Under consideration for publication in J. Fluid Mech. 1

Convective/absolute instability in misciblecore-annular flow. Part 1 : Experiments

By M. d ’OLCE1 J . MARTIN1 N. RAKOTOMALALA 1

D. SAL IN1 AND L . TALON1†,1Univ Pierre et Marie Curie-Paris6, Univ Paris-Sud, CNRS, F-91405.

Lab FAST, Bat 502, Campus Univ, Orsay, F-91405, France

(Received 4 December 2007)

We address the issue of the convective or absolute nature of the instability of coreannular pipe flows, in experiments using two miscible fluids of equal density but differentviscosities, the core fluid being much less viscous than the wall one. We use a concentricco-current injection of the two fluids. An axisymmetric parallel base state is obtaineddownstream the injector. The core radius RI and the Reynolds number Re of the so-obtained base state are varied independently thanks to the control of the flow rate of eachfluid. However, a downstream destabilization of this base state was observed within theexplored range of the two control parameters RI and Re. Moreover, the fixed locationof this destabilization, observed for some particular parameters, suggests an absolutenature of the instability. We present a tentative delineation of the nature (convective orabsolute) of the instability and discuss the accessible measurements to experimentallyaddress this issue.

1. Introduction

First discovered in the context of plasma physics (Briggs (1964)), the concept of ab-solute and convective instability has been significantly used in hydrodynamic open flows(Huerre & Monkewitz (1990)) such as falling film down an inclined wall (Brevdo et al.(1999)) or a fiber (Duprat et al. (2007)), wake and jets (Chomaz (2005)). When a givenunstable flow is locally perturbed by a small disturbance, the perturbation can exhibittwo different types of evolution (Huerre & Monkewitz (1990)). For a convectively un-stable flow (CU), the disturbances are amplified and advected away from their initiallocation. Such a convectively unstable flow behaves like a noise amplifier. For an abso-lutely unstable flow (AU), although advected, the perturbation is so strongly amplifiedthat it invades the whole space (downstream and upstream). In such a case, one fre-quency mode will prevail at long times and the system will behave like a self-sustainedresonator (see Godreche & Manneville (1998) for more details).

Theoretically the transition corresponds to the pinching of two spatial branches of thedispersion equation. Therefore the transition can be either computed or obtained by thestudy of the impulse response propagation as carried out in the companion paper (Selvamet al. (2008)). However, except in devoted experiments where the so-called ”ribbon” de-vice can be used (Huerre & Monkewitz (1990)), the transition is experimentally difficultto determine. First, it is nontrivial to perturb the system at a particular location awayfrom the boundary conditions. Moreover, in an absolute case, the base state is experi-mentally unachievable as one can never avoid the presence of some noise which would

† To whom correspondence should be addressed. Email: talon@fast.u-psud.fr

2 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

trigger the instability. For those reasons, experiments usually only allow to build a bodyof evidence of the presence of a CU/AU transition from a series of different experimentalmeasurements (Duprat et al. (2007)).

Here we address this issue for a core-annular flow of two miscible non-buoyant fluids assketched on Figure 1. We note that the immiscible counterpart, subjected to the RayleighPlateau instability (Drazin & Reid (1981)), has been widely investigated (Hickox (1971);Joseph et al. (1984); Hu & Joseph (1989); Bai et al. (1992); Joseph & Renardy (1992a,b);Joseph et al. (1997); Kouris & Tsamopoulos (2001, 2002)) as well as other shear instabil-ities (Hinch (1984); Albert & Charru (2000)). Several authors (Petitjeans & Maxworthy(1996); Chen & Meiburg (1996); Kuang et al. (2003); Scoffoni et al. (2001); Balasubra-maniam et al. (2005)) have also investigated miscible, variable viscosity displacements incapillary tubes, though from a different perspective of analyzing the front propagationvelocity. Recently, Selvam et al. (2007) have performed the linear stability analysis of themiscible core-annular flow instabilities. They found that miscible core-annular flows aresubjected to both axisymmetric and corkscrew (first azimuthal mode) instability modesdepending on the Schmidt number, with the axisymmetric mode being the dominant athigh Schmidt numbers. The focus of the aforementioned authors were on the temporalnature of the core-annular instability with a spatial periodicity. However, the core-annularconfiguration is mostly encountered in open flows, where the instabilities develop spa-tially. These spatially growing instabilities can be classified as being either convective orabsolute (Briggs (1964)). Such characterization of miscible core-annular flows has, to ourknowledge, never been addressed. We use high enough flow rates in order to minimizediffusive mixing and to get a well-defined pseudo-interface. Our objective is to assess theeffect of the Reynolds number and of the core fluid radius on the convective/absolutenature of the instability for a fixed viscosity ratio.

In section 2, we present the experimental setup. In Section 3, we introduce all thedifferent measurements used to investigate the instability and the CU/AU transitionpresented in section 4. A discussion of the results follows in Section 5. In a followingcompanion paper (Selvam et al. (2008)), our results will be quantitatively discussed andcompared to the linear stability analysis and nonlinear numerical simulations.

