Transition to turbulence
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.
Turbulence is characterized by highly enhanced transfers of momentum, heat and chemical species, when compared to molecular transfers in laminar flow. Understanding the transition in view of its control is an important problem. 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 .
The concept of transition scenario was implicitly introduced by Landau in 1944 [2a] and later revised by Ruelle and Takens in 1971 [2b]. 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 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 only after a finite and small number of bifurcations.
From the general 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 example is the continuous bifurcation between two steady states, one replacing the other, the distance between them increasing steadily as the control parameter is further increased, often as the square root of the distance to the bifurcation threshold (fork bifurcation). The next step is often the birth of oscillations the amplitude of which progressively increases with the control parameter (Hopf bifurcation). Local in state space-the space whose "points" serve to represent the state of the system-, linear stability analysis governs the evolution of mathematically infinitesimal perturbations. It is the natural starting point of a perturbation approach to the nonlinear problem. When unstable, the base state is replaced by the primary bifurcated state, the stability of which is studied in the same way. The bifurcation cascade then proceeds through a secondary bifurcation, next tertiary, etc., up until chaos. The concept of globally supercritical scenario thus emerges 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 at a given value of the control parameter, and an hysteretic behavior as the control parameter is varied. In essence, this is non-local in phase space. Coexistence and discontinuity are the most relevant features of a globally sub-critical scenario. The situation is radically nonlinear from the start and information about the local structure of phase space of lesser relevance.
From a physical viewpoint, instabilities and the transition to turbulence occur in systems driven far from equilibrium. At equilibrium, a macroscopic system stays in a time-independent, spatially uniform state. Departures from that state regress spontaneously as an effect of microscopic fluctuations, resulting in dissipation. When driven out of equilibrium, the system may respond in some unexpected way, as a result of the competition between driving and restoring forces.
In fluid mechanics, the distance to equilibrium is measured by the Reynolds number which compare the effects of applied shear disturbing the fluid to those of viscous dissipation ironing out velocity inhomogeneities. The state directly stemming from equilibrium is referred to as the base flow. Let \(L\) be the length scale over which velocity variations of magnitude \(U\) are imposed, and let \(\nu\) be the kinematic viscosity. Viscous dissipation operates on a time scale \(\tau_v=L^2/\nu\) while the shear introduces its own time scale \(\tau_s=L/U\ .\) When the Reynolds number \( Re=\tau_v/\tau_s=U L/\nu\) is small, i.e. \(\tau_v\) much shorter than \(\tau_s\ ,\) viscous dissipation rapidly irons out velocity disturbances and the flow responds smoothly to the perturbation, laminar flow prevails. On the contrary, when \(Re\) is large, viscosity has no longer a sufficient time to damp fluctuations that may be amplified by the shear. The fluid becomes unstable and ultimately turbulent when driven sufficiently far from equilibrium. The structure of the Reynolds number, measuring the relative intensities/time-scales/length-scales of different processes, is typical of a control parameter. Here the transition to turbulence in simple flows is reviewed but the same approach applies to more general situations in systems experiencing the emergence of complexity.
A basic distinction has to be made between open and closed flows [4a,5]:
- Open flows are characterized by a global transfer of matter from upstream to downstream, with the consequence that the 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 (absolute instability), or
- are wiped out by the stream while developing (convective instability); observing sustained turbulence at a given place then depends on the amplitudes of perturbations, either controlled or due to residual noise; reducing their amplitudes delays the transition 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 physical quantities. Such instabilities usually introduce an intrinsic length scale in the flow, leading to the formation of dissipative structures [6,7]. The scenarios leading to turbulence depend on the relative width of the experimental cell compared to this length scale, its aspect ratio, which measures the strength of confinement effects.
Scenarios in open flows
The nature of the transition is, for a large part, controlled by the presence of walls possibly bounding the sheared region:
- 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 linked to the presence of an inflection point in the base flow profile [4,8]. Viscosity plays a stabilizing role (plain mechanical friction damping). This inertial instability is linear. It develops at relatively low values of \(Re\) and is generally the starting point of a globally supercritical scenario: The primary instability sets in as a series of vortices with axes perpendicular to the direction of flow (streamwise modulation). Beyond a second threshold, these rolls become unstable against spanwise modes which induce transversal inflection points. A third instability takes place when the corresponding shear is large enough and small scale turbulence follows soon after.
