Transition to turbulence
| Paul Manneville and Yves Pomeau (2009), Scholarpedia, 4(3):2072. | doi:10.4249/scholarpedia.2072 | revision #91885 [link to/cite this article] |
Transition to turbulence is the series of processes by which a flow passes from regular or laminar to irregular or turbulent as the control parameter, usually the Reynolds number \(R\), is increased.
Understanding the transition in view of its control is an important problem because turbulence is characterized by transfers of momentum, heat, chemical species, highly enhanced when compared to corresponding molecular transfers in the laminar regime. Though qualitative descriptions might be found earlier, the history of the problem at a quantitatively level begins with Reynolds' experiments of flow along a long pipe with circular cross-section (1883) [5].
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General setting
The concept of transition scenario was implicitly introduced by Landau (1944) and later revised by Ruelle and Takens (1971) [1]. According to Landau, turbulence is the flow regime reached at the end of an indefinite superposition of successive oscillatory bifurcations, each bringing its unknown phase into the dynamics of the system. In contrast, Ruelle and Takens mathematically showed that Landau's assumption of quasi-periodicity is not generic when nonlinearities are acting. They identified turbulence with the stochastic regime of deterministic chaos characterized by long term unpredictability due to sensitivity to initial conditions and reached after a finite and small number of bifurcations only. Transition to space disorder can be put in the same framework upon assimilating a spatial coordinate with time in the limit of large systems.
From the viewpoint of the theory of nonlinear phenomena, a major difference is between supercritical and sub-critical scenarios:
- At a supercritical transition a continuous substitution of states is observed as the control parameter is increased. The simplest examples are the forward fork bifurcation between steady states and the forward Hopf bifurcation between a steady state and an oscillatory state. Local in phase space, linear stability analysis governs the evolution of (mathematically) infinitesimal perturbations and is the natural starting point of a perturbation approach to the nonlinear situation. This can be generalized into the concept of globally supercritical scenario when at each step the bifurcated state remains close to the bifurcating sate and no hysteresis is observable when the control parameter is varied back and fro.
- The sub-critical transition is in contrast characterized by the coexistence of several possible locally stable states at a given value of the control parameter and an hysteretic behavior as it is varied. By essence it is non-local in phase space. Obvious examples are the inverse fork bifurcation or the inverse Hopf bifurcation, which can be generalized into the concept of globally sub-critical scenario when coexistence and discontinuity are the relevant features.
From a physical viewpoint, one should next distinguish between open and closed systems:
- Open flows [5,8] are characterized by a global transfer of matter from upstream to downstream, with the consequence that transition depends on whether perturbations:
- can resist to the global flow rate and develop into turbulence while staying a fixed location in the laboratory frame, or
- are wiped out by the stream and can at most amplify perturbations at entrance, either controlled or uncontrolled (residual noise); observing sustained turbulence at a given place depends on the amplitudes of the perturbations and reducing their levels delays turbulence farther downstream.
- Closed flows are characterized by the presence of lateral boundaries in all space directions, thus stopping mass transfer. Instability mechanisms must involve feedbacks coupling the fluid's velocity field to other fields. Such instabilities usually introduce some intrinsic length scale in the system, leading to the formation of dissipative structures [2]. The nature of the scenarios leading to turbulence then depends on whether the width of the experimental cell is large or not compared to this length scale, which measures the strength of confinement effects.
Scenarios in open flows
Open flows may be unbounded or bounded [5,8]:
- Examples of unbounded flows are the free shear layer that develops downstream a splitting plate separating two streams parallel to one another with different speeds, the jet, the wake of a blunt obstacle. If present, walls are far from the sheared region and play a marginal role. The major instability follows from the Kelvin-Helmholtz mechanism linked to the presence of an inflection point in the base flow profile, while viscosity plays a stabilizing role (plain mechanical friction damping). This inertial instability is linear and takes place at relatively low values of \(R\). It is generally the starting point of a globally supercritical scenario: The primary instability sets in rolls perpendicular to the direction of flow (stream-wise modulation). Beyond a second threshold, these rolls become unstable against a span-wise mode which induce transversal inflection points. A third instability takes place when corresponding shear is large enough. Small scale turbulence follows soon after but large scale coherent structures organizing the flow can still be detected even in the highly turbulent flow observed far downstream [5].
- The presence of solid walls is essential to the dynamics of bounded flows [5,8]. Standard examples are the Blasius boundary layer, the plane Poiseuille flow driven by a pressure gradient between two walls, the plane Couette flow diven by two walls moving parallel to one another at opposite velocities, or the pipe Poiseuille flow studied by Reynolds. Absence of inflection point in the base flow profile explains that the instability, if any, must rely on the Tollmien-Schlichting mechanism, a counter-intuitive linear feedback coupling in which viscosity plays a destabilizing role. Involving infinitesimal perturbations, such an instability is only possible at large values of \(R\). This leaves room for sustainable nonlinear finite amplitude departures from the base state at more moderate values of \(R\). The general mechanism sustaining this nonlinear non-trivial state involves streamwise vortical perturbations generating alternatively slow and fast streamwise streaks [9]. This linear lift-up mechanism is next closed by a nonlinear feedback that regenerates the vortices. The transition in bounded shear flows typically follows a globally sub-critical scenario marked by the -not yet fully understood- coexistence of spots filled with turbulent flow scattered amidst laminar flow. The same regeneration cycle is expected to hold inside the turbulent spots [4,9].
