =2800 to 2200 in a comoving frame with the mean flow. The local turbulence intensity is plotted versus aixal position in a logrithmic color scale.

As pointed out in [14] at Reynolds numbers close to 2000 turbulent puffs extract energy from the adjacent (actually upstream) laminar parabolic flow. The plug like turbulent profile in the central part of the puff is unable to sustain turbulence (or rather to feed turbulence in the downstream direction) consequently the turbulence intensity decreases along the leading edge of the puff in the downstream direction and the flow eventually relaminarises. Due to the action of viscosity the laminar profile now begins to recover its parabolic shape so that at sufficient distance a second puff can be sustained. If on the other hand the distance between puffs is too short the downstream puff will border onto fluid with a flatter plug like profile in the upstream direction. Consequently it cannot extract sufficient kinetic energy from the flow upstream and decays as shown in [14]. The interaction distance between two puffs is approximately [15]. As a result turbulent puffs have a minimum spacing of that same distance. While this argument qualitatively explains why fully turbulent flow cannot be sustained at these low Re, the energetic aspects of this process are not understood in full detail.
A key attribute of puffs is their highly chaotic dynamics. This gives rise to a loss of memory and limits the prediction of the flows future evolution. To illustrate the sensitive dependence on initial conditions we simulated two puffs with velocity fields that only deviate by . As usual for chaotic systems this deviation grows exponentially and as can be seen form the energy time series shown in Fig. 2 the signals notably depart and become completely unrelated after about 200 advective time units (the time the puffs takes to travel 200 pipe diameters downstream). The loss of predictabiliy becomes especially clear when one puff suddenly decays (green curve in Fig. 2) while the other continues unchanged (i.e. the average quantities remain unchanged).
It is a typical feature of puffs that they live for very long times and decay suddenly (see Fig. 2). Extensive statistical studies have shown that the survival probability is exponentially distributed and the decay is memoryless. This behaviour is in line with the escape from chaotic repellers observed in lower dimensional systems [20, 21]. Here a chaotic attractor turns into a chaotic saddle after an unstable periodic orbit within the attractor and one on the basin boundary collide (unstable-unstable pair bifurcation). Above the bifurcation point chaotic transients persist for very long times before they eventually decay.

Figure 2: Sensitive dependence on initial conditions of a puff at . These two lines are the time traces of the kinetic energy of two runs starting with two very close initial conditions separated by . While one decays after about 300 time units, the other persists. Eventually the second one will also decay (not shown in the figure) due to the transient nature of puffs

A similar scenario has recently been observed for pipe flow[12]. In this numerical study the dynamics were confined to a symmetry subspace (imposing a mirror and a 2-fold rotation symmetry with respect to the pipe axis). While this somewhat simplifies the dynamics the flow still exhibits turbulent motion for large enough Re. By following the laminar turbulent boundary (the so called “edge state” [16], which in this case is a localised periodic orbit) to lower Re the saddle node bifurcation where the periodic orbits originates was reached (at =1430). The upper branch of this saddle node however is a stable periodic orbit(see Fig. 4) and again localised and its length is comparable to that of puffs.

Figure 3: Stable periodic orbit in direct numerical simulations of pipe flow with an imposed 2-fold rotation and a reflection symmetry. (Note that this is necessarily also a numerical solution of the full Navier Stokes equation.) This orbit arises in a saddle node bifurcation and is originally stable in the sub-space. At higher Re a bifurcation sequence leads to chaos and transient turbulent puffs. The periodic orbit is shown at ( where , is the kinematic viscosity, the pipe diameter and the mean velocity). Isosurfaces in red/blue show velocity deviations of from the laminar parbolic profile. Yellow/ cyan mark isosurfaces of positive/negative streamwise (i.e.axial) vorticity. Upstream and downstream of the periodic orbit the velocity filed quickly approaches the laminar parabolic profile.

