Download Experimental Study of the Faraday Instability: Parametric Excitation of Surface Waves and more Schemes and Mind Maps Dynamics in PDF only on Docsity!
J. Fluid Mech. (1990), vol. 221, pp. 383409 Printed in &eat Britain
Experimental study of the Faraday instability
By S. DOUADY
Laboratoire de Physique de 1’Ecole Normale Superieure de Lyon, 46 Allee d’Italie, 69364 Lyon Cedex 07, France
(Received 1 June 1988 and in revised form 16 May 1990)
An experimental study of surface waves parametrically excited by vertical vibrations is presented. The shape of the eigenmodes in a closed vessel, and the importance of the free-surface boundary conditions, are discussed. Stability boundaries, wave amplitude, and perturbation characteristic time of decay are measured and found to be in agreement with an amplitude equation derived by symmetry. The measurement of the amplitude equation coefficients explains why the observed transition is always supercritical, and shows the effect of the edge constraint on the dissipation and eigen frequency of the various modes. The fluid surface tension is obtained from the dispersion relation measurement. Several visualization methods in large-aspect-ratio cells are presented.
1. Introduction
Several simple hydrodynamic instabilities have been intensively studied in the past ten years in order to find universal basic mechanisms of transition from a perfectly ordered system to temporal and spatial disorder. We study here an instability first reported by Faraday in 1831 : under vertical periodic motion, the free surface of a fluid layer is unstable to standing surface waves. Faraday noticed that the waves oscillate at half the excitation frequency, but Mathiessen, in 1868, claimed that they oscillate at the excitation frequency. This led Rayleigh (1883) to do his own experiment. In two different ways, he found the results to be in accordance with Faraday’s statement. Benjamin & Ursell (1954) developed the linear theory of this instability, and found that each eigenmode of the free surface is parametrically excited. Experimentally, we found that this parametric excitation is a simple way to study the surface wave modes of a closed vessel, as one measures the eigen frequency and the dissipation from the marginal stability curves. But Benjamin & Ursell could not describe the excited modes as they used the usual inviscid boundary condition for the free surface, which is unrealistic for these experiments where the meniscus dynamic is important. To avoid the presence of 4 meniscus, we pin the free surface at the edge of the lateral walls. This simple boundary condition leads to the theoretical problem studied by Benjamin & Scott (1979) and Graham-Eagle (1983) in the inviscid limit. They concluded that the dispersion relation is changed in a significant way, and we indeed found an important effect on both the eigen frequency and the dissipation of the free-surface wave modes. Since the linear and nonlinear theory is difficult to derive in a realistic way from the hydrodynamics, it is tempting to write directly the normal form of the equation for the amplitude of an unstable mode by using only symmetry arguments. We used a new method to measure small wave amplitudes, and observed behaviour
A
S. Douady
A
t = O 2T
A
1 t = T tT
A
FIGURE1. Excitation at half the excitation frequency of a fluid layer undergoing a vertical oscillation. When the vessel goes down, the fluid inertia tends to create a surface deformation, as in the Rayleigh-Taylor instability. This deformation disappears when the vessel comes back up, in a time equal to a quarter-period of the corresponding wave (T).The decay of this deformation creates a flow which induces, for the following excitation period T, the exchange of the maxima and the minima. Thus one obtains = 2T.
corresponding to the predictions of this amplitude equation. The fit of the curves allows us to estimate the amplitude equation coefficients. The significant nonlinear dissipation explains why we did not observe the possible subcritical transition. Recently, experiments were performed to study the transition to chaos of these parametrically excited surface waves in various geometries. Keolian et al. (1981) studied the temporal chaos occurring in an annular cell when the excitation amplitude is increased far above the instability threshold. Gollub & Meyer (1983) stlgdied a small circular cell, and Cilliberto & Gollub (1984, 1985) the interaction of two different modes. Wu, Keolian & Rudnick (1984) report the observation of soliton-like surface deformation in a long and narrow channel for particular values of the excitation parameters. Ezerskii et al. (1986) studied the spatio-temporal chaos in very large cells, square and circular. They found that this chaos occurs via the dynamics of lines transversely modulating the standing plane wave amplitude. In a large square cell, Douady & Fauve (1988) looked a t the pattern selection and the interaction between two symmetric modes. They showed that these lines naturally appear for the eigenmode of a rectangular cell and could correspond to defects lines
joining two regions oscillating with opposite phases. We check this interpretation
with one of the visualization methods described in $3.2.
