Experimental and analytical study of the shear instability of a gas-liquid mixing layer Jean-Philippe Matas, Sylvain Marty, Alain H. Cartellier

To cite this version:

Jean-Philippe Matas, Sylvain Marty, Alain H. Cartellier. Experimental and analytical study of the shear instability of a gas-liquid mixing layer. Physics of Fluids, American Institute of Physics, 2011, 23, pp.094112. ￿10.1063/1.3642640￿. ￿hal-00644002￿

HAL Id: hal-00644002 https://hal.archives-ouvertes.fr/hal-00644002 Submitted on 3 Jan 2012

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Experimental and analytical study of the shear instability of a gas- liquid mixing layer Jean-Philippe Matas, Sylvain Marty, and Alain Cartellier

Citation: Phys. Fluids 23, 094112 (2011); doi: 10.1063/1.3642640 View online: http://dx.doi.org/10.1063/1.3642640 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v23/i9 Published by the American Institute of Physics.

Related Articles Modal versus nonmodal linear stability analysis of river dunes Phys. Fluids 23, 104102 (2011) On the transition between distributed and isolated surface roughness and its effect on the stability of channel flow Phys. Fluids 23, 104101 (2011) Optimal excitation of two dimensional Holmboe instabilities Phys. Fluids 23, 074102 (2011) Coupling of Kelvin–Helmholtz instability and buoyancy instability in a thermally laminar plasma Phys. Plasmas 18, 022110 (2011) Competitions between Rayleigh–Taylor instability and Kelvin–Helmholtz instability with continuous density and velocity profiles Phys. Plasmas 18, 022704 (2011)

Additional information on Phys. Fluids Journal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions PHYSICS OF FLUIDS 23, 094112 (2011)

Experimental and analytical study of the shear instability of a gas-liquid mixing layer Jean-Philippe Matas,a) Sylvain Marty, and Alain Cartellier Laboratoire des Ecoulements Ge´ophysiques et Industriels, CNRS, Grenoble University, 38041 Grenoble Cedex 9, France (Received 23 September 2010; accepted 16 August 2011; published online 29 September 2011) We carry out an inviscid spatial linear stability analysis of a planar mixing layer, where a fast gas stream destabilizes a slower parallel liquid stream, and compare the predictions of this analysis with experimental results. We study how the value of the liquid velocity at the interface and the finite thickness of the gas jet affect the most unstable mode predicted by the inviscid analysis: in particular a zero interface velocity is considered to account for the presence in most experimental situations of a splitter splate separating the gas and the liquid. Results derived from this theory are compared with experimentally measured frequencies and growth rates: a good agreement is found between the experimental and predicted frequencies, while the experimental growth rates turn out to be much larger than expected. VC 2011 American Institute of Physics. [doi:10.1063/1.3642640]