For increasing Re the orbit first undergoes a secondary Hopf bifurcation followed by the formation of a chaotic attractor. This is the point where chaotic motion originates in this subspace. The basin of the attractor increases rapidly with until it reconnects with the unstable periodic orbit on the basin boundary. At this moment the chaotic dynamics turn into long lived transients (presumably following an unstable-unstable pair bifurcation). The memoryless nature of the decay and the loss of predictability is a direct consequence of the sensitive dependence on initial conditions characteristic for deterministic chaos.
While the classical picture of turbulence is that of a chaotic attractor, dating back to the landmark paper of Ruelle and Takens, this approach only takes the temporal dynamics into account and neglects the spatial complexity which is however intrinsic to turbulent flows. The importance of spatial aspects and the spatio temporal intermittent character of the turbulence transition have been emphasized in studies of model system [17, 19]. Also the chaotic attractor hypothesis has been questioned in 1980’s by Crutchfield and Kaneko [18] who propose that chaotic super transients are more relevant to turbulence. A first observation supporting this view came from direct numerical simulations of pipe flows where transients were observed [22]. A number of later studies were concerned with long lived transients at low Reynolds numbers in pipe flow [23, 24, 25, 26, 27, 28, 29] and found that the decay is a memoryless process with a characteristic lifetime which is a function of Re. There was however no consensus if or if not the lifetimes of individual puffs became infinite or remained finite. A number of studies proposed that the turbulent flow decay rates (inverse characteristic lifetimes) scale as [13, 23, 24] or at least as a power law [13]. This view was questioned by Hof [25] who found that instead implying that turbulence remains transient. In a refined study for much longer observation times than in any previous study (spanning almost 8 orders of magnitude in time) was found to scale super exponentially with [27] 2. This scaling was confirmed by another experiment [29] and in direct numerical simulations [28]. It is remarkable that based on their studies of spatially coupled maps Crutchfield and Kaneko had not only proposed that fluid turbulence could evolve around spatially coupled transients but also that the lifetimes under certain conditions (type 2 supertransients) are memoryless and increase superexponentially with system size. While it has been argued that an increase in Re in turbulent flows is analogous to an increase in system size in coupled chaotic maps this correpondence is however not entirely clear. The lifetime studies in pipe flow were carried out for single turbulent puffs (not many spatially coupled ones) and the puff size hardly changes with . One may argue that the smallest scales of turbulence decrease with and hence the system size based on this smallest scale increases. Nevertheless the Reynolds number range over which the lifetime increase is observed is relatively small () and hence this size effect will be only very moderate.

A more direct analogy can be drawn to another model system of coupled chaotic maps which is motivated by bistable excitable media [31]. A key difference to the above mentioned map models is that here the susceptibility of a “laminar” site to perturbations from neighbouring chaotic sites (and hence the minimum perturbation amplitude to trigger chaotic dynamics at a laminar site), decreases with (where is a the control parameter analogous to the Reynolds number). This model input reflects experimental observations of pipe flow where the minimum perturbation amplitude to trigger turbulence was found to scale with [32, 9, 33]. In the model just like in the experiments localized excited states, i.e. puffs, with transient lifetimes are observed and with an increase in the control parameter lifetimes of individual puffs scale faster than exponential and hence remain transient.
Consequently in pipe flow the increase in the temporal complexity alone does not lead to sustained turbulence. As proposed by Moxey and Barkley [34] and later explicitly shown by Avila et al. [35] turbulence becomes sustained by a spatial growth process called puff splitting. Puff splitting is commonly observed at Reynolds numbers of around 2300 [36, 37] and while here turbulence is still confined to puffs typically to in length puff sizes fluctuate and can occasionally reach larger values. In these instances in a small number of cases a segment of turbulent fluid at the leading edge of the puff can escape further downstream beyond the puff-puff interaction distance and a new puff develops here (see Fig. 4). This splitting process leads to an increase in turbulent fraction.

