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Temporal Forcing of Traveling Wave Patterns
Joceline Lega, Jean-Marc Vince
To cite this version:
Joceline Lega, Jean-Marc Vince. Temporal Forcing of Traveling Wave Patterns. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1417-1434. �10.1051/jp1:1996155�. �jpa-00247255�
Temporal Forcing of Traveling Wave Patterns
Joceline Lega (~,*) and Jean-Marc Vince (~)
(~ Institut Non Linéaire de Nice (*), 1361 route des Lucioles, 06560 Valbonne, France (~) CEA, Service de Physique de l'État Condensé, Centre d'Études de Saclay,
91191 Gif-sur-Yvette Cedex, France
(Received 11 January 1996, revised 31 May 1996, accepted 29 July 1996)
PACS.47.20.-k Hydrodynamic stability
PACS.47.27.-i Turbulent flows, convection, and heat transfer PACS.47.54.+r Pattern selection
Abstract. We experimentally and numerically study one-dimensional, temporally forced
wave patterns and analyze trie transition from unforced traveling waves to forced standing waves
when a source defect is present in the system. Dur control pararneter is the amplitude of the
forcing. Two scenarios are identified, depending on the distance from the bifurcation threshold of traveling waves.
1. Introduction
Traveling wave patterns are commonly observed in nonlinear systems driven far from equilib-
rium il,2]. But for a few exceptions [3-5], standing waves are however rarely stable, unless
some external forcing is applied to the system. This forcing takes in general the form of a
spatial [6,7] or temporal [8-15] modulation of a control parameter. In both situations, trie mechanism is fairly simple: by resonating with one of trie waves, trie forcing drives trie wave
propagating m trie opposite direction and therefore stabilizes standing waves.
This paper is an account of trie experimental observation and analysis of forced traveling-
wave patterns in a one-dimensional system consisting of a bot wire immersed in a bath of silicon off. When the temperature of the wire gets above some critical threshold, a source
of traveling waves appears in the system. In one dimension, a source defect is a point from
which two waves propagating in opposite directions emanate, the group velocity of each wave
being such that information is transported away from the core of the defect. The transition towards standing waves (SW) is then studied, in the presence of the defect, as a function of the amplitude of the external forcing. We identify two distinct regimes, depending on the
distance from the threshold of appearance of traveling waves in the system. Dur results are
confirmed and supported by a theoretical analysis of forced standing waves, performed in the framework of amplitude equations. The paper is organized as follows. We first describe
(Sect. 2) the experimental setup, vanous control parameters and measuring techniques. We then tutu (Sect. 3) to the description of the growth of standinj wave patterns in the presence of
(*) Author for correspondence (e-mail: lega@doublon.unice,fr) (*) UMR 129 CNRS UNSA
@ Les Éditions de Physique 1996
Parallel hght beam
1
~~~
Silicon off
Chromel wire
Glass
Thermostated brass ~
Mi«or ~""~"'' '~"'~'~Î
CCD Camera
~
Ground Glass Screen
Fig. l. Experimental setup (schematic).
temporal forcing. Section 4 deals with the theoretical approach of this problem, and describes SW and "false traveling wave (TW)" patterns, whose stability is then analyzed. Finally.
Section 5 summarizes our results and comments
on the role played by the non-zero group velocity of each wave. Links with absolute and convective [16,17] instabilities of the zero state
are discussed.
2. Experimental Setup
One-dimensional traveling wave patterns
can be produced by a long hot wire located under- neath and parallel to the free surface of a hquid [18-20]. These waves take the form of a spatio-temporal modulation of the free surface above the hot wire, and also involve thermal variations in the fluid layer. Dur experimental system, fully detailed in [19], consists of a hori- zontal rectangular container 60 cm long, 1 cm wide, 2 cm high with a 60-cm-long chromel wire stretched in its center (see Fig. l). The sides of the cell are made of brass and kept at
constant temperature. The bottom is made of glass, and the top is a plexiglas plate which
restrains the evaporation of the fluid. The cell is filled with silicon oil up to a level h above the wire, leaving some space between the free surface of the liquid and the plexiglas cover of
the container.