2. Theoretical introduction
2.1. Parametric excitation To understanding the parametric excitation of surface waves, we consider a layer of fluid subjected to a vertical periodic motion. If S(z,y) is an eigenmode of the free surface, with an amplitude a($), the surface deformation is
5 = W ) S ( z , Y)
386 S.^ Douady
0 & O( 2w, 2w; w
FIQURE 2. Schematic of the parametric tongues for three eigenmodes of a given vessel, as functions of the external parameters : gravitation modulation amplitude B and excitation pulsation w. One observes that the 4 tongues mask the other tongues, because as the critical values increase quickly with the excitation number (i, 1, t ,.... +n),their critical value E, increase with frequency, the tongues are larger. The curve E, ( w ) is the threshold when the density of the mode is large. The
minimum of the i tongue gives the eigen frequency and the dissipation of the mode.
observed experimentally. One can also notice that (4) gives approximately the marginal stability curve in the limit of an unbounded cell.
2.2. Boundary conditions
For the linear inviscid problem, there are only two possible boundary conditions for the fluid free surface : (i) if the normal vector at the lateral walls, n, is defined at the contact line, one can apply the kinetic boundary condition simultaneously at the free surface and at the wall. The result is that the angle of the free surface and the lateral wall must be constant, and so equal to in for vertical walls and an initially horizontal surface
_ -ac - constant = 0; an
(ii) if the normal vector at the lateral wall is not defined at the contact line, for instance at the edge of the lateral walls, then the kinetic boundary conditions impose no condition on the free surface. But this happens only for this contact line position, thus
5 = constant = 0. (6)
The first boundary conditions allow simple algebra, and in rectangular cells the eigenmodes are products of cosines (with the origin at a cell corner). The flow is not perturbed by the walls, so the dispersion relation is the same as for an unbounded layer. But this boundary condition is not compatible with surface tension, so the corresponding inviscid solution is quite different from the experimental one when the capillary length is not negligible compared to the wavelength.
Experimental study of the Faraday instability (^387)
On the other hand, the second boundary condition makes the problem difficult, and the eigenmodes are no longer sinusoidal (cf. Benjamin & Scott 1979; Graham- Eagle 1983), but they can be roughly approximated by the product of two sine waves. If we neglect the first quarter-wavelength near the edges, we find the previous situation and the flow is the same. But now the first quarter-wavelength corresponds to a flow over the whole cell, and that depends on the number of wavelengths on each side, and on their parity (Douady 1988). Now, the dispersion relation results from a balance between the variation of kinetic and potential energy. Even if the surface is not exactly sinusoidal, the potential is nearly the same as in the previous case, but, because of this additional flow, the kinetic energy is different. So the dispersion relation is changed. The value of this boundary condition is that it is compatible with the introduction of surface tension and viscosity, and is easily obtained experimentally. These modes are thus close to the experimental ones.
2.3. Nonlinear problem Standard asymptotic methods were used in the inviscid case to obtain the nonlinear amplitude equation for the wave, and with boundary condition ( 5 ). The case of the second boundary condition (6) was not studied : this would be much more difficult as the flow is no longer simply related to the surface deformation. A realistic description of a closed cell is thus difficult. However, one can find the form of the amplitude equation directly using symmetry arguments, whatever the modes and the boundary conditions are. First we assume that only one mode, S(z, y ) , is unstable. This mode has an eigen frequency wo close to half the excitation frequency w , and is forced parametrically to oscillate at &.With a complex amplitude, the surface deformation is then
f; = AS(x, y ) exp (iiwt)+ C.C.+ h.0.t.