Figure 4: Puff splitting process. At the puff leading edge vortices are shed in the downstream direction. While normally they decay, occasionally in rare cases they manage to escape beyond the interaction distance of the upstream puff. The first puff and the vortex patch are now separated by a region of laminar fluid and the vortex patch grows to a new puff. The isosurfaces correspond to the axial vorticity component (positive in blue, negative in yellow).

As shown by Avila et al. [35] puff splitting is also intrinsically memoryless and can already be found at much lower Re as previously expected. The characteristic time for such an event to occur decreases super-exponentially with Re. The argument for turbulence to become sustained is now straightforward. If the characteristic time for turbulent puffs to decay is smaller than the time for new puffs to be created (i.e. by splitting) turbulence will eventually decay. If on the other hand new puffs are created faster than existing ones decay in the thermodynamic limit turbulence becomes sustained. The critical Reynolds number where turbulence changes from a transient to a sustained state can be estimated by the intersection point of the characteristic time scales of the two processes shown in Fig. 5.
This transition is analogous to non equilibrium phase transitions and as we will argue below, bears close resemblance to directed percolation (DP) and related contact processes. First speculations about a possible connection between transition in linearly stable shear flows and DP date back to Pomeau in 1986 [38]. Just like in DP, pipe flow has a unique absorbing state which is the laminar flow. Once turbulence has decayed the flow cannot by itself return to turbulence unless it is disturbed from the outside. The recent studies of puff decay and splitting [27, 35] also suggest that the interaction is only short range, which is another requirement for DP [39]. No observations indicate that a localized puff would create a second one in a part of the pipe not adjacent to it (i.e. more than 25 D or so away). Equally it has never been observed that puffs would influence the lifetimes of other puffs that are sufficiently far away. These recent studies also infer that if such a realtion to DP exists a single unit (i.e. a lattice point in DP) must correspond to a turbulent puff and not for example to a single vortex. To explore this analogy further hence requires much larger system sizes allowing to follow the evolution of many turbulent puffs(/spots). While a number of earlier studies (e.g. [40]) looked at such aspects retrospectively the system sizes used were too small.

Figure 5: The intersection point between characteristic life- and splitting times determines the onset of sustained turbulence in pipe flow. To the right of the intersection point on average new puffs are created faster (by puff splitting) than existing puffs decay and hence (in the thermodynamic limit) spatio-temporally intermittent turbulence persists. The data is taken from [27, 35]. Note that both the decay and the splitting process are memoryless and described by characteristic timescales (see original papers for details

In a numerical investigation [41] of plane Couette we consequently chose a narrow but very long domain so that a large number of turbulent stripes (analogous structure to puffs in pipes) could be accommodated. This study gave more direct evidence that the transition is indeed a non-equilibrium continuous phase transition. Here just like in pipe flow super-exponential lifetime and splitting statistics were observed for single stripes, and a critical point for the onset of sustained turbulence could be determined in the same manner described above. The much shorter time scales at the intersection point between decay and splitting curves allowed to resolve size distributions at this point. The distributions of laminar gaps exhibit scale invariance supporting the proposition of a continuous phase transition. Also here the same transition scenario between transient localized chaos and sustained spatio temporal intermittent chaos was found.
Analogies to DP have also been explored in recent theoretical studies, e.g. [31, 42, 43] In particular for the coupled map model presented in [31] close agreement was found to the experimental results for pipe flow: the turbulent state becomes sustained when the splitting outweighs the decay of individual puffs. In addition the critical exponent for the increase of the turbulent fraction above onset was found to agree well with the universal one for DP in dimension.
While at present a final answer to the question if the laminar turbulence transition is a non-equilibrium phase transition in accordance with DP is outstanding experiments and numerical simulations to clarify this question are under way. One of the main challenges here is to resolve the extremely long time scales relevant in the vicinity of the transition point (note that characteristic splitting and decay times in pipes correspond to almost advective time units!). In this time puffs trave a distance correpsonding to pipe diameters. Equally an accuracy in the Reynolds number of about 0.1% is required setting a further challenge for experiments. A further open issue is the transition from spatially intermittent turbulence (i.e. puffs) to expanding space filling turbulent structures which takes place somewhere between Reynolds numbers 2300 and 3000.