The temperature of the chromel wire is controlled by a stabilized continuous voltage. The electric power Q supphed to the wire is measured through a GPIB IEEE interface and mon- itored on a computer. Surface deformations and teInperature gradients m the fluid layer are
visualized by shadowgraphy. The image (Fig. 2a) is observed on a ground glass screen, recorded
by a CCD camera and digitized on the computer. Data acquisition is generally performed along
a single 512-pixel hne parallel to the wire. By adjusting the samplmg rate, one can produce spatio-temporal diagrams, an example of which is given m Figure 2b. The diagram shown
is obtained when the electric power Q is greater than a threshold Qc (whose value depends
fk).( ''~'r".$~
iii h >'~
.,~~';~ ~,
'. .' ~" ~7 ~
' Mi ] ~ '~ ç'~~' ',j ' ) ~"~« ~Î
~ ,,<(
, ,
~.ii'~>,
a)
b)
Fig. 2. Experimental image (a) of the traveling waves formed above the hot wire, and corresponding (b) spatio-temporal diagram. Time goes downward.
on h) [19], above which traveling wave patterns are observed. These waves always emanate
from a source defect, which can be located anywhere in the cell. Since the amplitudes of the
left- and right-gomg waves are equal at the core of such a defect, we expect standing waves to
start growing from this locus when the forcing is turned on. Note that this diflers from the
2d experiments of [loi, where SW were forced from an initial state showing no defect. Here, the presence of the source will ease the transition towards SW, and therefore limit hystereti- cal behaviors. On spatio-temporal diagrams, standing waves correspond to a checked domain, whereas traveling waves, or rather, as we will see later, false traveling waves, give rise to slanted rolls. In order to stabilize standing waves, we need to add an altemating component to the continuous voltage. The voltage across the wire is then given by Uw = Uo + Uasm(uJt), and the intensity m the wire by Iw
= Io + Ia sm(uJt). As a consequence, the electric power Q reads Q = Uwlw
= Uolo + (Uola + Ualo sin(uJt) + Uala sm~(uJt).
Since the values of Ua and Ia we use are of the order of one tenth of Uo and Io, we can write
Q = Qo + Qasin(uJt),
where Qo
" Uolo and Qa
= Uola + Lfalo. The value of uJ is chosen very close to 2uJo, where uJo is the frequency of the traveling waves at threshold (in the absence of forcing).
The experimental procedure is always as follows: we first obtain a source of traveling waves by supplying a constant power Qo > Q~ to the wire. This defines a control parameter e =
(Qo Q~)/Q~, which measures trie distance from the TW threshold. Keeping e fixed, we then
slowly mcrease trie value of Qa, starting froIn Qa " 0, and always satisfying Qa < Qo Qc in order to remain above threshold. Trie parameter /t = Qa/Qc is trie actual control parameter
of trie experiment. After having reached values of /t high enough to establish SW, we decrease /t back to zero, in order to analyze trie reversibility of trie transition.
3. Experimental ILesults
3.1. LOW VALUES OF
e. We first set trie value of
e to e = 0.10. When ~t is increased from zero, standing waves do not appear immediately. In fact, although SW start growing from trie
core of trie source, trie extent of trie latter depends on trie value of /t, as can be seen on trie
spatio-temporal diagrams of Figures 3a-e. We call standing wave a state where trie amplitudes of trie left- and right-going waves are equal. Each amplitude is obtained by demodulating trie
pattern recorded on trie computer. We then define trie size of the source as the width of the region where the two amplitudes are within a few percents of each other. This value is plotted
as a function of /t in Figure 4. If we keep increasing /t slowly, the core of the source eventually extends to the whole cell (Fig. 3e). For e
= 0.10, this first happens at /t
= 0.088 (highest value of /t in Fig. 4).
If starting from a state where SW bave invaded trie whole cell, we now slowly decrease trie value of /t, a source re-appears in trie system (Fig. 3f), and its core gradually shrinks as /t is
reduced. At /t
= 0, we are left with a source of traveling waves of trie same size
as trie source
we started with. This process is reversible in trie sense that trie source is present in trie system when /t is both increased or decreased. However, a slight hysteresis can be observed in trie
behavior of trie size of trie defect (Fig. 4).
In order to analyze trie role played by trie value of trie detuning between
uJ and 2uJo, we bave repeated trie above experiment for diflerent values of
uJ, centered about trie value uJo
= 1.128 Hz.
It tutus out that almost no SW are observed if trie detuning is more than a few percenis of
uJo, and that trie extent of trie SW domain is of course maximum for an uJ close to 2uJo.