2. Experimental setup

2.1. Core-annular flow setup

The experimental setup is sketched in Fig. 1. Experiments are performed in a verticaltransparent Perspex cylindrical tube of length 1 m and internal radius R = 1 cm. Anupwards injection of the two fluids is implemented, using a concentric inner nozzle tube ofinner and outer radius 0.75 cm and 0.85 cm. The two fluids, namely the core fluid and thewall fluid, are simultaneously injected with two pumps, at the constant flow rates, Qcoreand Qwall, respectively. Note that for the experiments of harmonic excitation (Sec. 4.2),an electronic device enabled to impose an oscillating Qcore, while maintaining the totalflow rate, Q = Qcore + Qwall constant. The two fluids are salted water for the core and awater-natrosol mixture for the wall, of viscosities equal to ηcore = 10

−3 Pa.s and ηwall =(25±2)×10−3 Pa.s, respectively. Salt (calcium chloride) was added to match the densitiesof the two fluids to ρ = 1000 kg/m3, with an accuracy of ±10−1 kg/m3. These mixturesare perfectly Newtonian in the range of shear rates used in our experiments (Torrest(1982); Le Bars & Davaille (2002)), and have a viscosity ratio M = ln(ηwall/ηcore) =3.2± 0.1. To ascertain the validity of our results, some additional experiments were alsoperformed with water-glycerol mixtures with the same viscosity ratio.

Convective/absolute instability in miscible core-annular flow: Experiments 3

As can be seen on Fig. 2, a parallel core annular flow is achieved beyond an inletlength Lν (∼ 1 cm in our experiments). This length results from the competition betweenadvection and diffusion of mass momentum. The core radius in the parallel flow regionis set by the ratio Qcore/Q. Indeed, under parallel flow conditions and assuming thatthe two fluids do not mix, the base Poiseuille Parabolic-like velocity profile expressed interms of the relative core fluid radius R̃I = Rcore/R and the relative radial coordinater̃ = r/R reads :

Uwall =2

(

1 − r̃2)

1 + (eM − 1) R̃I4 Ū (2.1)

for the wall fluid (R̃I < r̃ < 1) and

Ucore =2

(

1 +(

eM − 1)

R̃I2− eM r̃2

)

1 + (eM − 1) R̃I4 Ū (2.2)

for the core fluid (0 < r̃ < R̃I), where U = Q/(π R2) is the average flow velocity. Thus,

the flow rates ratio reads

QcoreQ

=2 R̃I

2+

(

eM − 2)

R̃I4

1 + (eM − 1) R̃I4 (2.3)

As the above function is monotonic for R̃I ∈ [0, 1], the core radius is uniquely de-termined from the flow ratio Q̃core. In the following, we will use R̃I as a control vari-able, together with the Reynolds number, defined by Re = ρU R/ηcore. We explored therange R̃I ∈ [0, 0.6] and Re ∈ [2, 60], for the fixed viscosity ratio e

M = 25. We note,that our experiments were performed with a fixed Schmidt number Sc = ηcore/ρDm.The molecular diffusion coefficient, Dm, estimated as in Torrest (1982); Le Bars &Davaille (2002) is Dm ≃ 10

−10 m2/s, and thus Sc ∼ 104. Since the Péclet number readsPe = U R/Dm = Sc Re, the control parameter Re also quantifies the relative magnitudeof convective and diffusive effects. As a consequence, keeping a well defined interfacerequires a high enough Pe, and thus Re (Re > 2 in our case). On another hand, foreither high Re values (Re > 60) or high R̃I values (R̃I > 0.6), the instability patternsexhibit some asymmetry. One may note that in our range of R̃I , the difference between

the shears on both sides of the interface ∆γ̇ = 4r (eM−1)

1+(eM−1) r4UR

is almost linear in r (for

r4 < 13 (eM−1) , i.e. r < 0.35 for eM = 25), so that r quantifies also the shear difference,

∆γ̇ ≃ 4r (eM − 1)UR

, in most of our experiments.In the following, we will use characteristic length and time R and R/Ū respectively to

define nondimentionalized physical quantities (marked with a tilde).

2.2. Optical setup

2.2.1. Concentration measurement

Measurements were achieved using a video camera. To minimize any optical distortioninduced by a variation of optical indices between the tube and the atmosphere, we en-closed the apparatus in a square box filled with water. We observed a small magnifyingeffect with no distortion.

Preliminary experiments were performed using a blue dyed core fluid (Fig. 2 left).However, to analyze more precisely the pseudo interface between the two fluids, rI(x, t),we used fluorescin as dyer and illuminated the axis of the tube with a thin, 1 mm wide,

4 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

laser sheet as sketched in Figure 1 and showed in Fig. 2 (right). This technics provides ameasurement of the interface radius r̃I(x̃, t̃) with an accuracy less than 5%.

In these pictures, we can observe streamwise from the inlet: 1) the viscous entry lengthLν , of the order of the tube radius; 2) a quasi-parallel length L̃‖ where the pseudo interfacebetween fluids is parallel; 3) an axisymmetric instability pattern the shape of which ismushroom like. We have already reported on these nonlinear axisymmetric patterns(d’Olce et al. (2008)).

