FRONT MOTION, METASTABILITY AND ... - ScienceDirect

Physica 23D (1986) 3-11 North-Holland, Amsterdam

FRONT MOTION, METASTABILITY AND SUBCRITICAL BIFURCATIONS IN HYDRODYNAMICS Y. POMEAU Service de Physique Thbrique,

Centre d’Etudes Nuclkzires de Saclay, 91191 Gif-sur- Yvette Cedex, France

Metastability occurs whenever, under given external conditions (as defined by the Reynolds number in hydrodynamics) more than one linearly stable solution may exist. In variational problems, the preferred state is usually claimed to be the one with the lowest energy. But in hydrodynamics, and except for very special conditions no energy functional exists. From an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations (as turbulent and laminar in pipe flows or boundary layers) moves with a constant mean velocity, depending on the control parameter. The sign of this velocity allows to decide which state is stable and which is metastable. Possible physical implications of these ideas for various problems of hydrodynamics are discussed.

1. Introduction The aim of this paper is to present some ideas related to the nonlinear development of subcritical instabilities in fluids. Owing to the rather broad spectrum of questions that will be examined, there is no claim of generality and even less of reviewing a sizeable part of the literature in the field. The emphasis will be mainly on some points that seem to be original, at least in the present form. Moreover, as this stems from a talk presented in front of a rather broad audience, many details pertinent to fluid mechanical problems have been omitted. We refer the interest reader to a forthcoming Monography [l] that will consider (among others) those questions in a far more detailed fashion. Most of the equations that will be written will be simply models and no attempt will be made to derive them in a rational fashion from the equations of fluid mechanics (if this were possible). Recently, a great deal of attention has been focused on the onset of supercritical instabilities [l]. It can be seen as the mechanical (= no thermal fluctuations) partner of second order phase transitions in thermodynamics. In particular the gross features of the transition are easily understood on the basis of the so called Landau theory

[2]: below the instability threshold only one steady stable solution of the equations of motion exists and a little above threshold this steady solution becomes linearly (but weakly) unstable, although another solution branches off from this one, which is linearly stable [3]. There are many examples of this in hydrodynamics: Rayleigh-Benard instability, rolls in Taylor-Couette flow, longitudinal fluctuations of the vortex shedding in the wake of cylinders [4] are all examples of supercritical instabilities. But this does not exhaust by far every possible flow behavior. For instance the laminar Poiseuille flow in a circular pipe is known to be linearly stable at any Reynolds number, although there is experimental evidence that for Reynolds numbers larger than a few thousands [5], the flow is always turbulent unless very special conditions are met at the entrance of the pipe. This leads one quite naturally to seek to understand in what sense this laminar flow at large Reynolds number is metastable, although the turbulent flow is “stable” under the same conditions. This work is devoted to an attempt to understand this point. We shall recall in the second section how metastability is usually introduced for systems with a finite number of degrees of freedom: one computes the energy of all possible steady and linearly

0167-2789/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

4

Y. Pomeau/

Front motion, metastability

stable solutions and defines as stable the one with the lowest energy, every other one being metastable. Then we shall extend this to a slightly more general case, i.e. the one described by reactiondiffusion equations. There the physical quantities depend on time and space variables and one may study the following problem: given the fact that one possible linearly stable solution fills a half space and another one the other half space, what will ultimately happen? It turns out that the answer to that question allows to introduce a new definition of metastability (vs stability) which can be extended in principle to rather general situations, in particular wherever the equations of motion, as the Navier-Stokes equations have no built-in variational structure. However, even in the framework of problems with a variational structure the dynamics of the interface separating two linearly stable states may be more complicated than expected from the analysis of reaction-diffusion equations. If one “phase” bears a regular modulation in space, locking phenomena between the interface and this structure may appear, and make the analytical description of the interface motion rather complicated, as explained in section 3. Finally, section 4 is devoted to the presentation of a possible connection between directed critical phenomena and the marginal growth of turbulent domains in a laminar flow.