Footnotes

  1. The first such solution was discovered for Couette flow by Nagata in 1990 [5].
  2. The super exponential scaling has been related to extreme statistics theory [30].

References

  1. Reynolds O, An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels, 1883 Proc. R. Soc. London 34 84-99
  2. Drazin P G and Reid W H, Hydrodynamic Stability, 1981 Cambridge University Press
  3. Ruelle D and Takens F, On the nature of turbulence, 1971 Comm. Math. Phys. 20 167-192
  4. Gollub J P and Swinney H L, Onset of Turbulence in a Rotating Fluid , 1975 Phys. Rev. Lett. 35 14
  5. Nagata M, Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity, 1990 J. Fluid Mech. 217 519-527
  6. Kerswell R R, Recent progress in understanding the transition to turbulence in a pipe, 2005 Nonlinearity 18 R17
  7. Eckhardt B, Schneider T M, Hof B, and Westerweel J, Turbulence Transition in Pipe Flow, 2007 Annu. Rev. Fluid Mech. 39 447-68
  8. Kawahara G, Uhlmann M, and van Veen L, The Significance of Simple Invariant Solutions in Turbulent Flows, 2012 Annu. Rev. Fluid Mech. 44 203-25
  9. Hof B, van Doorne C W H, Westerweel J, Nieuwstadt F T M, Faisst H, Eckhardt B, Wedin H, Kerswell R R, and Waleffe F, Experimental Observation of Nonlinear Traveling Waves in Turbulent Pipe Flow, 2004 Science 305 1594-1598
  10. Hof B, van Doorne C W H, Westerweel J, and Nieuwstadt F T M Turbulence Regeneraation in Pipe Flow at Moderate Reynolds Numbers, 2005 Phys. Rev. Lett. 95 214502
  11. de Lozar A, Mellibovsky F, Avila M, and Hof B, Edge State in Pipe Flow Experiments, 2012 Phys. Rev. Lett. 108 214502
  12. Avila M, Mellibovsky F, Roland N, and Hof B, Building blocks of turbulence, 2013 Phys. Rev. Lett. 110 224502
  13. Willis A P and Kerswell R R, Turbulent dynamics of pipe flow captured in a reduced model: puff relaminarisation and localised ’edge’ states, 2009 J. Fluid Mech. 619 213-233
  14. Hof B, de Lozar A, Avila M, Tu X, and Schneider T M, Eliminating Turbulence in Spatially Intermittent Flows, 2010 Science 327 1491-1494
  15. Samanta D, de Lozar A, and Hof B, Experimental investigation of laminar turbulent intermittency in pipe flow, 2011 J. Fluid Mech. 681 193–204
  16. Schneider T M, Eckhardt B, and Yorke J A, Turbulence Transition and the Edge of Chaos in Pipe Flow, 2007 Phys. Rev. Lett. 99 034502
  17. Kaneko K, Spatiotemporal Intermittency in Coupled Map Lattices, 1985 Prog. Theor. Phys. 74 1033-1044
  18. Crutchfied J.P., Kaneko K, Are Attractors Relevant to turbulence?, 1985 Phys. Rev. Lett. 60 2715-2718
  19. Chaté H and Manneville P, Transition to Turbulence via Spatiotemporal Intermittency, 1987 Phys. Rev. Lett. 58 112-115
  20. Grebogi C, Ott E, and Yorke J A, Fractal Basin Boundaries, Long-Lived Chaotic Transients, and Unstable-Unstable Pair Bifurcation, 1983 Phys. Rev. Lett. 50 935-938
  21. Do Y and Lai Y C Extraordinarily superpersistent chaotic transients, 2004 Europhys. Lett. 67 914-920
  22. Brosa U, Turbulence without strange attractor, 1989 Journal of Statistical Physics 55 1303-1312
  23. Faisst H, Turbulence transition in pipe flow, 2003 Ph.D., thesis Universität Marburg