3.2. HIGH VALUES OF
e. We now perform the same type of experiment with e
= 0.25. For small values of /t, the observed pattern is very similar to what we had for e
= 0.10. In other words, we mainly see a source of (false) traveling waves. For larger /t's, the emitted waves
start showing wavelength modulations, as can be seen in Figure 5. These modulations do not
seem periodic and can take place anywhere in trie cell. When /t reaches trie value 0.21, SW
suddenly appear throughout trie cell (for a spatio-temporal diagram of SW, see for instance
Fig. 3e). This diflers from trie previous scenario in which we had a progressive increase of the size of the SW region, and therefore a smooth transition towards SW. Besides, the value of /t at which SW appear is strongly dependent on e.
When decreasing the value of /t, the system does not go through the same steps as when /t was increased. Standing waves persist for quite
a large range of /t, and then mixed states, involving both SW and false TW, are observed (Fig. 6). The smaller the /t, the larger the area occupied by TW. As a consequence, sourçes and sinks are created in the system. Finally, when /t is equal to zero, these defects annihilate by pair or vanish at one edge, and we generally
observe a single traveling wave: the source we started with has disappeared.
To summarize, the transition from TW to forced SW strongly depends on the distance from threshold, and two scenarios can be identified. Trie first one is "qualitatively" reversible and shows a smooth transition towards SW characterized by a progressive increase of trie size of
trie source with trie amplitude of trie forcing. The second one exhibits a sudden appearance of
SW all over the cell, and is no longer reversible: mixed states of SW and false TW are created when /t is decreased back to zero, and a single TW is eventually stabilized at /t = 0. We will see that these two scenanos
are captured by the amplitude equations describing forced traveling
wave patterns.
a) b)
' , .<~
l t~ l''
t r (~ "
4 j
>, l
~.
'
/~ ~.,
é
~ i
?
.~« ' l )
-. ~
~
' '
. , ~
~ ~
j
.- ~ ,4
~ #
', i
_
~
' ''
~ i
~, ~
j ~
' '
.
' 7
' ? '
f z'
Fig. 3. (a-e) Spatio-temporal diagrarns showing the behavior of the wave pattern during trie
mcrease of /t, for
e = 0.10. (a) /t
= 0, b) ~ = 0.029, c) /t = 0.058, d) p = o.073, e) /t = o.088). (f) Spatio-temporal diagram obtained when /t is decreased, at /t = o.073.
soc
A increasing~
~
~~~
o decreasing ~
@ o
~ 300
fi~
~
8 200 °
1 ~
o o ~
ioo °
o
0 0.02 0.04 0.06 o.08 0=1
it
Fig. 4. Size of the SW demain
as a function of /t, for e
= 0.lo.
Fig. 5. Spatio-temporal diagrarn obtained during the increase of /t (here, ~t
= o.19), for
e = o.25.
3.3. DIscussIoN. We believe that the discrimination between the two
scenarios is related to an instability of the source. As we will see later, the forced defect tends to select a wavenumber for which the temporal frequency of each traveling wave is equal to the threshold frequency.
This corresponds to a locking of the system onto the external forcing, and is also a required condition for forced standing waves to exist. If the selected wavenumber remains stable when /t is increased, the source persists till SW have invaded the whole box. If not, the source is
destroyed and a single false TW is stabilized. When /t is further increased, the stable false TW is progressively transformed into a standing wave pattern, which then appears at once
over the cell.
Fig. 6. Spatio-temporal diagram showing the mixed states observed when ~t is decreased. Here,
~t = 0.15.
These two scenarios are detailed in the following sections, where we consider the amplitude equations descnbing forced traveling wave patterns. Previous theoretical studies [8, 9] have focussed on the transition towards SW either when e is mcreased, keeping /t fixed, or when trie
detumng is varied, keeping /t and e constant. These results were obtained for wavenumbers
equal to the wavenumber at threshold, and it was shown that, depending on the sign of the detumng, the transition could be sub- or super-critical. Here, we know that the presence of
SW at the core of the source will suppress any localized eflect due to a subcritical bifurcation
(assuming /t is of course increased slowly enough), and that a proper tuning is crucial in order to observe standing waves. We will therefore restrict our analysis to zero detuning. Besides, the existence of the source forces us to consider traveling waves of wavenumber diflerent from the wavenumber at threshold. It is indeed wellknown that the source selects a nontrivial
wavenumber, whose value depends on the properties of the system under consideration. In
terms of amplitude equations, this amounts to including spatial dependence of the amplitudes
of the left- and right-going asymptotic waves. To simplify our analysis, we only consider initial conditions showing a source defect, in agreement with the experimental situation.