- The transition of a cylinder's wake illustrates the cascade of symmetry breaking associated to the different bifurcation steps of such a scenario: consider a cylinder of diameter \(D\) placed transversally in a uniform flow of speed \(U\ .\) At \(R=U D/\nu < 1 \ ,\) flow lines turn the obstacle while sticking to it; the flow pattern is upstream/downstream-symmetric. As \(Re\) is increased, this symmetry is first broken and a time-independent recirculation bubble appears downstreams. This flow configuration, made of two steady vortices parallel to the cylinder and symmetrically placed with respect to the plane of the wake, persists up to \(Re_c\simeq49\ .\) At \(Re_c\) the wake enters a regime of periodic parallel vortex shedding well described as a Hopf bifurcation. The two initially steady vortices are alternatively shed above and below the plane of the wake (von Karman vortex street). Secondary instabilities next develop when \(Re>180\ ,\) breaking spanwise translational invariance. A periodic series of streamwise vortices appear, introducing strong three-dimensionality in the flow. Two such modes (A,B) can occur, generated by different mechanisms with different wavelengths . This complicated flow structure then decay into smaller scales, understood as turbulent flow and better described by its statistical averages. However large scale coherent structures can still be detected even in the highly turbulent flow observed far downstream, as if the mean turbulent flow could be considered as an effective base flow experiencing the same global scenario.
- The presence of solid walls is essential to the dynamics of bounded flows [4,5,8]. 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 . The 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 perennial 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 . This linear lift-up mechanism is next closed by a nonlinear feedback that regenerates the vortices. The same cycle is expected to hold inside the turbulent spots which are long-lived domains filled with turbulent flow scattered amidst laminar flow universally observed in plane wall-bounded flows. The transition typically follows a globally sub-critical scenario observable in a wide enough range of Reynolds numbers between stable laminar flow and developed turbulence.
- In the flow along a cylindrical tube , for which \(Re\) is defined as \(U D/\nu\) with \(U\) the mean velocity of the base flow and \(D\) the diameter of the pipe, laminar flow can be maintain up to \(Re\sim10^4-10^5\) in particularly clean experiments but sustained turbulence can be observed down to \(Re\sim2,000\ .\) Below 2,000 localized chaotic structures of well-defined length called turbulent puffs are observed in the flow, analogue to the turbulent spots seen in plane wall-bounded flows. As \(Re\) is increased, they transform themselves into more turbulent structures called turbulent slugs expanding both downstream and upstream of the point where they appear, thus invading the whole pipe. This explains that without special precautions, turbulence is generally observed in pipes for \(Re>2800\) .
Scenarios in closed systems
Instability mechanisms at work in closed systems generate dissipative structures. Turbulence develops in these systems by progressive disordering of initially regular spatiotemporal patterns when they are driven farther from equilibrium.
- Some pattern forming 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 relevant control parameter is the Rayleigh number. The wavelength \(\lambda_c\) of the resulting convection rolls is about twice the cell's height \(h\)[6,7,12]. Close to the instability threshold, the pattern is made of time-independent straight rolls.
- Convection in binary mixtures (solute in a solvent) 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 diffusivity of the components .
- Taylor-Couette instability develops in a fluid sheared by two coaxial cylinders rotating at different speeds. This instability is due to the interplay of centrifugal and viscous forces . It produces axially periodic toroidal flow patterns called Taylor vortices. The width of the vortices is on the order of the gap between 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 .
- ... e.g. cellular automata, predator-prey systems and population dynamics
- Besides the nature of the instability mechanism, the most important feature of the transition is linked to confinement effects . 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 patterns are no longer well organized; spatiotemporal chaos takes place.
- Scenarios in confined systems: On theoretical grounds, 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 [2,3]. 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.
- 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 [2c,6]. A similar situation holds for other systems, like the Taylor-Couette system when the height of the cylinder is comparable a few gap widths, or the stirred BZ reaction. These scenarios have universal features of mathematical origin but each system has its own physical traits.
- 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. 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 [6,13,14]. 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 . The equation governing the long wavelength modulations of the envelope is generically called a Ginzburg-Landau equation (GLE) . It is a partial differential equation governing the space-time dependent amplitude accounting for slow space-time modulations brought to a uniform reference solution. Its specific form is dictated by the nature of the latter, the symmetries of the system (translational and rotational invariance, possibly additional Galilean invariance). It 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 \(\ne 0\)) .
- Envelope description of typical patterns:
- 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 \(\omega_c\ne0\ ,\) \(k_c=0\ .\) It is described by the standard CGLE, i.e. a GLE with complex coefficients.
- 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 modes). They often give a sufficiently good understanding of the transition to turbulence in such cases [13,15].
- Phase and defect turbulence: 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). It is obtained by a gradient expansion of the CGLE around uniform waves solutions. It is valid as long as the phase instability is weak enough, so that phase gradients stay small and the modulus of the amplitude remains bounded away from zero. Its solutions are stochastic and account for phase turbulence. In contrast, \(2\pi\)-phase defects nucleate at locations when the phase instability is too strong so that the gradient of the phase can diverge and the modulus of the amplitude can reach zero. In two-dimensions, these defects are topologically stable. They control the disorganization of the system which then 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 .
- 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. 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 . The latter is defined as a probabilistic automaton describing contamination such as epidemics or forest fires. The plane Couette flow, in this respect, behaves as a closed flow.
Relevance to control
Understanding the early transition steps in detail is important in view of efficient turbulence control. Knowing how, where, and when to modify the base flow to delay the transition when turbulence is harmful or to promote instability when better mixing is beneficial relies on the study of specific mechanisms at play in given flow configurations .
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