Typical examples of closed systems
- Rayleigh-Benard convection (RBC) develops in a horizontal fluid layer heated from below and originally at rest. When the temperature gradient exceeds a critical value, convection rolls develop in the cell because viscosity and thermal diffusion are unable to damp out the buoyancy energy release from overturning. The wavelength \(\lambda_c\) of resulting convection rolls is about twice the cell's height \(h\)[6,2].
- Convection in binary mixtures (solute in a solvant) adds a coupling of the velocity and the temperature fields to the concentration of the solute. Resulting thermohaline convection, of interest to e.g. oceanography, generates either steady or oscillatory patterns depending of the relative diffusivities of the components [2].
- Standard Taylor-Couette instability develops in the flow of a pure fluid inbetween two coaxial cylinders rotating at different speeds due to the interplay of destabilizing centrifugal forces induced by rotation and stabilizing viscous forces [2]. It produces axially periodic toroidal flow patterns called Taylor vortices. The width of the vortices is of the order of the gap, to be compared with length of the cylinders.
- The Belousov-Zhabotinsky system (BZ) is an oscillatory chemical reaction which develops uniformly in space. In a stirred reactor, the concentrations of the reactants are homogeneous in space (small aspect-ratio) but still functions of time. In contrast, when performed in a wide Petri dish, a thin layer of reactants displays oscillations that do not stay uniform in space but generate spiraling reaction fronts [7].
- ...
As already noted, besides the nature of the instability mechanism, the most important feature of the transition is linked to confinement effects [2]. They are best appreciated in terms of aspect-ratios \(\Gamma\), i.e. for convection-like systems, especially RBC which has been much studied and is globally well understood, the typical number of rolls in the experimental cell of lateral extent \(\ell\): Confinement is strong when \(\ell \sim \lambda_c\). The instability modes have then frozen spatial structures and the dynamics of the system is best described through the temporal evolution of the amplitudes of these modes, a transition to temporal chaos is observed. In contrast, extended systems correspond to the limit \(\ell \gg \lambda_c\). Confinement is weak and the spatial dependence of the modes participating in the dynamics cannot be evacuated, so that spatiotemporal chaos obtains.
Scenarios in confined systems
Transition theoretically follows from standard reduction to the center manifold (Haken's slaving principle for physicists) yielding a putative low-dimensional dynamical system and a la Ruelle-Takens scenarios [1,2]. Primary, secondary, tertiary... instability modes interact in a complicated way to give its specific structure to the associated phase space and to govern the possible routes to chaos as the control parameter is varied. For example, all scenarios based on the instability of a limit cycle have been observed in Rayleigh-Benard convection, the quasi-periodic route, the sub-harmonic scenario, Type I and Type III intermittency, depending on the fluid's physical properties and the shape of the experimental cell [2]. A similar situation holds for other systems, e.g. the Taylor-Couette wide-gap case when the height of the cylinder is of the order of a few gap widths, or the stirred BZ reaction. Though scenarios have deep universal features of mathematical origin, it does not make sense to classify them since each system has its own physical specificities.
Scenarios in extended systems
- Instability mechanisms work at a local scale to produce structures which are coherent over a length scale \(\xi\). At a supercritical bifurcation this length scale diverges as the threshold is approached [3,2]. However, modulations remain allowed.
- Topological defects are also possible. In convection-like systems, examples are dislocations corresponding to the termination of a pair of rolls, or grain boundaries between differently oriented roll domains.
- Imperfect dissipative structures with modulations and defects are called patterns [2]. The evolution of the system can then be reduced to that of an envelope describing the pattern. The envelope formalism takes advantage of the existence of the two length scales \( \lambda_c \) and \(\xi\gg\lambda_c\) by averaging the dynamics over the small wavelength [2].
- The equation governing the long wavelength modulations of the envelope is generically called a Ginzburg-Landau equation (GLE) [2,5]. It is a partial differential equation governing the space-time dependent amplitude describing the primary bifurcation. Its specific form is dictated by the nature of the latter, the symmetries of the system (translational and rotational invariances, possibly additional Galilean invariance), depends on whether the most unstable mode has finite or infinite critical wavelength (\(k_c=1/\lambda_c\) finite or \(=0\)) and whether it is steady or oscillatory (\(\omega_c=0\) or \(\ne 0\)) [3].
- On practical grounds, the GLE relevant to some given system can be derived only when the primary instability is supercritical. Phenomenological extensions, mostly based on symmetry considerations, are the necessary starting points of more complicated cases (sub-criticality, coupling with large scale flows). The transition to turbulence of patterns is most often well understood using such phenomenological extensions [2,7].
- Rayleigh-Benard convection is an example of instability with \(k_c\ne0, \omega_c=0\), the corresponding GLE, the Newell-Whitehead-Segel equation, has real coefficents.