2.2.2. PIV measurements of the velocity profile

In order to measure the flow field in the inlet region, we used the LaVision (GmbH)Particle Image Velocimetry software. The camera has a resolution of 1280 × 608 pixelsin the plane of the laser sheet (Figure.1). The PIV algorithm requires a window of either16 × 16 or 32 × 32 pixels. Measurements over the whole cell diameter (2 cm) providetypically from 30 to 60 data points across the tube section. The spatial resolution istherefore of the order of 300 µm. Fig. 3 (left) gives a typical velocity field. Fig. 3.b displaysthree velocity profiles, Ṽ (r̃), at three locations from the inlet (2.2, 2.7 and 3.2 cm). Thethree solid lines correspond to the theoretical velocity profile (Eqs. 2.1 and 2.2). Theagreement of the latter with the measurements supports the contention that a parallelflow is achieved and that the diffusive mixing is actually negligible.

3. Experimental observations and quantities of interest

In this section, we introduce the different quantities that will be used to investigatethe CU/AU transition.

3.1. Interface Spatio-temporal diagram

On the pictures of the axisymmetric patterns (see Fig. 2 right), the inflexion point ofthe light intensity provides the space and time dependent radial position of the pseudo-interface, r̃I(x̃, t̃). In Fig. 4.a, we plot the time dependence of r̃I(x̃, t̃) for four differentlocations x̃. As expected for a spatially unstable regime, we observe, downstream fromthe inlet, an increase of the amplitude of the fluctuations. It is convenient to plot thesevariations r̃I(x, t) on a spatio-temporal diagram : the vertical axis is the time t̃, thehorizontal axis the distance x̃ from the inlet, the position r̃I(x, t) of the interface isencoded into a grey level. A typical spatio-temporal diagram is displayed on figure 4.b.The homogeneous grey level close to the inlet corresponds to a region of constant R̃I(quasi-parallel flow base state). Further downstream, one can observe dark and lightstripes, the slope and contrast of which represent the velocity and the amplitude ofinstability waves.

3.2. Temporal power spectrum of the interface fluctuations

As mentioned above, a key difference between convective and absolute states is theirspatial response to noise. Therefore, it might be useful to analyze the spatial evolutionof the frequency spectrum with x̃. For that purpose, we decompose at each location x̃the temporal fluctuations of r̃I(t) into Fourier modes characterized by their amplitude

and their phase, i.e. A(x̃, f̃)ei(φ(x̃,f̃)−2πf̃ t̃). From this formulation, one can compute thelocal wave number k̃r(x̃, f̃) =

∂φ∂x̃

and the local wave speed c̃(x̃, f̃) = 2πf̃/k̃r(x̃, f̃) foreach mode.

A typical frequency spectrum, A2(x̃, f̃), obtained at a given x̃ position is plotted inFig. 4.c. Such frequency distribution is characterized by the frequency at which the

Convective/absolute instability in miscible core-annular flow: Experiments 5

distribution is maximum, f̃max(x̃), and by spectrum width characterized by the rootmean square deviation from the mean rms(x̃):

rms(x̃) =

√

∫

p(x̃, f̃)f̃2df̃ −

(∫

p(x̃, f̃)f̃df̃

)2

with p(x̃, f̃) = A2(x̃, f̃)/

∫

A2(x̃, f̃)df̃

(3.1)The spatial evolution of the maximum amplitude of the power spectrum, A2

m(x̃) is plottedin Fig. 4.d. One can observe an increase of the amplitude followed by its saturation. Closeenough from the inlet, the spatial growth appears to be exponential, revealing a linearregime. We will measure the characteristics of the instability patterns in this linear range,namely the spatial growth rate −k̃i, the phase velocity, c̃ and the real part of the wavenumber, k̃r.

The spatial evolution of the RMS is plotted in Fig. 4.e. Near the inlet, no instabilityis detectable, so the spectrum is a random noise one (infinite RMS), and the measuredRMS is fixed by the size of the measurement window. Downstream, the RMS decreasesas the instability grows and reaches a minimum value, denoted by rms(f̃ )min, at largex̃.

Since a convectively unstable flow behaves like a noise amplifier and because our in-herent noise is broadband we can expect that the convectively unstable state will have alarger rms(f̃)min than an absolutely one which resonates at a single frequency.

To account for the spatial variations of these spectra, we built a frequency f̃ versusspace x̃ diagram where the amplitude of the power spectrum is encoded in grey scale.Such a typical diagram is given on figure 4.f. Like on the spatio-temporal diagram, onecan observe that a minimum distance from the inlet is required for the most unstablemode to be amplified. One can also note, further on, the appearance of the correspondingharmonic modes.

3.3. Quasi-parallel length L̃‖, filtering length L̃P and filtering rate α

An important feature of the observed instability pattern is the presence of a parallel flowregion at distances from the inlet between the viscous entry length Lν and the positionwhere the instability appears. The extension of this region might be a relevant featureto discriminate the nature of the instability. Indeed, in a very precise theoretical context(no background noise in a semi-infinite domain), this region is expected to extend overan either infinite or finite length for a CU or AU instability, respectively (see Couairon& Chomaz (1997a,b)).