As usual for any asymptotic method, we assume that the unstable-mode amplitude varies slowly compared to all the other characteristic times, and so we look for a relaxation equation for A , a,A = F ( A ). We also assume that this amplitude is small so P ( A ) can be expanded in Taylor series, and that only the first terms need be considered. Without any excitation, there is no imposed time origin, so the equation
must be invariant under any translation in time t * t + to. This corresponds for the
amplitude to the phase rotation A * A exp (&to). So the amplitude equation must have the form, up to the third order in A,
3,A = -(h+iv)A+(a+ip)IAi2A,
where A , v, a, p are real constants. I n first approximation the solution is
[ = AoS(x,y)exp(-At)exp(i(v+&~)t).
Thus one recognizes h as the dissipation of the mode, and v as the detuning: v = wo -@. a and / 3 can directly be interpreted as the nonlinear dissipation and detuning : A’ = A+alAI2, v’ = V+,L?IA~~. This nonlinear detuning results from a nonlinear dispersion relation: wk = w0+/3IA(*.We assume /3 to be negative as the frequency of
an oscillator usually decreases when its amplitude increases, and also h > 0 and a <
0. The interpretation of A and a as linear and nonlinear dissipation can also be checked by making the assumption that the fluid is inviscid. The system is then
invariant under the time-inversion symmetry, t a - t , which corresponds to A =d
(the bar denotes the complex conjugate) ; and the invariance of the equation imposes h = a = 0.
Experimental study of the Faraday instability 389
by the condition that A , v and p are small compared to oo. But this assumption upon A simply corresponds to the fact that the system is close to the minimum of the resonance tongue, which indeed implies that v is small, but only that p and h are similar. So we think that it would be possible to justify this equation with h and p not small, which is always the case experimentally. If we look a t the stationary solutions of this equation, the arithmetic is easier with the notation a,A = y e i ~ A + p J + c e i ~ ~ A ( 2 A. (7b) We also decompose A into amplitude and phase: A = aexp (is),which gives
(7 c) I
a = ( y cos (q) +p cos (28))a + c cos ($) a3, i"' = (ysin (q)-psin (28)) fcsin ($)a2.
For stationary solutions, we found by eliminating 0 that
ca2 = - y cos (q-$) (p2-y2 sin ( q ~ - $ ) ~ ) ;. (8) By drawing this solution, figure 3, one sees that the transition can either be supercritical or subcritical, depending on the sign of cos (q-$), i.e. on the detuning
(i) if cos (v-$) > 0, which corresponds to v > -ha/P, the transition is super- critical. In this case, the characteristic time of the wave decay below the threshold, with the assumption that the phase remains constant, is
V :
1 /70 = - y cos (q)) k (p2 - y2 sin (P)~);; (9)
(ii) if cos (q-$) < 0, which corresponds to v < - A alp, the transition is subcritical. The stability of the different stationary states can be found without any further calculation by simply using the conservation of the topologic degree, i.e. the conservation of the number of stable solutions minus the number of unstable solutions (Leray & Schauder 1934), which applies to ( 5 a ). Figure 4 summarizes the various cases in the external parameter plane. By increasing the detuning (i.e. w ) , we continuously go from a subcritical transition to a supercritical one. The existence of hysteresis is due to the nonlinear detuning coefficient p. For an excitation frequency that is too small, the mode is non-resonant for a small amplitude; but if it jumps to a significant frequency, this diminishes its eigen frequency, and thus it can resonate and become unstable. Note that this subcritical bifurcation does not need a higher-order term to saturate: if the amplitude continues to increase, the eigen frequency keeps on decreasing, so the mode is again non-resonant and the amplitude saturates. If the sign of / 3 is changed, the situation is symmetric with the previous one with the hysteresis located at the right. The importance of this subcritical region is given by the ratio a l p (i.e. $): (i) if this ratio is small ($ - -in), the hysteresis is observed for w < 2 w 0 (Hsu 1977; Nayfeh & Mook 1979; Meron 1987; Gu & Sethna 1987), i.e. for all the left side of the tongue, and the hysteresis is maximum ;
(ii) if this ratio is significant ($ h- -n), the frequency below which the hysteresis
is observed is pushed to the left, and the hysteresis becomes small (cf. figure 4). At the limit $ = -n, the transition is always supercritical (Hsu 1977).