2. Metastability

in reaction-diffusion

equations

In this section I review briefly some classical material related to ordinary differential equations defining a gradient flow and then partial differential equations usually denoted as “reaction-diffusion” equations, again with a variational structure. More details as well as relevant references may be found in the review article by Clavin and Linan [6]. Let us imagine that the full hydrodynamic equations for a given instability problem may be reduced to a set of nonlinear ODES relating the evolution of a finite set of time dependent ampli-

and subcritical bifurcations in hydrodynamics

tudes denoted collectively as A. We shall moreover assume that this dynamics can be put into a variational form, that implies that this set of ODES can be written as: A, = -dV/dA,

(1)

where the subscript t denotes the time derivative, although the r.h.s. of (1) is the derivative of a function of A, denoted as V, with respect to the component of A that is on the 1.h.s..(unnecessary proliferation of indices will be avoided as much as possible). In hydrodynamics, as said before there is little if no chance to get equations of motion in a variational form as in eq. (1). Nevertheless there are a few cases where this can be done in a rational fashion, besides the obvious case where there is a single real amplitude A. Later on this possibility will be used for a description of hexagonal patterns in the weak amplitude limit. On the other hand in bifurcation problems related to elasticity, as in the buckling of plates [7] an energy functional exists due to some fundamental physical principle. Finally it must be noticed that dynamics described by PDE may be reduced to ODE as eq. (1) either by “a la Landau” perturbative methods or by the powerful method of the reduction to inertial manifolds presented by B. Nichols at this conference. But no attempt will be made to calculate the potential V(.) for an explicit physical problem, and thus we shall restrict our considerations to some qualitative remarks based upon the shape of this function when A represents only one real function. Metastability appears typically when V(A) is a double well potential (fig. 1). The comparison of the energies of the two minima of V allows to decide which one is stable and which one is metastable: the local minimum with the lowest (/highest) value of V is dubbed the stable (/metastable) one. However this denomination is a little misleading in the framework of deterministic dynamics: the final evolution toward a local minimum of V depends on the initial condition only and cannot be decided by looking at the final

5

Y. Pomeau/ Front motion, metastability and subcritical bifurcations in hydro&namics VA

initial conditions such that A + A k as the space coordinate x tends to + co, A + being two local minima of the potential V. Then under fairly general conditions this will evolve toward a moving interface, that is to a function of space and time that is stationary in a moving frame. Let u be the velocity of this frame, that is also the one of the front. Thus, in this asymptotic state A depends on x and t through the combination c = x - ut and is the solution of the ODE:

W

I

L

A

Fig. 1. Double well potential V(A).

value of the potential. Thermal fluctuations change obviously this conclusion but we shall not take them into account. There is another way however for comparing metastable with stable steady states than by computing their energies: that is by watching their behavior when they coexist. Theoretically this can be described by adding to the r.h.s. of (1) a diffusion term. This leads to what is usually called a reaction-diffusion equation: A,= DAA - aV/aA,

(2)

where D is a (positive) diffusion coefficient, A is the usual Laplacian in the r space, and A(t, r) is a real scalar field. Equations like eq. (2) are called reaction-diffusion equations, because one has in mind that the last term on its r.h.s. models the relaxation toward equilibrium by the nonlinear equation of the chemical kinetics, although the diffusion term represents the molecular diffusion of chemicals due to an eventual lack of homogeneity of their distribution in space. Now we shall use a formal analogy between reaction-diffusion equations as (2) and amplitude equations derived by the method of Segel and of Newell and Whitehead for thermoconvection near the onset of instability [8]. Later on we shall give a more concrete example of this connection, in the case of weakly nonlinear hexagonal patterns. For the moment we shall merely use the formalism of reaction-diffusion equations. The comparison between “stable” and “metastable” states, i.e. between different minima of the potential V can be thought in the following fashion. Let us look for solutions of (2) with smooth

-uA,=

DA,,-

aV/dA.