In our experimental context, in which background noise cannot be avoided, this parallelregion remains finite in any case. In order to evaluate it, we compute, for each positionx̃, the time average of the magnitude of the interface oscillation. We define the quasi-parallel length L̃‖ as being the location at which this average magnitude is equal to agiven threshold.

An alternative characterization of the parallel region can be obtained from the plot ofthe root mean square of the frequency power spectrum (rms(x̃)) (Fig.4.e). We denote byL̃P the intersection location of the tangent at the inflexion point with the minimum ofthe RMS (rms(f̃ )min). Physically, as the rms(f̃ )min evaluates the inverse of the qualityfactor of the noise filter, L̃P quantifies the distance needed for the filter to be efficient.The latter may indeed be different from the length needed for the instability to appear,L̃‖.

One may note that L̃‖ and L̃P are defined as distances from the inlet nozzle (x̃ = 0),which does not match at all the theoretical semi-infinite domain origin (i.e. a locationwith a non-noisy unstable basic state). Another characterization of the spatial decrease

6 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

of the rms(f̃ ), which is not linked to the inlet location, may be obtained through thevalue of the slope at the inflexion point as sketched in Fig.4.e. This RMS maximum slope,α, will be called the filtering spatial rate. We note that −1/α is a typical length, whichis supposed to compare with L̃P but which is measured locally.

We have already pointed out the differences between the theoretical frame of work andthe experimental constraints which may raise difficulties to delineate the change of natureof the instability. However, since an absolutely unstable system behaves like a frequencyresonator and since global modes saturate at a finite distance from the inlet, we expect,for a small enough background noise, that rms(f̃ )min, L̃P , L̃‖ and α drastically changefrom a CU to an AU state.

4. Delineation of the transition between the convectively andabsolutely unstable states

In this section, we shall first measure the characteristics of our experimental instability,as defined in the previous section. Then, the response of the system to a harmonicperturbation will be analyzed.

4.1. Natural instability

The two control parameters of the experiment are the Reynolds number, Re and the rel-ative core radius R̃I of the quasi-parallel base state. As mentioned above, the interface iswell defined and the patterns remain axisymmetric in limited ranges of these two param-eters, R̃I ∈ [0, 0.6] and Re ∈ [2, 60]. In these ranges, the parallel core-annular flow neverextended all along the tube, whatever the attention paid for the injection conditions.We have investigated various features of the instability which develops naturally in thesystem, in the parameter space (R̃I , Re). For the sake of clarity, we will first analyze theevolution of our different measurements, with one of the two control parameters fixed (Re = 48 and R̃I = 0.48 for the left and right column of graphs of Fig. 5 and 6, respec-tively). The variations of the quantities of interest in the plane Re-R̃I are then discussedand displayed in Fig.7.

Fig. 5 displays, from top to bottom, the measures of the frequency f̃max, of the wavenumber k̃r(f̃max) and of the phase velocity c̃(f̃max) of the most amplified mode.

The latter decreases continuously with R̃I (Fig. 5e) whereas it remains almost constantas Re is varied (Fig. 5f). Moreover, the measures of c̃(f̃max) compare very well with thephase velocity c̃SW (dashed line) obtained in the short wave limit of the temporal linearstability analysis (Joseph & Renardy (1992b)). The latter equals the fluid velocity atthe interface in the parallel base state and can be obtained by equation 2.1 or 2.2:

c̃SW = Ṽwall(R̃I) = Ṽcore(R̃I) =2 (1−R̃2I)

1+(eM−1) R̃4I

. We note that in our experiments, this

expression c̃SW is found to hold for the phase velocity of the most amplified mode,although the latter has a normalized wave number k̃r(f̃max) in an intermediate range(between 3 and 10, typically). Thereby, the frequency and the wave number of the mostamplified mode are linked by the relation 2πf̃max/k̃r(f̃max) = c̃SW (R̃I). As a matterof fact, the ratio of f̃max/k̃r(f̃max) does not change as the Reynolds number is variedfrom 5 to 50, keeping R̃I = 0.48 constant † , and the frequency (Fig. 5b) and the wavenumber (Fig. 5c) exhibit the same smooth increase (i.e. an overall increase by a factorof 2). On the contrary, as the core radius is increased at a fixed Reynolds Re = 48

† It has to be noted that the ratio 2πf̃max/k̃r, obtained by linear stability analysis and directnumerical simulation by Selvam et al. (2008), was also found to be independent of Re althoughslightly higher than c̃SW .

Convective/absolute instability in miscible core-annular flow: Experiments 7

(left column of Fig. 5), the frequency exhibit a strong decrease whereas the wave numberremains nearly constant. In other words, the rather strong variations of the phase velocity,c̃(f̃max) = c̃SW (R̃I), with the core radius are accounted by the frequency variations. Tosummarize, the wave number of the most amplified mode is fixed by the Reynolds numberwhereas its phase velocity is set by the core radius. And despite a small jump in f̃maxand k̃r(f̃max) in the vicinity of R̃I ∼ 0.45 and Re ∼ 45, we do not observe any drasticchange in the most amplified mode which could characterize a change of the instabilitynature.