1=2 I. INTRODUCTION q 1 k 1:5 l (1) Airblast atomization is used to turn a liquid jet into a  qg! dg spray with a fast cocurrent air stream. This process is exploited in particular in aeronautics and space applications 1 The velocity of the waves can be well approximated by the for the injectors of several propulsion systems. The mecha- following convection velocity, estimated from a pressure nisms responsible for the break-up of the liquid jet are only balance between the liquid and gas in a frame moving with partially understood, see reviews by Lasheras and Hopfin- the waves7,12,14: ger,2 and Eggers and Villermaux.3 However, an improve- ment of injection techniques is needed to notably limit the pqlUl pqgUg þ emission of pollutants and to increase the reliability of these Uc (2) ¼ pqg pql engines. ffiffiffiffi þ ffiffiffiffiffi It has been shown that the initial destabilization of the The frequency of the wavesffiffiffiffiffi can thenffiffiffiffi be approximated by interface in airblast atomization is caused by a shear instabil- f U =k, with k is the wavelength. Experimental results show ¼ c ity akin to a Kelvin-Helmholtz instability, but whose most wavelengths scaling with dg and frequencies scaling as Uc=dg 6–8,12,13 unstable mode is controlled by the thickness dg of the gas as predicted above, but measured frequency values are vorticity layer. See for example the original work by Ray- consistently larger than predicted. There is also a disagreement 4 leigh with the inclusion of a velocity profile in the shear between existing experimental data: Raynal et al.7,12 find ex- 5 instability, or the work of Lawrence et al. where both a ve- perimental frequencies about 50% larger than predicted ones, locity and density profile are taken into account. Several whereas Ben Rayana et al.9,13 on the same planar geometry 6–9 recent studies have focused on the case of gas-liquid shear and Marmottant and Villermaux8 on a coaxial geometry meas- layers, relevant to atomization applications. When a stability ured a factor two or three between the theory and experiments. analysis is carried out with both a gas and a liquid boundary If viscosity is taken into account in the temporal analy- layer (see velocity profile of Fig. 1(a)), the most unstable sis, an additional unstable mode is found,15,16 with a much mode turns out to be directly controlled by the liquid vortic- shorter wavelength than both the inviscid and the experimen- 10,11 ity layer dl, and not the gas one. This is not what is tal modes. For density ratios corresponding to air=water 7,12 observed in the experiments of Raynal, Marmottant and experiments Boeck and Zaleski15 find that this viscous mode 8 9,13 Villermaux, and Ben Rayana. This inconsistency has is the most unstable one. However, contrary to the inviscid been resolved by pointing out that the time needed for the and the experimental modes, this mode does not scale with liquid shear layer to grow by viscous diffusion is large com- the gas vorticity thickness dg: we therefore choose not to 3,8,12 pared with the time needed for the gas mode to grow. include viscosity in our stability analysis, and will focus Under the assumption that there is a vorticity layer only instead on the improvement of the inviscid analysis. We will on the gas side (see Fig. 1(b)), an explicit expression for the discuss later on what our results suggest as to how viscosity dispersion relation is then found; in the limit of large Weber may be included in the analysis. and small Richardson numbers the wavenumber of its most In order to clarify the discrepancies between the inviscid 3 unstable temporal solution is found numerically to scale as : stability analysis and experimental results, we have extended the analysis to integrate key aspects of the injection: a more realistic initial velocity profile accounting for the necessary a)Electronic mail: [email protected]. presence of a splitter plate in the experiments, and the finite

1070-6631/2011/23(9)/094112/12/$30.00 23, 094112-1 VC 2011 American Institute of Physics

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-2 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011)

FIG. 1. (a) Velocity profile with two vorticity thicknesses; (b) Velocity profile with a gas vorticity thickness only; and (c) Profile exhibiting a velocity deficit at the interface.

thickness of the gas stream. We will show in Sec. II that face lower than the bulk liquid velocity, has been observed in these changes significantly modify the predictions of the our experimental set-up with particle image velocimetry (PIV) inviscid stability analysis. We will then compare in Sec. III measurements (seeding with rhodamine and Nd-YAG laser these predictions with both existing and new experimental slice). Only a couple of velocity vectors could be obtained results. within the liquid boundary layer, so its full velocity profile could not be measured, but we could check that even in the II. INVISCID ANALYSIS presence of the fast gas flow the liquid velocity remained smaller in the liquid boundary layer than in the bulk of the The stability analysis is carried along the lines of the 7 12 liquid, up to several centimeters downstream the injection. inviscid analysis carried out by Raynal et al. and Raynal. An error function velocity profile can be used in the We note U the base flow, and u (u, v, w) the velocity per- ¼ analysis instead of the linear velocity profiles of Fig. 1: how- turbation. The position of the interface is given by g. The ever, this more realistic profile yields exactly the same pre- pressure perturbation is p. We note Ui the base flow velocity dicted unstable modes for our inviscid case, provided the gas at the interface. The influence of the surface tension r and of D dU vorticity thickness dg U= dz max is the same in both cases. gravity will be neglected in this section. We therefore choose to¼ present here results obtained with the