(3)

As often noticed this is the equation of the motion for a particle of mass D in a potential (- V) and submitted to viscous friction of coefficient U. The boundary condition for (3) is A + A + as 5 + & 00. As shown in fig. 2, the search of a solution of (3) is equivalent to the one of the value of the damping u such that a point starting with zero velocity from A _ will ultimately reach the other extremum of V, i.e. - V( A +) and stop there. The uniqueness of the solution follows from elementary considerations: given a damping u that solves the problem a larger damping will not allow to reach A+, although a smaller damping will not allow to stop on A +. Let us now quote a few more properties of this moving front solution: i) It is robust, in a loose sense. Which means that replacing (3) by a slightly changed equation one still gets an equation that has the same property of having moving front solutions. How-v A‘

m +b

I

w

A

Fig. 2. The research of the velocity u in eq. (3) amounts to find the unique viscous damping such that a particle of mass D starting at rest from the highest maximum of (- V) rolls down, moves up and finally stops on the other extremum of (- V).

6

Y. Pomeau/

Fronr motion, metastabiliiy


U

Fig. 3. Deformation of V(.) when an extremum disappears. On top is represented a generic case without symmetry, and at bottom an even V(.). singularity of u(.) near the inflection point of V(., g).

ever, this perturbation cannot be singular, that means that it has to remain small everywhere in the complex extension of A. As already noticed [9] those singular perturbations are responsible of new physical phenomena that cannot be analysed by usual perturbative methods. Typically those singular perturbations include higher derivatives and are thus often present in problems leading to amplitude equations, because these ones are basically adiabatic theories. ii) All the above analysis could be repeated for reaction-diffusion equations for more than one scalar field. The main difference being that there is no more simple proof of unicity of the front velocity U. iii) The one dimensional front given by the solution of (3) is linearly stable, including cases with more than one spatial dimension. iv) Let us consider now the behavior of u as a function of a control parameter g describing the deformations of the potential V. One interesting problem in this framework is to look at the marginal case when one of the equilibria A + disappears, as shown in fig. 3. Possible application of this could be the propagation of turbulent flashes near the onset of linear stability in parallel flows or in BCnard-Marangoni thermoconvection. Thus let g, be the value of the control parameter for which one of the competing states goes from stable to unstable or disappears. Bifurcation theory tells us

\/” ti i+ Fig. 4. Generic form of u(g) as the potential V( A, g) changes. Note the square root singularity of u at the limit points when two singular points of the flow collapse.

that under “generic” conditions the crossing of g, by g changes the structure of the phase space in a region of size of order ]g - gC]1/2. On the other hand a numerical solution of (3) may be found by a shooting method and the above remark implies that the goal to be reached by this shooting moves of an amplitude of order (g - g,(“’ near g,, so that the function u(g) has generically a square root singularity near g,. For g exactly equals the critical value the front can move with any velocity larger than the limit value of u(g) when the corresponding equilibrium of V disappears beyond the critical point (see fig. 4).

3. Locking phenomena in the dynamics of fronts As said before the motion of fronts separating two linearly stable solutions of the equations of motion can be analysed in some cases by amplitude equations that look quite similar to the reaction-diffusion equations presented in the previous section. Those equations can be derived in a systematic manner from the original fluid equations in the limit where linear theory predicts slow

Y. Pomeau/ Front motion, metastability and subcritical bifurcations in hydrodynamics

growth of the fluctuations, and where the nonlinearities always remain weak. To make our considerations more concrete let us introduce the amplitude equations for an hexagonal pattern, as the one produced by Benard-Marangoni convection. These patterns are made by nonlinear superposition of three sets of straight rolls at directions at angle of 2a/3 from each other. Let A,(r, t) (i = 1,2,3) be the complex amplitude of rolls in each of the three possible directions, r being the position in the two dimensional planform. At the dominant order in the weak instability-small amplitude expansion, the corresponding amplitude equation reads: &

= CAi + (ei -g2(lAi+112

l

VJ2Ai

-

(AfAi

+ lAi+212)AiY

-

g,AT+;,A:+;, (4)

where the addition in the index set is understood mod. 3, and where the other symbols have the following definitions: z is the small rate of growth (or decay) of the unstable fluctuations in the linear approximation, ei is the unit vector perpendicular to the rolls of index i, g,,, are real interaction coefficients and the star denotes the complex conjugation. This amplitude equation is consistent if the smallness parameter c can be formally eliminated by convenient resealing, which implies that g, is of order cl/*, as we shall assume it. For e small positive, the uniform steady solutions of (4) are: (1) Ai = 0 for any i. This represents the undisturbed state. (2) A. = cl/* for one i, the other A’s being 0. This solution corresponds to parallel rolls. (3) Ai = u, where a is a real root of the algebraic equation: 6-a’

- g,a - 2g2a2 = 0.