To address the issue of the nature of the instability, we have defined quantities whichare expected to be more discriminating as rms(f̃)min, L̃P , L̃‖ and α. Fig. 6 displaysfrom top to bottom:

(a,b): The minimum of the root mean square frequency rms(f̃ )min which characterizesthe oscillator at work. The frequency distribution width decreases significantly with bothR̃I and Re and levels off at a value corresponding to the minimum measurable width ofour frequency window (∼ 0.02), for RI & 0.45 and Re & 40, respectively.

(c,d): The quasi-parallel length L̃‖ (△) and the filtering length L̃P (•). The two lengthsexhibit the same trend: A strong decrease from about 7 to a value smaller than 1 as eitherR̃I or Re is increased. Note that the minimum measured values are smaller than 1, whichis of the order of the viscous length. In other words, both lengths are so small that theinstability develops before the parallel state is completely achieved.

(e,f): The spatial filtering rate, α. As expected α behaves as −1/L̃P and accordinglyexhibits the same trend as L̃P , with, however, steeper variations at RI ∼ 0.5 and Re ≃ 35,respectively. We recall that α which gives a local estimation of L̃P , has been introduced toget rid of the difficulty to define the location of the ”entry” in an experimental system. Asa matter of fact, −1/α could a priori reach values lower than the normalized viscous entrylength L̃ν . However, such values,(corresponding to α < −1), have not been measured.In anyway, we should point out that in the absence of the parallel base state at somepoint in the system ( which gave the limitation applying to L̃‖ and L̃P measurement),the physical meaning of the measure of α is also less clear. This point may also explaina dispersion in the corresponding α measurements (as may be noticed from the valuesat high Re displayed in Fig. 6f). However, a steep decrease toward low dispersed valuesof α measurements is still a good candidate for the signature of a convective to absolutetransition.

To summarize, in this series of figures 6, we clearly observe a strong decrease of theminimum of the rms frequency, of the stable and filtering lengths and of the filtering ratewith either R̃I or Re, while keeping the other parameter constant. Although, the decreasesare steep, the experimental constraints, as the frequency window width for rms(f̃)minand the viscous entry length, L̃ν for L̃‖ and L̃P and α, raise difficulties to measure thethreshold of the transition. Still, the variations of the four measured quantities, presentedin Fig. 6, all support a transition towards an absolutely unstable state above RI ≈ 0.45and Re ≈ 30. Although an agreement for the RI ≈ 0.45 transition is obtained in bothlinear stability analysis and nonlinear simulations (Selvam et al. (2008)), our so-obtainedtransition Re ≈ 30 is significantly higher than the one obtained numerically (Re ∼ 10).

Fig. 7 collects , in the (Re-R̃I) plane, all the measures obtained for: the frequencyof the most amplified mode, f̃max (Fig. 7a), the inverse of rms(f̃ )min, i.e. the qualityfactor (Fig. 7b), the inverse of the filtering length, L̃P (Fig. 7c) and the absolute valueof the spatial filtering rate, |α| (Fig. 7d). In every diagram, the size of the data point isproportional to the value of the depicted quantity. As a steep increase of rms−1(f̃)min,L̃P and |α|, is expected at the absolute instability transition, absolute states should bepointed out by large dots. Note that the results presented in Fig. 5 and 6 correspond

8 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

to data points on two straight lines parallel to either axis. The general trends obtainedfrom Fig. 5 and 6 are confirmed in the (Re-R̃I) plane of Fig. 7. The frequency of themost amplified mode f̃max ((Fig. 7a) increases with Re and decreases with R̃I , but doesnot exhibit any steep variation. This is not true for the three other quantities displayedin Fig. 7: the three of them exhibit a steep increase at large R̃I and Re. The threecorresponding plots help estimating the reliability of an experimental delineation of theCU/AU states: The frontiers will slightly vary from one plot to the other, and with thearbitrary threshold. However, they are very similar and agree qualitatively to the linearstability analysis carried out by Selvam et al. (2008).

A last check of the CU/AU transition accessible with our experimental device is theinvestigation of the response to a harmonic time perturbation forcing.

4.2. Response to time harmonic excitation

In terms of an oscillator, the response of the system to a disturbance at a given frequencyis a key issue (Chomaz et al. (1988)). Thanks to the control of the two pumps by anelectronic device (Fig. 1), the experimental set up enables us to keep the total flow rateconstant, while modulating sinusoidally the flow rates of the core and annular fluids.The overall effect of such forcing is an almost sinusoidal modulation of the interface atthe inlet. The spatio-temporal and spatio-frequency diagrams obtained at Re = 38,R̃I =0.48, for a frequency of modulation f̃f = 0.43 are displayed in Fig. 8(b, d), together withthe ones obtained without forcing (Fig. 8(a, c)). In the last case, the spatio-frequencydiagram (c) contains a single, natural, fundamental frequency f̃n = 0.73, whereas, withforcing, the diagram exhibit two peaks, at f̃n and f̃f . It is important to note that theforced mode appears very close to the injector, whereas the natural mode appears beyonda distance of the order of L̃‖. One can also observe, downstream the establishment ofthe fully developed natural mode, the damping of the forced mode. These trends can bealso inferred from the forced spatio-temporal diagram Fig. 8b which can be compared tothe similar unforced, ”natural” one, Fig. 8a: The forced mode develops much closer tothe inlet, and the resulting straight lines appearing downstream in the spatio-temporaldiagram are more regular than those observed for the natural mode.