We linearize the classical equations of momentum and more straightforward linear velocity profile. Note that though mass conservation, and look for normal mode solutions viscosity is not included in the derivation of the stability of u~ i kxx kyy xt kx; ky; z; x e ð þ À Þ of the resulting system. Solving for the velocity perturbation, it is of course implicitly needed theð verticalÞ component of the perturbation velocity w~,we 17 upstream to generate the base flow profiles of Fig. 1. obtain the following equation : For all three base flows of Fig. 1, Eq. (4) then reduces to ~ d dw dU ~ 2 ~ d2w~ q x kxU qkx w qk x kxU w 0 2 ~ dz ðÀ þ Þ dz À dz À ðÀ þ Þ ¼ k w 0  dz2 À ¼ (3) kz Two solutions are built in the gas phase: w1 AeÀ in the 2 2 2 ¼ kz kz where k k k . Away from the interface, q is constant unbounded constant velocity region and w BeÀ Ce in ¼ x þ y 2 and this equation reduces to: the boundary layer; similarly two solutions¼ are builtþ in the kz kz kz liquid phase: w DeÀ Ee and w Fe . These solu- 2 ~ 2 3 ¼ þ 4 ¼ d w 2 d U tions are built under the assumption k > 0 (waves propa- q x kxU k w~ qkx w~ (4) ðÀ þ Þ dz2 À ¼ dz2 <ðÞ  gating downstream). Continuity of w at the limit between constant flow and linear flow zones in each phase, and across As discussed in the introduction, the nature of the solution the interface, i.e., at z d , 0 and d , gives three relations ~ g g for w depends on the velocity profile chosen for U(z). If a lin- between the six integration¼À constants. Three additional jump ear profile is adopted, then the term on the right hand side of conditions are obtained by integrating Eq. (3) across the Eq. (4) is zero: an analytical solution can easily be provided same three locations (i.e. across corner points of the velocity ~ for w, and with proper jump conditions at the interface yield profile)17: an expression for the dispersion relation.

kdg kdg kdg kdg x kxUg kAeÀ BeÀ Cke kxc AeÀ 0 ðÀ þ Þ À þ À g ¼ A. Effect of a velocity deficit kdl kdl kdl kdl x kxUlÂÃk DeÀ Eke Fe kxclFe 0 ðÀ þ Þ À þ À À ¼ We have studied the stability of the base flow showed x ÂÃkxUi k q B C q D E on Fig. 1(c). This base flow typically exists just at the exit of ðÀ þ Þ gðÀ þ ÞÀ lðÀ þ Þ hi the injector in the experimental set-up, due to the solid plate qgkxcg B C qlkxcl D E 0 separating the liquid and gas flows. Viscosity will lead to a À ð þ Þþ ð þ Þ¼ diffusion of momentum towards the interface, and the profile where c (U U )=d and c (U U )=d . We have g ¼ g À i g l ¼ i À l l of Fig. 1(c) is expected to hold only over a short distance, taken ky 0, meaning that we look for modes propagating in and evolve eventually towards the profile of Fig. 1(a). How- the x direction.¼ The dispersion relation is obtained by writing ever, a velocity deficit, meaning a velocity close to the inter- that the determinant of this system must be zero:

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-3 Experimental and analytical study Phys. Fluids 23, 094112 (2011)

01 11 10 1 1 e2K À00À 0 À À 2K=n 00 0e 11 r 1 ai n al ai r 1 ai n al ai À À 0 r ð À Þþ ð À Þ r ð À Þþ ð À Þ 1 10 À À X Kai À X Kai À 0 À þ À þ ¼ 1 ai 1 À 1 e2K 00 0 À þ X K À a a À þ 2K=n i l 00 0e 1 1 n X À À À À Kal À þ