4. Theoretical Approach
In the vicinity of their bifurcation threshold, one-dimensional wave pattems can be descnbed by the superposition of two counter-propagating waves, whose amplitudes may vary slowly in
time. Any relevant physical quantity is then written as
U(t) = Uo(t) + -4 exp[1(-kox + uJot)] + A exp[-1(-kox + uJot)]
+ B exp[1(koz + uJot)] + É exp[-i(kox + uJot)] +..
,
(l)
where uJo and ko are respectively trie frequency and wavenumber at threshold, and trie dots stand for higher order corrections. Assuming some symmetry properties of trie system, trie amplitudes A and B obey trie following scaled amplitude equations [3,21-23]
~~
= eA + ii + iaj c~ (1 + iàjjAj2A
(~T + iôjjBj~A
~
= eB + Il + io)
(
+
ci
il+ ifl))B)~B
(~y + iô))A)~B,
where e measures the distance from the bifurcation threshold, c is the group velocity of each
wave, o corresponds to dispersion, fl to nonlinear renormalization of the temporal frequency,
and ~y + iô describes the coupling between the two counter-propagating waves. Standard sta- bility analysis shows that TW (for which either A or B is zero) are preferred to SW iiAi = )B)
when ~y > 1. We will then assume that
~y > 1, 1-e- that TW are stable in the absence of
temporal forcing.
If we now apply a multiplicative ii-e. of the form U(t fit temporal forcing of frequency ~J to this wave-forming system, it is easy to see that the coupling of the forcing with the right-going
wave will give rise to components of the form -kox + (uJo + uJ)t and kox + (-uJo +uJ)t, whereas the left-going wave will induce modes of the form kox + (uJo + uJ)t and -kox + (-uJo + uJ)t.
As a consequence, if
uJ ci 2uJo, the right-going wave will drive the left-going wave, and vice-
versa. This resonance will then stabilize standing waves [8,9]. Trie corresponding amplitude
equations [8,9] are
~
= je + iv)A + Il +
ia)j ci
il + ifl))A)~A (~y + iô))B)~A + ~tÉ (2)
)
= je + iv)B + Il + ia) +
ci
il+ ifl))B)~B
(~y + iô))A)~B + ~tÀ, (3)
where u measures trie detuning between uJ and 2uJo and ~t is trie amplitude of trie forcing. Given trie experimental fact that trie detuning must be less than a few percents in order to observe
standing waves, we will consider u
= 0 in trie following. Note that it was shown in previous works [8,9] that u = 0 corresponds to a change in trie nature (supercritical or subcritical) of trie bifurcation from traveling to standing waves when e is increased at fixed /t. As mentioned
before, trie presence of trie source acting as a seed of standing waves will hinder subcritical eflects, and trie sign of u is therefore not relevant for this particular study. We nevertheless
numencally checked that very small (compared to e) values of
u do not modify trie scenarios
we are describing.
4.1. STANDING WAVES AND FALSE TRAVELING WAVES. A quick look at equations (2,
3) show that TW'S ii-e- A
= 0 or B
= 0) are not possible solutions when /t # 0. Besides, because of the forcing, the phases of A and B will be coupled. Une can then look for solutions
in the form A
= Rexp[1(kx + Qt)] and B
= Qexp[-1(kx + Ht + qJ)], where R and Q are
real amplitudes, and qJ is the relative phase of A and B (the equations are still invanant by
trie transformation A
- Aexp(i~fi), B
- Bexp(-i~fi), so only trie relative phase matters).
Plugging these expressions into (2) and (3), one gets
je k~)R R~ ~yQ~R + /tQ cos(qJ)
= 0 (4)
in + ck + ak~)R + flR~ + ôQ~R /tQsin(qJ)
= 0 (5
je k~)Q Q~ ~yR~Q + /tR cos(qJ) = 0 (6)
(-Q + ck + ak~)Q + flQ~ + ôR~Q /tR sin(qJ)
= 0. (7)
We then obtain two types of solutions, namely standing waves, for which both amplitudes
are equal, and false traveling waves which correspond to left- and right-going components of diflerent amplitudes.