Fig. 9 shows the spatial growth rate, −k̃i, versus the forcing frequency, f̃f , measured

between L̃ν and L̃‖. The left figure (Fig. 9a) displays the measures obtained for Re = 48,

and for three values of R̃I . The response of the system to the forcing frequency looks likea resonance curve, which becomes sharper and sharper as R̃I increases and presents acusp-like shape for R̃I = 0.43. A similar evolution is observed on the curves of the rightfigure (Fig. 9b), as Re is increased at a fixed R̃I = 0.48. Note that the results obtainedfrom the natural instability analysis are recovered: At constant Re, the frequency ofthe most unstable mode decreases as R̃I is increased (Fig. 9a and 5a) and the widthof the frequency peak decreases (Fig. 9a and 6a). At constant R̃I , the frequency of themost unstable mode increases as Re is increased (Fig. 9b and 5b) and the width of thefrequency peak decreases (Fig. 9b and 6b).

We could not determine the dispersion curves for values larger than Re = 31 andR̃I = 0.43, as L̃‖ becomes too small to enable reliable measurements. The vicinity ofthe threshold is once again difficult to investigate. Nevertheless, according to Godreche& Manneville (1998), the appearance of a cusp should be taken as a strong warningsignal of a pinch of the two spatial branches k̃+i (f̃) and k̃

−i (f̃). This infers that a non-

cuspy dispersion curve is a proof for a CU state. This is the case for the dispersioncurves measured for R̃I ≤ 0.36 at Re = 48 and for Re ≤ 20 at R̃I = 0.48, which gives,respectively, two lower bounds of the transition. Moreover, one can suspect from Fig. 9

Convective/absolute instability in miscible core-annular flow: Experiments 9

that the thresholds are close to R̃I ∼ 0.43 at Re = 48 and Re ∼ 31 at R̃I = 0.48. Thesetransitions are comparable with the estimations obtained from Fig. 6.

5. Discussions/Conclusion

We performed experiments of miscible non-buoyant core annular flows, which alwaysdeveloped an axisymmetric instability. Some features of the resulting flows suggested thatthey might be either convectively unstable (CU) or absolutely unstable (AU), dependingon the control parameters Re and R̃I . We have attempted to delineate experimentallythe transition CU/AU. To address the difficulties inherent to experiments, we discussthis experimental approach first, without the support of the numerical study presentedin the following companion paper (Selvam et al. (2008)). The results obtained from thepresent experimental work may be summarized as follows:

The standard measurements used to characterize instabilities, such as the wavelength,the frequency and the wave speed of small perturbations of the pseudo-interface, do notexhibit any steep variation which could indicate the transition. And yet, a jump couldhave been expected, as the selection process differs from CU to AU states. Indeed, themost spatially unstable mode, which dominates in a CU state should differ from theabsolute mode, observed in a AU state. Actually, although different, these two modesmerge at the transition. Therefore, the latter should not correspond to a discontinuity, butto a change of slope which is difficult to localize on the experimental curves characterizingthe evolution of the most amplified wave (Fig. 5).

As an absolutely unstable system behaves like a self-frequency resonator, we haveanalyzed the frequency content of our system in terms of the minimum of the frequencywidth rms(f̃)min, the filtering length L̃P (which was found almost identical to the quasi-parallel length L̃‖) and the rate of filtering α. As a result, rms(f̃)min, L̃P , L̃‖ and α

are good indicators of the states which are likely to be AU, at large values of R̃I andRe. But because their evolution are relatively smooth, we have difficulties to determineexperimentally a threshold for the transition.

According to the theoretical approach (see Couairon & Chomaz (1997a,b)), the healinglength, which denotes the length needed for a global mode to grow from the inlet, isinfinite at the absolute transition, and decreases with the distance to criticality. Thisproperty should provide a good criterium to analyze the evolution of either L̃P or L̃‖.However, it is important to stress the reasons why the lengths measured in experimentsmay not be comparable with the healing length, defined in a theoretical context. First,the latter was introduced considering a semi-infinite domain, for which the base stateholds at x̃ = 0. In our experiments, the boundary conditions are perturbed by the viscousentry length. Secondly, the characterization of the healing length assumes that no noiseis injected in the system, once the latter has been initially perturbed. In presence of apermanent noise, any CU system (absolutely unstable or not) will amplify it over a typicalfinite length. As a matter of fact, L̃P and L̃‖ do not diverge as the control parametersapproach the transition CU/AU.