where r qg=ql is the density ratio, n is the vorticity thick- growth rate increases, while for larger al an increase in Ul ness ratio¼n d =d , a U =U , and a U =U are the inter- causes a decrease in the growth rate. Figure 4 compares the ¼ g l i ¼ i g l ¼ l g face and liquid velocities nondimensionalized by Ug. The variations of the frequency and growth rate of the most unsta- frequency and wavenumber are made nondimensional by set- ble mode as a function of the velocity ratio al in the case ting X xd =U and K kd . where there is a velocity deficit (solid curve, a 0) and when ¼ g g ¼ g i ¼ This equation is solved numerically for spatial solutions: the interface velocity is equal to Ul (dotted line, ai al). The the variable is the real x, and the dispersion relation is difference between the two configurations increases¼ with solved for the complex wavenumber k kr iki. The fre- increasing al, up to a factor three in the frequency when al quency retained is the one which minimizes¼ þ the (negative) becomes close to 0.1. The variations of the growth rate show spatial growth rate ki. Note that with the form of the pertur- that for low al the velocity deficit enhances the instability, 3 bation taken above, the perturbation is unstable if ki < 0. Fig- while for al > 5.10À it lowers the magnitude of the growth ure 2 shows typical variations of kr and ki, as a function of rate compared with the configuration without a deficit. When nondimensionalized frequency xdg=Ug. The dotted curves the velocity deficit is progressively reduced, i.e., 0 < Ui < Ul, represent the curves obtained with the velocity profiles of the results obtained without any liquid boundary layer (Fig. Fig. 1(b) (corresponding to ai al), while the solid curve is 1(b)) are continuously recovered. If a gravity field perpendicu- obtained with a velocity profile¼ exhibiting a velocity deficit lar to the interface is included, it will damp large wavelength at the interface (ai 0, Fig. 1(c)). It can be seen that for the modes and therefore increase the most unstable frequency. ¼ 3 2 conditions of Fig. 2 (r 10À , al 10À , and n 1) the For the conditions of Fig. 4, which are typical of the experi- unstable mode obtained¼ with a velocity¼ deficit has¼ a larger ments presented in Sec. III, the frequency would typically be frequency. The absolute value of the (negative) growth rate increased by about 10%. If surface tension is included it will is also slightly larger. Note also that when the velocity deficit on the contrary decrease slightly the most unstable frequency is taken into account the unstable mode scaling with dl, (by about 2% for the same typical conditions). which would be dominant3 if the analysis were carried out We now plot on Fig. 5 the results of the stability analysis with the profile of Fig. 1(a), is not observed: a large viscous when the ratio n of the gas to liquid vorticity layer is varied diffusion time therefore need not be invoked anymore to jus- in the range 0.15–10. It can be seen that when n is larger tify that the most unstable mode scales with dg. than 0.5, the frequency, growth rate and velocity of the most We plot on Fig. 3 the influence of the velocity ratio al on unstable mode are roughly independent of n. When n is the spatial wavenumber and growth rate. The interface veloc- decreased below 0.5, i.e., when the liquid vorticity layer ity is taken to be zero (a 0). The dimensionless frequency becomes more than twice as large as the gas vorticity layer, i ¼ of the most unstable mode increases with increasing al. It can the growth rate and wavenumber are significantly increased, 4 2 be seen that when al is varied in the range 5.10À 4.10À the while the group velocity decreases significantly (steeper À slope of the wavenumber curve). A consequence of this

FIG. 2. (a) Dimensionless wavenumber kr and (b) growth rate ki obtained by stability analysis, as a function of dimensionless frequency xdg=Ug. The solid curve is obtained with a base flow having a full velocity deficit at the FIG. 3. Dimensionless wavenumber kr and growth rate ki obtained by stabil- interface (ai 0), and the dotted curve with an interface velocity equal to ity analysis, as a function of dimensionless frequency xdg=Ug, for ratios ¼ 3 2 4 3 3 3 2 2 2 2 the liquid velocity (ai al). Density ratio r 10À , velocity ratio al 10À Ul=Ug 5.10À ,10À , 2.10À , 5.10À ,10À , 2.10À , 4.10À , 6.10À , and ¼ ¼ ¼ 2¼ 3 and vorticity thickness ratio n 1. 9.10À (from left to right). a 0, n 1, and r 10À . ¼ i ¼ ¼ ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-4 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011)