390 S. Douady
I -Aa/p 0
(cos (c, - $) = 0) U
FIGURE4. Stability boundaries as a function of the detuning v ( ( w , -? p ~ ) / w , , and the forcing p. The linear threshold corresponds to the hyperbola p = y and the nonlinear threshold reveals a hysteresis in the grey region. This hysteresis is at its maximum for a = 0 and disappears for /? = 0 (the transition is then always supercritical).
FIQURE 5. A meniscus, the typical length of which is the capillary length I = (T/pg$, is always excited by vertical oscillation. When the cell goes up, the effective gravity is increased and the meniscus length decreases. So it emits a surface wave in order to preserve mass. For a vertical oscillation of the vessel, the meniscus thus produces an isochronous wave.
3. Experimental procedure
3.1. Boundary condition and mode shapes Experimentally, the boundary condition for the free surface with a meniscus is not clear. The contact line can move and the angle of contact depends on its speed. In some cases, the line can remain nearly fixed if the angle oscillation corresponds to that of the wave. Experimentally, the maxima of the patterns are never located at the lateral walls, and the boundary condition seems to be a surface pinned at a fixed contact line. Moreover, an experimental problem arises : a vertically oscillating meniscus always emits surface waves at the excitation frequency. This emission happens even if the contact line is pinned at the edge of the lateral walls, if there is still a meniscus, i.e. if the surface is not perfectly flat. This can be physically understood by recalling that the typical meniscus length is the capillary length 1 =
392 8. Douady
FIGURE7. An example of the patterns observed with oil with a significant meniscus, for the same conditions as figure 6, but above the instability threshold, They are nearly perfect squares near the centre of the cell, even though they are turned through a certain angle, but completely deformed at the boundaries, where the meniscus wave is still visible.
FIGURE8. Mode observed without a meniscus, at 70 Hz, for water containing Kalliroscope. Its regularity shows that it is quite near sinusoidal.
Experimental study of the Faraday instability 393
Bright region of nearly
Zero of the sinusoidal mode
FIQURE9. Schematic of the bright regions observed for a product of two sine waves illuminated vertically. They correspond to the horizontal points, extrema and saddle points. When the reflection is not purely vertical, one observes segments or surfaces joining these points.
nor the observer are at infinity, so one also observes lines or surfaces joining these points (cf. figure 9). The interest of this method is the independence of time of the observed structure (so it does not need any stroboscopic observation), and the direct identification of the pattern using simple rules. The modes in a rectangular cell are described by the number of nodes in each direction, and figure 10 presents the patterns in a square cell corresponding to the mode (9,13), and the superpositions of (14,2), (14,4), and (13,7) with their respective symmetrical modes. Figure 11 shows computer drawings of the nearly horizontal regions of the surface deformation for the corresponding sinusoidal modes. In the case of a single mode, for instance figure lO(a), the two node numbers are obtained by counting the number of oblique lines along each side. In the case of a superposition of two symmetrical modes, for instance in figure 10(b,c), a larger number of modulations is obtained from the number of rapid modulating lines parallel to one side, the second number being given by the number of slow modulations. Figure 10(d) presents a case of diagonal modulations as 7/13 is larger than tan($), and corresponds to a transient state leading to one mode only, (7,13) or (13,7) (Douady & Fauve 1988).
Flow visualization
Another visualization method for thin layers (Kh < 1) consists in looking at the
fluid motion revealed by small particles (for instance Kalliroscope in water). For stationary states, we principally observe a steady convection-like motion related to the surface waves: the fluid rises a t the maxima of the waves, and falls at the nodal lines (cf. figure 12). This flow is the rectified flow generated a t the viscous boundary- layer limit by the oscillatory fluid motion due to the surface waves. The acoustic streaming escapes from the maxima of horizontal velocity and converges to the minima (Batchelor 1967). This streaming had already been observed by Faraday (1831) with some sand accumulated at the cell bottom. Figure 13(a) presents a typical example of a streaming pattern, and figure 13 ( 6 ) a computer drawing of the extrema of the corresponding sinusoidal mode. This method is more complicated than the previous one, as the prediction of the extreme positions for the superposition of two sinusoidal modes is not always simple, so pattern identification is harder.