(9

This last solution corresponds to a hexagonal pattern, and it is subcritical [lo] if (1 + 2g,) is positive (as we shall assume it), because it exists for E L -g:(l + 2g,)-‘, that includes negative values of E. The bifurcation diagram is sketched

A

t /Hexagons

* E

Fig. 5. Bifurcation diagram for r independent solutions of eq. (4).

steady state

on fig. 5. As eq. (4) has a variational structure, one applies to it the general method explained previously for reaction-diffusion equations. The value of the potential for the undisturbed state is 0, it is ~*/4 for rolls and (E - ugi)( ag,(3 - 2g,) + c(2g2 - 1))/(4 + 8g,) for hexagons, a being still the root of (5). As noticed by Palm [lo] this implies that hexagons represent the state of lowest energy in the range pi -Ce < e2. For c greater than e2 rolls have a lower energy although for c less than pi the undisturbed state has the lowest energy. Accordingly it is relevant to try to compute the velocity of the hexagons vs. the undisturbed state in the range -gf/(l + 2g,) -CE -C0 and vs rolls in the range 0 < z < + cc. This velocity will vanish for c = zi and E*, where the hexagons can stay in equilibrium with the undisturbed state and the rolls respectively. We shall not go on with this problem of computing this front velocity. It can be transformed into the one sketched before for a single amplitude A but for the fact that one has in the present case three coupled ODES to solve instead of the single equation (2) written previously. Moreover the front velocity depends on the orientation of the front with respect to the symmetry axis of the hexagonal pattern. Below I consider a question that does not appear in the framework of this amplitude equation approach, viz. the pinning of the interface on the microscopic periodic structure of the hexagons. This is an example of a general class of problems

8


that appear whenever a structured pattern grows. Indeed this has been studied already for understanding how a crystal grows at the expanse of its melt. Besides the fact that it is difficult to have direct experimental access to a moving solidification front (see however ref. [ll]), thermal fluctuations are believed to play an important role in crystal growth [12]. And, as said before we are concerned here with “mechanical” phenomena (= without thermal fluctuations). It was known from studies on crystal growth [12] that, in the absence of thermal fluctuations, a disequilibrium of finite amplitude must exist between the two phases to allow the growth. This is because the interaction between the interface and the periodic crystal structure gives rise to a periodic potential and because the disequilibrium between the two phases amounts to add to this potential another potential proportional to the distance, the constant of proportionality being the change of total energy brought by displacing the interface of one unit length. As shown in fig. 6 the addition of this disequilibrium amounts to tilt the periodic potential. Accordingly a finite amount of desequilibrium between the two phases is needed to make the interface moving: this is also represented by the finite tilting needed to get rid of local minima in the potential. Thus there is a finite range of values of the control parameter around which the velocity of the interface should stay equal to zero. Indeed this contrasts a lot with reaction-diffusion equations, where the front velocity vanishes for isolated values of the control parameter only. But that does not mean that the interface can stay unmoving over a whole range of values of the control parameter. Actually, for problems with a variational structure, as for instance the one posed by the buckling of plates [7], the equations for the steady state have an Euler-Lagrange form, and one can derive from them a Noether invariant, that is a function of the local field quantities independent on the position along some space dimension. It is worth noticing too that such invariants may exist even in the absence of the Euler-Lagrange structure [14]. And this invariant

Locking potential t

t-

Position of interface

Fig. 6. Potential due to the interaction between the fast spatial modulation and the interface position. An unequal energy density on the two sides of the interface amounts to tilt this potential.