sion in powers of r, we introduce the dynamic pressure ra- tio M q U2= q U2 , and assume M to be of order 1 or ¼ g g l l larger. We then set Ul Ug r=M in the dispersion relation. ÀÁ¼ pffiffiffiffiffiffiffiffiffi • Based on the scaling of Eq. (1) for the wavenumber, we look for a dimensionless wavenumber K of the form K K0pr, where K0 O(1). Given the scaling of the phe- nomenological¼ convection¼ velocity (equation 2), we there- ffiffi fore introduce X X0r with X0 O(1) as well. ¼ ¼ Only the lower order terms in r are kept in the dispersion FIG. 4. (a) Variations of the dimensionless frequency and (b) growth rate of relation, yielding the simplified dispersion relation: the most unstable mode as a function of the liquid to gas velocity ratio al. Solid curve: velocity deficit at the interface; dotted curve: no velocity deficit. 3 2 2 2 K0 K0 K0 2 The vorticity thickness ratio n dg=dl is fixed at n 1, and the density ratio X0 K0 2 X0 K0 0 3 ¼ ¼ at r 10À . À þ pM þ pM þ M þ ¼ ¼  ffiffiffiffiffi ffiffiffiffiffi result is that since experimental conditions on our set-up cor- We solve this equation for temporal solutions and find 2 respond to situations where dg > dl, the stability analysis pre- X0 K0=2 K0 2=M6pK 4 . An unstable solution ¼ð Þð þ 0 À Þ dicts that for these conditions the thickness dl of the liquid will appear for K < 2. The most unstable solution when K is 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 vorticity layer will not affect the nature of the unstable mode varied is found for K0 p2, corresponding to ¼ when there is a velocity deficit. X0 1 2=M i . The most unstable mode dimensional wavenumber,¼ð þ growthþ Þ rate, frequency,ffiffiffi phase velocity v , and pffiffiffiffiffiffiffiffiffi u B. Asymptotic analysis and influence of the liquid on group velocity vg are therefore characterized by: the gas mode • k p2pr Motivated by the fact that previous studies found a rela- ¼ dg • Ug Ug 1=2 tively simple dependence of the most unstable mode wave- xi rffiffiffi ffiffiand xr r 1 p2MÀ ¼ dg ¼ dg þ length upon flow parameters in the case without a velocity • p2 v/ prUg Ul andÀÁvg pffiffiffi 2prUg Ul deficit (see Eq. (1)), and in order to find how this scaling is ¼ 2 þ ¼ þ modified when there is a velocity deficit, we now carry out Theseffiffi expressionsffiffi agree extremelyffiffiffi ffiffi well with the results an asymptotic analysis on the dispersion relation, around the found when the complete dispersion relation is solved unstable mode identified in Sec. II A. We first make this numerically. They are also consistent with the coefficients expansion for the case where there is a single vorticity layer found numerically in previous studies (Eq. (1)). The phase in the gas phase (base velocity profile of Fig. 1(b). We make velocity is slightly smaller than the convective velocity of several assumptions: Eq. (2). A new and interesting result is the correction in M to the frequency of the most unstable mode: it analytically pre- • We assume that r 1. This is true in the air=water case, dicts how the liquid velocity will affect this mode. but will also hold in any gas=liquid case at moderate abso- We now apply the same method to the case of Fig. 1(c), lute pressures. Only lower order terms in r will therefore when there is a velocity deficit at the interface. The same be retained in the expansion. • assumptions are made on the magnitude of the liquid We assume Ul Ug: this is true in configurations where Kelvin-Helmholtz instability is observed in air=water con- ditions. In order to include this assumption in the expan-