Experimental study of the Faraday instability 395
FIQURE11. Computer drawings of the nearly horizontal regions of the surface for the sinusoidal modes corresponding respectively to the patterns of figure 10.
FIQURE12. Schematic of the steady flow generated by the parametric standing waves. The fluid rises at the maxima and goes down at the nodal lines. Thus the bright regions, i.e. regions of horizontal motion, correspond to level lines of the wave pattern. This flow is the steady streaming created by the viscous boundary layer a t the bottom of the cell.
396 S.^ Douady
FIQLJRE13. (a) Photograph of the flow, visualized by Kalliroscope, due to the parametric wave pattern, in a square cell at f = 64 Hz for water. The surface deformation corresponds to the superposition of the mode (21,4) and a symmetric one. It shows the ‘level lines’ of this pattern. ( b ) Computer drawing of the ’level lines ’ for the corresponding sinusoidal pattern, black corresponding to a surface deformation greater than a certain value.
Experimental study of the Faraday instability
f* Light beam ,#
1 Free surface Cell bottom
Top view
t
........... Red
t -k $To
FIGURE 14. Principle of bicolour stroboscopy visualization. It consists in looking at the reflection of a light beam which alternately changes colour at each excitation period T. For half a resonant
wave period 4% (= T), the wave is illuminated in green, and for the other half-period in red. Thus
we observe alternating green and red bands.
turned or not to the light (cf. figure 14). For the other half-period, we illuminate for instance in red, and the black and bright regions are exchanged. Thus, for a plane wave, we observe an alternation of green and red bands with a periodicity of one wavelength (cf. figure 15a, plate 1). The phase of the local structure is given by the fact that a point is green or red, and we observe the phase opposition when one crosses each transverse modulation line of figure 15 ( a ) , as red is changed into green and green into red. This phase opposition is also verified when one crosses each vertical or horizontal modulating line in figure 15b, plate 1). In fact, for these figures, the modes are nearly the product of two sine waves, and what seems to be a transverse modulation of a plane wave in figure 15(a) merely corresponds to the slow and rapid modulation of these products : figure 15 ( a ) presents the transient of the mode (21,5) growing alone, and figure 15(b) the final state of the superposition of the two symmetric modes. So we cannot strictly speak of defect lines in these cases, even if they behave like during transient states. But we think that for a larger cell (or for a small wavelength), this method could be used to show that the observed transverse modulation corresponds to the appearance of these defect lines.
3.3. Amplitude measurement method The method consists in looking at the image on a vertical screen of a thin laser beam reflected a t the free surface. In the geometrical approximation, and if the screen is far enough away, the position of the reflected beam depends only on the surface slope at the reflection point. Figure 16 presents the principle and notation. In two
(
8. Douady
Incident laser beam
L
*.--
-*' ay[ Tangent plane
at the surface
if aY[ = 0
M
Z'
b = (cotan 8 + D/z' sin2 S )
... ...---......---..... - Y ,
Screen
if = (^0) A
FIQURE16. ( a ) The principle of the measurement method of the surface wave amplitude: a laser beam is reflected at the free surface, and one looks at the image point M on a vertical screen far away. The origin is chosen as the reflection with a horizontal surface. In the geometrical approximation, M only depends on the slope of the surface a t the reflection point. ( b ) and ( c ) The image points on the screen in two cases : the relation can be easily inverted to give the local slopes, and thus to measure the amplitude.