  24. Peixinho J and Mullin T, Decay of Turbulence in Pipe Flow, 2006 Phys. Rev. Lett. 96 094501
  25. Hof B, Westerweel J, Schneider T M, and Eckhardt B, Finite lifetime of turbulence in shear flows, 2006 Nature 443 59-62
  26. Willis A P and Kerswell R R, Critical behavior in the relaminarization of localized turbulence in pipe flow, 2007 Phys. Rev. Lett. 98 014 501
  27. Hof B, de Lozar A, Kuik D J, and Westerweel J, Repeller or Attractor? Selecting the Dynamical Model for the Onset of Turbulence in Pipe Flow, 2008 Phys. Rev. Lett. 101 214501
  28. Avila M, Willis A P, and Hof B, On the transient nature of localized pipe flow turbulence, 2010 J. Fluid Mech. 646 127-136
  29. KUIK D J, Poelma C, and Westerweel J, Quantitative measurement of the lifetime of localized turbulence in pipe flow, 2010 J. Fluid Mech. 645 529-539
  30. Goldenfeld N, Guttenberg N, and Gioia G, Extreme fluctuations and the finite lifetime of the turbulent state, 2010 Phys. Rev. E 81 035304
  31. Barkley D, Simplifying the complexity of pipe flow, 2011 Phys. Rev. E 84 016309
  32. Hof B, Juel A, and Mullin T, Scaling of the Turbulence Transition Threshold in a Pipe, 2003 Phys. Rev. Lett. 91 244502
  33. Hof B, Transition to turbulence in pipe flow, 2005 In Laminar-Turbulent Transition and Finite Amplitude Solutions, ed. T. Mullin, R.R. Kerslwell, Springer 221-231
  34. Moxey D and Barkley D, Distinct large-scale turbulent-laminar states in transitional pipe flow, 2010 Proc. Natl. Acad. Sci. USA 107 8091
  35. K. Avila, D. Moxey, A. de Lozar, M. Avila, D. Barkley and B. Hof, (2011), The onset of turbulence in pipe flow, 2011 Science 333 192-196
  36. Wygnanski I J, Sokolov M, and Friedman D, On transition in a pipe. Part 2. The equilibrium puff, 1975 J. Fluid Mech. 69 283-304
  37. Nishi M, Ünsal B, Durst F, and Biswas G, Laminar-to-turbulent transition of pipe flows through puffs and slugs, 2008 J. Fluid Mech. 614 425-446
  38. Pomeau Y, Front Motion, Metastability and Subcritical Bifurcations in Hydrodynamics, 1986 Physica D 23 3-11
  39. Hinrichen H, Non-equilibrium critical phenomena and phase transitions into absorbing states, 2000 Adv. Phys. 49:7 815-958
  40. Bottin S, Chate H,, Statistical analysis of the transition to turbulence in plane Couette flows, 2000 Eur. Phys. J B 6 143-155
  41. Shi L, Avila M, and Hof B, Scale Invariance at the Onset of Turbulence in Couette Flow, 2013 Phys. Rev. Lett. 110 204502
  42. Sipos M and Goldenfeld N, Directed percolation describes lifetime and growth of turbulent puffs and slugs, 2011 Phys. Rev. E 84 035304(R)
  43. Allhoff K T and Eckhardt B, Directed percolation model for turbulence transition in shear flows, 2012 Fluid Dyn. Res. 44 031201
  44. Eckert M, The troublesome birth of hydrodynamic stability theory: Sommerfeld and the turbulence problem, 2010 Eur. Phys. J. H 35 29-51
100339
This is a comment super asjknd jkasnjk adsnkj
""
The feedback cannot be empty
Submit
Cancel
Comments 0
""
The feedback cannot be empty
   
Add comment
Cancel

You’re adding your first comment!
How to quickly get a good reply:
  • Offer a constructive comment on the author work.
  • Add helpful links to code implementation or project page.