FIG. 6. Frequency of the most unstable mode found when the full dispersion FIG. 5. (a) Dimensionless wavenumber kr and (b) growth rate ki as a func- relation is solved numerically for spatial solutions (solid line), and predicted Ug 5 1=2 tion of dimensionless frequency xdg=Ug, for gas to liquid vorticity thickness by the asymptotic expansion xr r 1 p2MÀ (dotted line), when ¼ dg þ 2 ratio n dg=dl 0.15, 0.2, 0.5, 1, and 10 (from left to right). The liquid to the velocity ratio al is varied. The base velocity profile is taken to have a full ¼ ¼ 2 3 ÀÁffiffiffi gas velocity ratio is fixed a 10À and the density ratio r 10À . velocity deficit at the interface (a 0). l ¼ ¼ i ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-5 Experimental and analytical study Phys. Fluids 23, 094112 (2011)

FIG. 7. Velocity profile used for the analysis: the gas jet has a finite thick- FIG. 8. Growth rate as a function of frequency, for different thicknesses of ness Hg. the gas stream Hg: solid line Hg=dg 125; dotted line Hg=dg 25; dashed line ¼ ¼ 3 Hg=dg 10; dash-dotted line Hg=dg 2.5; other parameters are r 10À , ¼ 2 ¼ ¼ a a 10À ,andn 1. velocity, and on the dimensional frequency and wavenum- l ¼ i ¼ ¼ ber. In addition, the liquid vorticity thickness has to be such 3 1 velocity ratio a is varied in the range 10À 10À . The solid that K=n 1, which is equivalent to kd 1: the liquid vor- l l line is the same curve shown on Fig. 4, andÀ the dotted line ticity thickness has to be small compared with the wave- corresponds to the expression derived above. It can be seen length of the instability. These approximations lead to the that the agreement is rather good as long as a < 0.04. This following simplified dispersion relation: l limit corresponds to M > 1, which is precisely the condition

3 2 4 on the liquid velocity needed to derive the analytical 3 2K0 2 2 2K0 K0 2 K0 X0 K0 X0 K0 X0 0 expression. þ pM þ þ pM þ M þ þ M ¼  The expressions given in this section correspond to a temporal instability: it would be tempting to deduce from ffiffiffiffiffi ffiffiffiffiffi We solve it for temporal solutions in the limit of large M and them a similar asymptotic expression for the spatial growth find: rate ki, from the temporal growth rate xi and the group ve- locity based on the Gaster’s relation18: k x =v . However, • 3 1=2 pr i i g k p2 MÀ ¼ ¼ð þ 2 Þ dg the expression found for ki by this method turns out to be • Ug 1=2 Ug 5 1=2 xi r ffiffiffi 1 p2MÀ ffiffi and xr r 1 p2MÀ quite different from the expression numerically deduced ¼ dg ð þ Þ ¼ dg þ 2 • p2 7 5 from the dispersion relation. This is because Gaster’s crite- v/ prUg ffiffiffi Ul and vg p2prUÀÁg Ul ffiffiffi 2 4 2 rion for this equivalence, namely xi xr, is not met here: ¼ þ ¼ þ  Theseffiffi expressionsffiffi agree veryffiffiffi wellffiffi with the numerical it is incompatible with the hypothesis that M is larger than solutions given when the full dispersion relation is solved, one. 1=2 including for spatial solutions. Note that the MÀ behav- iour is equivalent to a linear dependence on al, which is what was evidenced on Fig. 4. More precisely, Fig. 6 shows how C. Effect of a finite gas thickness the analytical expression for the frequency compares to the We now modify the base flow (see Fig. 7) in order to solution to the full dispersion relation, when the liquid to gas take into account the finite thickness Hg of the gas stream.