particular cases, when a, 6 = 0 and a, 5 = 0, one can simply find the slopes from the position of the reflected beam. I n the general case, the image on the screen is a smooth deformation of the (a, 5, a, c)-plane. As we choose the reflection location, this method allows the maximum slope of the unstable wave, and thus its amplitude to be measured. For instance, we consider a long cell in which the unstable wave is approximately descri bed by
and we also have a meniscus wave coming from the lateral boundaries:
a sin (nz/L,z) sin (z/L,y) sin (!pwt),
b sin (ky y) sin (ot + @)
400 8. Douady
3.4. Experimental procedure In spite of the simplicity of a vertically oscillating layer of fluid, experimental problems arise from the meniscus effects, the mechanical set-up, and the variation of the fluid properties : (i) We have seen that the meniscus is important as it generates surface waves which can perturb the parametric standing surface waves. There is no general solution for this problem. For a moderately wetting fluid, such as water, one can pin the free surface a t the edge of the lateral walls to obtain a perfectly flat surface. I n our experiments with oil, we used a long cell in order to separate easily the resonant wave from the meniscus one. (ii) Like the meniscus, horizontal motion generates propagating waves from the sidewalls. Rolling motion must also be eliminated in order to have the same amplitude excitation in the entire vessel, and thus the same amplitude of the resonant wave. So the motion must be purely vertical. The fluid depth must be constant (i.e. the vessel must be horizontal). If any of these conditions is not fulfilled, the waves can be non-stationary and slowly move, coupled to a large-scale steady flow. A simple test for the mechanical system is to use a vessel brimful of water, and to check that the surface remains perfectly flat just below the instability threshold. The excitation homogeneity is^ also easily visible just above the threshold. (iii) The stability of the experiment over a^ long time depends on the stability of
all the fluid-layer characteristics : the fluid depth h, surface tension T , and viscosity
(i.e. dissipation). Thus the system must be thermally regulated, and, for a volatile fluid, the atmosphere must be saturated in the fluid vapour to avoid evaporation.
4. Experiments
4.1. Experimental set-up We used a Briiel & Kjaer 4809 vibration exciter which produces a clean vertical acceleration waveform (the horizontal acceleration is less than 1 % of the vertical one). The vibration exciter is driven by a Krohn-hite frequency synthesizer (frequency precision better than The cell acceleration, which is the relevant bifurcation parameter, is measured with an accelerometer. The mechanical system was tested with a vessel brimful of water: the surface remained perfectly flat just below the threshold, and just above, the amplitude was everywhere the same. To study the threshold of the instability, we used a copper vessel of dimension 65 x 16 x 5 mm3, thermally regulated. We varied the excitation frequency between 20 and 30 Hz. The first fluid used was a silicon oil (Rhodorsil47V20), a t 30 "C, chosen for the stability of its surface tension and non-volatility. As it wets the cell walls, the oil has a significant meniscus. But the geometry of the cell, a long rectangle, was chosen in order to separate the meniscus and the parametric wave, as the meniscus wave is important only along the lateral walls (the wave due to the nearly round meniscus is quickly damped), contrary to the parametric wave (cf. figure 18), which presents several modulations in the long direction and only one perpendicularly. Even in this case, the modes shape are similar to the product of two sine waves, so the real boundary condition seems to be close to a virtual pinning a t the meniscus. We also used this cell filled up with water, then held at 5 "C, to study the influence of the boundary condition.
Experimental study of the Faraday instability
.- c) 2
d
3 1 0 -
-0-^ --^.
12 _-
_ - ._I-.
-__-- -
Meniscus
Resonant wave n = 12
4 D 65 mm FIQURE18. Experimental vessel used for the study of the threshold. In the case of a significant meniscus (with oil) the isochronous standing meniscus wave shows modulations along the small dimension, and the half-frequency parametric standing wave is seen along the large dimension.
l 3 1
h
R 13
15 _ -^ - I I-
8! a I I I I I 1 28 30 32 34 36 38 40 42 Excitation frequency (Hz) FIGURE19. Stability boundaries for oil for two fluid depths h. Each measurement corresponds to two points, one below and one above the threshold. The precision does not allow us to distinguish them on this figure. The 4 tongues of the modes with 11-15 modulations in the long direction are observed. The threshold increases with the frequency (cf. (2)) and decreases when the fluid depth is increased, showing that the dissipation principally depends on the friction a t the waIls, including the bottom.
4.2. Threshold The threshold is measured for a fixed frequency, by starting with an amplitude excitation corresponding to a parametric wave of large amplitude. Then, we slowly decrease it in order to obtain a small amplitude, corresponding to a few centimetres