would imply that the equilibrium between two different “phases” depends on the equality of its value computed on both sides of a plane interface. In other terms this equilibrium would be realized over a set of codimension 1 in the parameter space, in apparent contradiction with the previously mentioned locking of the interface over a finite domain of values of a control parameter. Actually we have at our disposal another parameter, that is the wavelength of the pattern. This leads one to the following “scenario” for the front behavior in the locking regime. If the wavelength of the pattern (that could be an hexagonal planform) is such that the Noether invariant has not the same value on both sides of the front, then the evolution toward equilibrium would take place by a change of the wavelength (either by dilatation or by contraction) near the interface. Then this change would diffuse inside the pattern by phase diffusion [14], leading to a displacement of the front with the square root of time only (compared to a displacement growing linearly with time in the amplitude equation limit). Indeed one should not expect any continuity between this diffusive behavior and the motion at constant velocity that should take place outside the pinning range. In particular it seems unlikely that this velocity can vanish exactly. If it were so, one would have at some period of the front motion the possibility of growing by phase diffusion an interface with a wavelength exactly in equi-


Fig. 7. Dependence of the front velocity on control parameter when there is pinning. The thick segment at u = 0 near g = g, represents the locking range, although the curve u(g) does not reach continuously the zero velocity range. The precise value of g, depends on the wavelength of the structure.

librium with the other phase. That would lower the local tilting of the potential and thus stop the motion forever. A sketch of the dependence of u as a function of a control parameter is shown on fig. 7, where the effects of pinning are included. To end up, let us mention that, as usual those locking phenomena are out of reach of ordinary perturbation expansions. As often noted the sort of adiabatic expansion leading to (4) makes appears a spurious phase invariance: nothing changes when one moves the fast phase with respect to the slowly varying amplitude, although the physical picture changes qualitatively. An obviously related property is that the higher order terms in the amplitude approximation are not uniformly small in the complex extension of solutions of this amplitude equations, because they involve higher derivatives. As already mentioned [9] this leads to the fact that those nonadiabatic phenomena are of a transcendentally small order in the small parameter. That should be true for the width of the locking region in the parameter space for instance.

4. Marginal propagation of turbulent regions In this section I present some ideas about the behavior of fronts separating turbulent regions

9

from laminar ones, with some emphasis upon phenomena occurring at the onset of propagation, i.e. when the front velocity becomes very small. As far as I know there is no detailed study of such an experimental situation. Perhaps, an example of it could be found in the transition to turbulence in the wake of a cylinder. As reported by Roshko [15] this transition occurs by bursts, so that it is well possible that the considerations presented below could apply to this. This could be checked, at least in principle by comparing the predicted “ universal” critical exponents of directed percolation with experimental data, as explained below. The difficulty in trying to understand the “microscopic” behavior of a front separating a turbulent region from a laminar one is in the randomness of turbulent fluctuations. Thus it does seem reasonable to focus on this randomness while skipping other features. To have a more or less concrete model in mind one might think first of an oscillation coherent parallel to one spatial direction as the vortex shedding in the wake of a cylinder. Let us imagine that this coherence is lost by a phenomenon similar to the intermittent transition of a single oscillation [16]. Intermittency coherent parallel to the cylinder is not possible. This is because synchronization of a turbulent behavior would imply that the interaction of neighbouring turbulent oscillators is strong enough to forbid exponential divergence of their trajectories. But this interaction becomes weak at large distances and thus cannot make vanish strictly positive Lyapunov exponent. This is quite different from the synchronization of regular oscillations where the interaction makes (eventually) negative vanishing Lyapunov exponent corresponding to phases of periodic oscillations. Thus one may think to a turbulent state similar to the one induced by the intermittent transition in oscillations. The interaction between neighbouring oscillators may thus be seen as a sort of contamination process: by this interaction the local state of an oscillator may jump from regular to intermittent, the interaction being equivalent to a change of the control parameter from its value in