FIG. 9. Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of Hg=dg, for differ- 4 4 ent density ratio r; h: r 10À ; : r 3.10À ; 3 ¼3 Â ¼2 o: r 10À ;*:r 5.10À ; : r 10À . Other ¼ ¼ 2þ ¼ parameters are a a 10À and n 1. l ¼ i ¼ ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-6 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011)

FIG. 10. Variation of the (a) real part and (b) imaginary part of the wavenumber of the most unstable mode as a function of Hg=k, for differ- 4 4 ent density ratio r; h: r 10À ; : r 3.10À ; 3 ¼3 Â ¼2 o: r 10À ;*:r 5.10À ; : r 10À . Other ¼ ¼ 2þ ¼ parameters are a a 10À and n 1. l ¼ i ¼ ¼

We introduce d , the thickness of the outer vorticity layer. and real part of the wavenumber in the limit H . Figure ge g !1 The linear stability analysis is then carried out similarly as 10 shows that the thickness of the gas stream Hg starts to before: solutions are found in the liquid, inner vorticity layer, affect the wavelength and growth rate of the most unstable gas stream, outer vorticity layer, and outer region (where mode when Hg=k < 0.1: seen the other way round, if the there is no gas flow). These solutions are matched at the wavelength is larger than 10 Hg then the wavenumber and boundaries, and the resulting dispersion relation is solved for growth rate will be larger than they would be for a gas spatial solutions. It can be seen on Fig. 8 that when Hg is stream of infinite thickness. The same behaviour is observed decreased, the growth rate is increased, and the frequency is for the frequency of the most unstable mode: it increases increased. The effect of the finite thickness appears below a when Hg is decreased below 0.1 k, though the values for dif- threshold value: on the results of Fig. 8, which were obtained ferent density ratios are less collapsed than for the wavenum- 3 for a density ratio r 10À the influence of H becomes rele- ber (see Fig. 11(a). All these results were obtained for the ¼ g vant for Hg=dg < 10. In order to investigate if Hg=dg is the velocity profile of Fig. 7, i.e., without a velocity deficit. If correct dimensionless parameter controlling this effect, we the velocity deficit introduced in Sec. II A is taken into 4 2 have varied the density ratio r in the range 10À 10À : this account (ai 0), the same collapse of the data are observed, allows to vary the wavelength of the instability independentÀ and the same¼ curves are obtained for the wavenumber and of the vorticity thickness. Figure 9 shows that for different frequency. Only the growth rate displays a slightly different ratios r, the threshold ratio Hg=dg below which the instability behaviour, with a larger ki=ki Hg at small Hg=k (compare is affected is not constant: the threshold ranges from Figs. 10(b) and 11(b)): the influenceð !1Þ of the finite thickness 4 2 Hg=dg 50 for r 10À ,toHg=dg 5 for r 10À . Our hy- on the growth rate is stronger when a velocity deficit is pres- pothesis is that the¼ relevant lengthscale is the¼ wavelength of ent. All these results were obtained for an external vorticity the instability k, which also controls the extension of the ve- thickness dge equal to the inner vorticity thickness dg.Ifdge locity perturbation in the gas phase (since the velocity pertur- is varied, the same frequencies and growth rates are kz bation is of the form w eÀ ). We show on Fig. 10 that the obtained: this parameter has no influence on the unstable  ratios ki=ki Hg and kr=kr Hg , respectively, collapse on modes. ð !1Þ ð !1Þ a single curve when plotted as a function of Hg=k. The val- In our experimental case Hg was kept constant equal to ues k and k are, respectively the imaginary 1 cm, which is also the typical order of magnitude of the iHg rHg ðÞ!1 ðÞ!1

FIG. 11. (a) Variation of the frequency of the most unstable mode with Hg=k, for different 4 4 density ratio r; h: r 10À ; : r 3.10À ;o: 3 ¼3 Â ¼2 r 10À ;*:r 5.10À ; : r 10À . Other pa- ¼ ¼ þ2 ¼ rameters are a a 10À and n 1. (b) Varia- l ¼ i ¼ ¼ tion of the growth rate with Hg=k, same parameters except a 0. i ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-7 Experimental and analytical study Phys. Fluids 23, 094112 (2011)

FIG. 12. Measurements of Ben Rayana9: (a) frequency of the Kelvin- Helmholtz instability as a function of the gas velocity, for different thicknesses Hg. (b) Same data plotted as a function of Ug=dg: the series are collapsed.