Nevertheless, the analysis of the response of the system to a time harmonic forcing hasenabled us to get the spatial branches of the dispersion curve. And the cusp like shapeof these curves, which becomes sharper and sharper in the area of large radius and highReynolds numbers is a strong warning of the occurrence of an absolute region.

Theoretical analysis for non-parallel real flow, have shown that in some absolute un-stable state (for the so-called pulled fronts), global modes may be selected by the lineardispersion relation at the inlet (x̃ = 0) (see Couairon & Chomaz (1997a,b); Chomaz(2003)). In such a case, the inlet conditions should control the instability features (fre-

10 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

quency, healing length,..). On the contrary, in a convectively unstable state, these fea-tures are controlled by the noise. Therefore one can expect a different response of thetwo states to change of the boundary conditions. This can be experimentally achieved byusing different inlet nozzles. Figure 10 shows the evolution of the maximum frequency fora constant Re = 48 and different R̃I (left) and a constant radius R̃I = 0.48 and differentRe (right) for two different nozzles inner radius (0.5cm and 0.75cm). We observe thatthe selected frequency is almost the same for small Re and R̃I but rather different forhigher values. From this observation one could have infer that the system is AU whenthe frequency is inlet dependent. Unfortunately, the frequency mismatch are obtainedin experiments for wich L̃‖ ∼ L̃ν (see Fig. 6). Hence, the amplification of the noise is

totally achieved in the viscous entry length L̃ν , where the dispersion curve depends onthe injection nozzle. Consequently, a frequency mismatch can be observed even in a CUstate and no conclusion can been drawn from Fig. 10.

In conclusion, rms(f̃ )min, L̃P , L̃‖ and α behaviors and the evolution of the cusp likeshape of the forced spatial growth rate build a body of evidence for an absolute unstableregion to develop at high R̃I and high Re-number. A more quantitative determinationof the transition threshold is not achievable due to the inherent imperfection of experi-ments. These experimental results are discussed and compared to linear stability resultsand nonlinear numerical simulations in the following companion paper of Selvam et al.(2008). In this paper it is shown that the CU/AU transition is indeed very close to ourexperimental results.

Acknowledgement The authors would like to thanks E. Meiburg and B. Selvam forfruitful collaboration. MdO is supported by a PhD grant from the french Ministère dela l’Enseignement Supérieur et de la Recherche. LT was partially funded by the FrenchMinistry of Foreign Affairs through the Lavoisier felloship program.

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12 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

Figure 1. Sketch of the experimental set up used to achieve a core annular flow. The 1 m longcylindrical tube has a radius R = 1.0 cm; the core fluid is injected in the smaller tube (innerradius 0.75 cm) and the wall fluid is injected in the remaining annular (between the outer radiusof the small tube (.85 cm) and the wall tube R). The flow rate of each fluid is monitored by apump.

Figure 2. Left: Image of the instability pattern for a relative core fluid radius R̃I = 0.37; thecore fluid is blue dyed. Right: Image of the instability pattern using a laser sheet fluorescentillumination of the tube with the core fluid dyed with fluorescin (R̃I = 0.48). The well definedpseudo-interface between the two fluids, r̃I(x, t) is determined from these images. We can observestreamwise from the inlet: 1) the viscous entry length Lν , of the order of the tube radius;

2) a quasi-parallel length L̃‖ where the pseudo interface between fluids looks parallel; 3) anaxisymmetric instability pattern the shape of which is mushroom like. M = 25 and Re = 20.

Convective/absolute instability in miscible core-annular flow: Experiments 13

0 1 2 3−1

−0.5

0

0.5

1

| u| (

norm

alis

ée p

ar v

itess

e m

oy)

1

2

3

4

5

|Ṽ |

x̃

r̃

0 2 4 6 8 10 12−1

−0.5

0

0.5

1

Ṽ (r̃)

r̃

Figure 3. PIV measurements of the velocity field in the vicinity of the inlet across the wholetube diameter (M = 25, Re = 20, R̃I = 0.36). Left: Velocity field; The direction and theintensity of the velocity is indicated by an arrow and by the color, respectively. Note that thepseudo-interface contour between the two fluids is barely visualized. Right: Three transversevelocity profiles measured at distances 2.2, 2.7, 3.2 cm from the inlet. For clarity, the profileshave been shifted. The lines through the data correspond to the theoretical Poiseuille flow fieldfor parallel flows of two non mixing fluids.