10

Y. Pomeau/


the stable to the intermittent range. But the comparison with intermittency leads also to consider that in the absence of triggering by a turbulent neighbour, a local oscillator has a finite life time in its turbulent state, if it is below the threshold of spontaneous firing, as assumed. If one imagines a string of such oscillators, one is led to the following picture: each oscillator if in a turbulent state may either relax spontaneously toward its quiescent state or contaminate its neighbours (if they are already turbulent this interaction changes nothing). This is precisely the definition of the process called “directed percolation” in statistical physics [17]. And it is known that there is a well definite onset of percolation as a function of the ratio of two probabilities (in the context that would be the probability of going back to regular oscillations before or after contaminating the neighbours). This number by itself is not very interesting for us because it depends on the details of the lattice geometry or whether second neighbours are contaminated directly or not etc. But the relevance of this to concrete experimental situations should follow from the fact that relevant critical exponents at the onset of directed percolation depend on very general features only as the space dimension for instance. Let g, be again the value of the control parameter for which the velocity of the front separating turbulent from laminar domains vanishes. Then the front velocity should vanish near this value of the control parameter as lg - g,l*, where (Y( > 0) is an universal critical exponent (one dimension of the directed percolation is the time, so that one should compare the results for the wake of the cylinder with directed percolation in a 2d lattice, the velocity being measured spanwise). As each oscillator is by itself linearly stable, subcritical turbulence would depend - in this framework - on random triggering by external perturbations (that would represent the root in the percolation picture). To avoid that one could imagine to have a localized and permanent source of turbulence in the flow (this could be due to a local deformation of the cylinder). Below the onset of extended turbulence

and subcritical bifurcarionr in hydrodynamics

that permanent source of fluctuations would trigger turbulence over a domain of spanwise diverging extent near the onset, following an universal power law. Other possibilities are certainly conceivable too.

5. Conclusion

Although a lot of attention has been devoted to supercritical instabilities, it seems worthwhile to study in some detail too the subcritical instabilities, which are so frequent in real flows. In that field there is perhaps a need to develop new concepts and ideas, as attempted in this work.

Acknowledgements This work was done mostly at the GFD program at Woods-Hole (USA) during Summer 1985 where the author was supported partially by the program. The author acknowledges also support from the Los Alamos National Lab (USA) where this work was completed in Winter 1985/1986. Discussions with the GFD staff have been helpful as well as fruitful exchanges with Louis Boyer, Paul Clavin and Pierre Coullet.

References PI Y. Pomeau,

E.A. Spiegel and S. Zaieski, Nonlinear Phenomena, to appear. 121L. Landau and E.M. Lifshitz; $27 in Fluid Mechanics (Pergamon, Oxford, 1959). [31 E.A. Spiegel and S. Zaleski, to appear in Ann. Rev. of Fluid Mech. [41 C. Mathis, M. Provansal and L. Boyer, J. de Phys. Lettres 45 (1984) L483. and F.H. Champagne, J. Fluid Mech. 59 151 I.J. Wygnanski (1973) 281. Cooperative [61 P. Clavin and A. Linan in Nonequilibrium Phenomena in Physics and Related Fields, M.G. Velarde, ed., NATO ASI series, vol. 116. [71 Y. Pomeau, J. de Phys. Lettres 42 (1981) Ll.

Y. Pomeau/


[8] L.A. Segel, J. Fluid Mech. 38 (1969) 203;

[9]

[lo] [ll] [12]

A.C. Newell and J. Whitehead, J. Fluid Mech. 38 (1969) 79. Y. Pomeau in Cellular Structures and Instabilities, J.E. Wesfreid and S. Zaleski, eds., Lecture Notes in Physics 210 (Springer, Berlin, New York, 1984). E. Palm, J. Fluid Mech. 8 (1960) 183. P. Boni, J.H. Biigram and W. Kanzig, Phys. Rev. A28 (1983) 2953. J. Woodruff, Crystal Growth (North-Holland, Amster-


11

dam, 1976). [13] Y. Pomeau, CRAS Paris 293 (1981) 241. [14] P. Manneville and Y. Pomeau, J. de Phys. Lettres 40 (1979) L609. [15] A. Roshko, Naca Tech. Note no. 2913 (1957); M. Susan Bloor, J. Fluid Mech. 19 (1963) 290. [16] P. Manneville and Y. Pomeau, Comm. in Math. Phys. 74 (1980) 189. [17] S.P. Obukhov, Physica 1OlA (1980) 145.