- Convection in binary mixtures generates dissipative waves with \(k_c\ne0, \omega_c\ne 0\). They are described by two coupled GLE with complex coefficients accounting for wave sources and sinks.
- the BZ reaction is characterized by \(\omega_c\ne0\), \(k_c\ne0\) and described by the standard CGLE, i.e. a GLE with complex coefficients [7].
- Important universal scenarios involve phase instabilities linked to the position and orientation of wavy structures, i.e. the phase of the complex amplitude in the relevant GLE.
- Of special interest is the Kuramoto-Sivashinsky equation (KSE) and associated self-generated phase turbulence provided that the system is large enough [2,3]. The KSE is valid as long as the modulus of the amplitude of the CGLE remains bounded away from zero.
- In contrast, \(2\pi\)-phase defects take place at locations where the modulus of the amplitude reaches zero. In two-dimensions, these defects are topologically stable and control the disorganization of the system which enters a regime of defect turbulence. Defect may take the form of spirals as seen in RBC in some specific cases or in the BZ system [7].
- At a sub-critical bifurcation, several states coexist in phase space at a given value of the control parameter. Furthermore, the instability mechanism generates short range spatiotemporal coherence only. This implies coexistence of states separated by fronts in physical space [4]. Front propagation between laminar states is regular but when one of the competing states is chaotic propagation becomes stochastic. The whole process, called spatiotemporal intermittency then looks similar to directed percolation [4]. The latter is defined as a probabilistic automaton describing contamination processes such as epidemics or forest fires. The transition problem in relevant systems can then be set within the framework of critical phenomena and universality in statistical physics. Obvious candidates are wall flows mentioned earlier., e.g. plane Couette flow, the flow between two parallel plates moving in opposite directions and producing a linearly stable simple shear flow [4].
Later stages
Turbulence level increases upon driving the system farther from the threshold of the primary instability. Secondary, tertiary... non-universal instabilities then set in, as if confinement at the scale of \(\lambda_c\) were effective. The dynamics thus becomes locally chaotic in the same way as in confined systems and a wide spectrum of spatiotemporal scales become active, as expected for developed turbulence.
References
[1] P. Cvitanovic, Ed., 1989, Universality in Chaos, Adam Hilger, Bristol.
[2] P. Manneville, 1990, Dissipative structures and weak turbulence, Academic Press, Boston.
[3] M.C. Cross, P.C. Hohenberg, 1993, Pattern formation outside equilibrium, Rev. Mod. Phys. vol.65, p.851-1112.
[4] P. Berge, Y. Pomeau, Ch. Vidal, 1998, L'espace Chaotique Hermann, Paris.
[5] C. Godreche, P. Manneville, Eds., 1998, Hydrodynamics and nonlinear instabilities; Cambridge University Press, Cambridge, UK.
[6] E. Bodenschatz, W. Pesch, G. Ahlers, G., 2000, Recent developments in Rayleigh--Benard convection, Ann. Rev. Fluid Mech. vol.32, p.708-778.
[7] M.I. Rabinovich, A.B. Ezersky, P.D. Weidman, 2000, The dynamics of patterns. World Scientific, Singapore.
[8] P.J. Schmid, D.S. Henningson, 2001, Stability and Transition in Shear Flow, Applied Mathematical Sciences vol. 142, Springer, Hiedelberg.
[9] T. Mullin, R. Kerswell, eds., 2005, Laminar-Turbulent transition and finite amplitude solutions, Fluid Mechanics and its applications vol. 77, Springer, Heidelberg.
<review> Transition is the dynamical process(es) via which a flow field changes from the laminar state to the turbulent one. It is a complex problem mathematically and analytical solutions can only be obtained for very few simple situations. Physical and numerical experiments are then needed to close the gap. The field is technologically important because of the potential to advance or delay transition. Early transition results in more vigorous mixing and resistance to boundary layer separation, which leads to lower pressure drag. Delaying transition keeps the skin-friction drag from the phenomenally high levels associated with the turbulent state.
Two prominent researchers in the field authored Scholarpedia’s entry on Transition. It is written from a dynamical systems theory viewpoint. Other aspects such as transition control or physics are not emphasized here. Although relatively well written, the piece still requires some editing for spelling, grammar, etc. Encyclopedias typically also provide a different level of editing to ensure that all entries are for the most part free of jargons and, as a result, accessible to the non-specialists. This was not done in this case. I presume that is why Scholarpedia is free and McGraw-Hill Encyclopedia of Science and Technology, say, is not.
The entry is free of technical errors, and is acceptable albeit with a slight concern. Who would benefit from reading the brief treatise? If it is the expert in this particular field, then he/she presumably already knows that stuff. If it is a scientist or engineer who is new to transition and even fluid mechanics, then this entry is not accessible to him or her. Writing the piece in terse bullet format does not help either. Witness as one example the sentence: “The simplest examples are the forward fork bifurcation between steady states and the forward Hopf bifurcation between a steady state and an oscillatory state.” Who would know what a forward fork bifurcation is? I can understand this writing style in a journal paper, but an encyclopedia entry?
Mohamed Gad-el-Hak, Virginia Commonwealth University </review>
