Transition to turbulence

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Author: Dr. Paul Manneville, Ecole polytechnique, Palaiseau, France
Author: Dr. Yves Pomeau, University of Arizona, Tucson, AZ

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 Re, is increased.

Understanding this flow transition in view of its control is an important problem. Turbulence is characterized by highly enhanced transfers of momentum, heat and chemical species, when compared to molecular transfers in laminar flow. Though qualitative descriptions might be found earlier, the history of the problem at a quantitative level begins with pipe flow experiments by Reynolds in 1883 [5].

Contents

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General setting

The concept of transition scenario was implicitly introduced by Landau in 1944 and later revised by Ruelle and Takens in 1971 [1]. According to Landau, turbulence is 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 universal 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 only after a finite and small number of bifurcations.

From the viewpoint of the theory of nonlinear phenomena, there is a major difference between a supercritical and a sub-critical scenario:

  • In a supercritical transition, a continuous evolution 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 steady and oscillatory states. Local in phase space, linear stability analysis governs the evolution of (mathematically) infinitesimal perturbations and it is the natural starting point of a perturbation approach to the nonlinear problem. This can be generalized into the concept of globally supercritical scenarios when, at each step, the bifurcated state remains close to the bifurcating state and no hysteresis is observable when the control parameter is varied.
  • In contrast, the sub-critical transition is characterized by the coexistence of several possible locally stable states for a given value of the control parameter, and an hysteretic behavior as the control parameter is varied. In essence, it is non-local in phase space. Obvious examples are the inverse fork and the inverse Hopf bifurcations, which can be generalized into the concept of globally sub-critical scenarios when coexistence and discontinuity are the relevant features.

From a physical viewpoint, one should distinguish between open and closed flows:

  • 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 at a fixed location in the laboratory frame, or
    • are wiped out by the stream and can, at most, amplify perturbations at the entrance, either controlled or uncontrolled (residual noise); observing sustained turbulence at a given place depends on the amplitudes of the perturbations. Reducing their amplitudes delays turbulence farther downstream.
  • Closed flows are characterized by the presence of lateral boundaries in all space directions. Instability mechanisms must involve feedbacks between the fluid's velocity field and other flow fields. Such instabilities usually introduce an intrinsic length scale in the flow, leading to the formation of dissipative structures [2]. The scenarios leading to turbulence depend on the relative width of the experimental cell compared to this length scale, which measures the strength of confinement effects.
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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 of a splitting plate separating two parallel streams with different speeds, the jet, and the wake of a blunt obstacle. If present, walls are far from the sheared region and they play a marginal role. The major instability follows from the Kelvin-Helmholtz mechanism due to the presence of an inflection point in the base flow profile. Viscosity plays a stabilizing role (plain mechanical friction damping). This inertial instability is linear and it happens at relatively low values of Re. 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 the corresponding shear is large enough. Small scale turbulence follows soon after but large scale coherent structures 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 driven by two walls moving parallel to one another with opposite velocities, or the pipe Poiseuille flow studied by Reynolds. Absence of inflection points in the base flow profile explains that the instability, if any, must rely on the Tollmien-Schlichting mechanism, a counter-intuitive linear feedback in which viscosity plays a destabilizing role. Involving infinitesimal perturbations, such an instability is only possible at large values of Re. This leaves room for sustainable nonlinear finite amplitude departures from the base state at more moderate values of Re. The general mechanism sustaining this 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].
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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 buoyant energy release from overturning. The wavelength \lambda_c of the 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 oceanography, generates either steady or oscillatory patterns depending on the relative diffusivities of the components [2].
  • Standard Taylor-Couette instability develops in the fluid flow between two coaxial cylinders rotating at different speeds. This instability is due to the interplay of centrifugal and viscous forces [2]. It produces axially periodic toroidal flow patterns called Taylor vortices. The width of the vortices is on the order of the gap, to be compared with the length of the cylinders.
  • The Belousov-Zhabotinsky reaction (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 they are 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. For convection-like systems, especially RBC which has been extensively studied and is well understood, confinement is strong when \ell \sim \lambda_c, where \ell is the lateral extent of the experimental cell. The instability modes have 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 evaluated, so spatiotemporal chaos takes place.

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Scenarios in confined systems

Theoretically, transition 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 a specific structure to the associated phase space. They also determine 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: quasi-periodic route, 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, like the Taylor-Couette wide-gap case (when the height of the cylinder is on the order of a few gap widths), or the stirred BZ reaction. Though scenarios have universal features of mathematical origin, it does not make sense to classify them since each system has its own physical traits.

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Scenarios in extended systems

  • Instability mechanisms work at a local scale to produce structures that 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), depend 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 (e.g., sub-criticality and coupling with large scale flows). The transition to turbulence of patterns is most often better 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. It is 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 its associated self-generated phase turbulence provided that the system is large enough [2,3]. The KSE is valid as long as the amplitude's modulus 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. They control the disorganization of the system when it enters a regime of defect turbulence. Defect may take the form of spirals as seen in the BZ system or in some specific cases of RBC [7].
  • At a sub-critical bifurcation, several states coexist in phase space at a given value of the control parameter. Furthermore, the instability mechanism only generates short range spatiotemporal coherence. 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, becomes similar to directed percolation [4]. The latter is defined as a probabilistic automaton describing contamination such as epidemics or forest fires. The transition problem can then be put within the framework of critical phenomena and of universality in statistical physics. Obvious candidates are wall flows mentioned earlier., e.g. the plane Couette flow (the flow between two parallel plates moving in opposite directions and producing a linearly stable simple shear flow) [4].
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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 was effective. The dynamics becomes locally chaotic as in confined systems, and a wide spectrum of spatiotemporal scales becomes active, as it is expected for developed turbulence.

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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.

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See also

Chaos, Dynamical systems, Turbulence

Action editor: Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Reviewer A: Dr. Mohamed Gad-el-Hak, Virginia Commonwealth University, Richmond, VA
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