14 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

0.5 1 1.5 2

0

20

40

60

80

r̃I(t̃)

t̃

(a)

0 2 4 6

10

20

30

40

50

60

70

80

0.4

0.5

0.6

0.7

x̃

t̃

(b)

0 0.5 1 1.5 2 2.5 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

f̃

A2(f̃)

(c)

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

x̃

A2m(x̃)

(d)

0 2 6 80

0.1

0.2

0.3

0.4

x̃

rms(

f̃)

rms(f̃)min

L̃P

α

(e)

0 1 2 3 4 5 6

0.5

1

1.5

2

2.5

3

x̃

f̃

(f)

Figure 4. a)Time (t̃) dependence of the interface location, r̃I(x̃, t̃), at four different locations,

x̃ = 1, 2, 3, 4, from the inlet. For clarity, the mean position (R̃I) has been shifted along thehorizontal axis x̃. b) Spatio-temporal diagram: The vertical axis is the time t̃, the horizontal

axis is the distance x̃ from the inlet and the grey level corresponds to the position R̃I(x̃, t̃) of the

interface. c) Power spectrum: Square of the amplitude of the modes, versus their frequency f̃ , ofthe Fourier transform of the temporal variations of the interface at x̃ = 4. d) Amplitude of themaximum frequency versus the position x̃ from the inlet. The linear spatial growth is measuredon the exponential growth part of this curve. e) Root mean square (RMS) of the power spectrum

versus x̃ and definition of the minimum of the RMS, rms(f̃)min, of the filtering length L̃p and

of the spatial rate of filtering α. f) Frequency f̃ versus x̃ diagram where the amplitude of the

power spectrum is coded in grey scale. M = 25, Re = 20, R̃I = 0.36.

Convective/absolute instability in miscible core-annular flow: Experiments 15

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

R̃I

f̃max

(a)

0 10 20 30 40 500

0.5

1

1.5

Re

f̃max

(b)

0 0.1 0.2 0.3 0.4 0.5 0.60

2

4

6

8

R̃I

k̃r

(c)

0 10 20 30 40 500

2

4

6

8

Re

k̃r

(d)

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

R̃I

c̃

(e)

0 10 20 30 40 500

0.5

1

1.5

Re

c̃

(f)

Figure 5. From top to bottom: Maximum frequency, f̃max (a,b), real part of the wave number

of the most amplified frequency, k̃r(f̃max) (c,d) and phase velocity c̃(f̃max) (e,f), versus R̃I at

constant Re = 48 (left figures) and versus Re at constant R̃I = .48 (right figures). The dashedline in the bottom two figures is the theoretical velocity for the short wave regime, c̃SW .

16 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

0 0.1 0.2 0.3 0.4 0.5 0.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

R̃I

rms(f̃)

(a)

0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Re

rms(f̃)

(b)

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

7

R̃I

L̃P,‖

(c)

0 10 20 30 40 500

1

2

3

4

5

6

7

Re

L̃P,‖

(d)

0 0.1 0.2 0.3 0.4 0.5 0.6

−0.8

−0.6

−0.4

−0.2

0

R̃I

α

(e)

0 10 20 30 40 50

−0.8

−0.6

−0.4

−0.2

0

Re

α

(f)

Figure 6. From top to bottom: (a,b): Mimimum of the RMS , rms(f̃)min (a,b), quasi-parallel

length L̃‖ (△) and filtering length L̃p (•) (c,d), and filtering rate α (e,f), versus R̃I at constant

Re = 48 (left figures) and versus Re at constant R̃I = .48 (right figures).

Convective/absolute instability in miscible core-annular flow: Experiments 17

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

R̃I

Re

(a)

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

R̃I

Re

(b)

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

R̃I

Re

(c)

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

R̃I

Re

(d)

Figure 7. Representation in the (Re-R̃I) plane of f̃max (a), the inverse of rms(f̃)min (b),the

inverse of the filtering length, 1/L̃P (c) and the absolute value of the filtering rate, |α| (d). Thesize of the data points is proportional to the depicted quantity.

18 M. d’Olce, J. Martin, N. Rakotomalala, D. Salin and L. Talon

1 2 3 4 5

10

20

30

40

x̃

t̃

(a)

1 2 3 4 5

10

20

30

40

x̃

t̃

(b)

0 1 2 3 4 5

0.5

1

1.5

2

2.5

3

x̃

f̃

(c)

0 1 2 3 4 5

0

0.5

1

1.5

2

2.5

3

x̃

f̃

(d)

Figure 8. Spatio-temporal and spatio-frequency diagrams obtained at Re = 38, R̃I = 0.48,in the absence of forcing (a, c) and with a modulation at f̃f = 0.43 (b, d). The forced modeis superimposed to the spectrums obtained without modulation (c), and appears closer to the

entry than the natural mode at f̃n = 0.73 (d).

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

f̃f

−k̃i

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

f̃f

−k̃i

Figure 9. Spatial growth rate −k̃i, versus the forcing frequency. Left: constant Re = 48 fordifferent R̃I (R̃I = .30 (•), R̃I = .36 (�), R̃I = .43 (N)). Right: constant R̃I = .48 for differentRe values (Re = 6 (•),Re = 10 (�) Re = 20 (N), Re = 31 (⋄)).

Convective/absolute instability in miscible core-annular flow: Experiments 19

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

R̃I

f̃max

0 10 20 30 40 500

0.5

1

1.5

Re

f̃max

Figure 10. Effect of the injection nozzle radius on the maximum frequency f̃max: R̃n = .75 cm(•), R̃n = .5 cm (◦). Left: at constant Re = 48, varying R̃I . Right at constant R̃I = 0.48, varyingRe.