wavelength: Hg is therefore not expected to modify the fre- III. EXPERIMENTAL RESULTS quency. In the coaxial set-up of Marmottant and Viller- Our experimental set-up is a mixing layer experiment: a maux,8 where H 1.7 mm which is smaller than the typical g liquid sheet of water (width 10 cm, thickness H 1cm)is k (of the order of¼ a centimeter), it should on the contrary l destabilized by a parallel gas flow of air (width 10¼ cm, thick- affect the value predicted by the linear stability analysis. ness H 1 cm). For the conditions considered here, gas ve- This effect might explain why their experimental growth g locity is¼ in the range U 10–30 m=s and liquid velocity rates were significantly larger than the ones observed on pla- g U 0.1–1 m=s. In order to¼ ensure a steady liquid flow, liq- nar experiments. We will not address this coaxial geometry l uid¼ is injected from an overflowing reservoir located above here, but will do so in a future study including also the effect the experiment, with gravity driven flow. The air flow goes of the cylindrical geometry on the instability. through a honeycomb, a porous plate and a convergent, and Ben Rayana9,13 carried out frequency measurements on the liquid flow through two honeycomb plates and a symmet- the same planar set-up used for this study (this set-up will be ric convergent, all aimed at reducing velocity perturbations described in Sec. III, see Fig. 13) for different values of H , g in each flow (see Fig. 13). The gas vorticity thickness the corresponding data are shown on Fig. 12. It can be seen D dU dg U= dz max has been measured with a hot-wire ane- that the data for the three different Hg investigated are col- ¼ 1=2 mometer, and was found to vary as dg 6HgReÀ for the lapsed on the same curve when plotted as a function of Uc=dg 12,13 ¼ conditions of our experiment , where Re U H = with (Fig. 12(b). This indicates that H does not modify the fre- g g g  the kinematic gas viscosity. A small amount¼ of fluores- quency of the unstable mode, other than via the vorticity ceine is added in the liquid phase, we make a longitudinal thickness d . For the data of Fig. 12, k varies in the range g laser sheet (Argon laser) of the liquid flow, and a fast camera 1.2 cm–2.5 cm (Ben Rayana), and three values of H were g (Phantom v12) records the section of the jet. Hence, the loca- investigated: H 5 mm, H 10 mm, and H 20 mm. g g g tion of the interface can be obtained by image processing as This makes for¼ a ratio H =k¼ in the range 0.2–1.7,¼ i.e., g a function of the downstream position, and of time. The above the threshold H =k 0.1 identified in the analysis of g height of the interface at a given downstream position is then Sec. II C. ¼ Fourier transformed using the Welch’s method with MATLAB. If the number of images is large enough (typically several hundred periods), a peak is observed to dominate the noise in the spectrum. Its maximum is then identified as the fre- quency of the instability. Figure 14(a) shows two typical spectra obtained by this method: depending on the gas=liquid velocities the aspect ratio of the maximum peak can vary. For given flow conditions, this spectrum can be made at dif- ferent downstream locations: Fig. 14(b) shows spectra com- puted every 2dg, up to a downstream position x 65dg (x is counted from the injector). The amplitude of the¼ maximum of the spectrum increases with downstream location. This is due to the increasing amplitude of the waves, as the liquid interface goes from undergoing a slight oscillation to being atomized. Note that though the value of the maximum peak is modified, the corresponding frequency remains unchanged. Beyond x 65dg the location of the highest am- plitude spectrum shown¼ of Fig. 14(b), the amplitude FIG. 13. (Color online) Sketch of the experimental set-up. decreases, but again the frequency of the maximum remains

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-8 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011)

FIG. 14. (Color online) (a) Example of a spectrum of the position of the inter- 1 face, for Ug 22 m sÀ : solid line ¼ 1 Ul 0.26 m sÀ ,dashedline ¼ 1 Ul 0.5 m sÀ . The insert, on a larger ¼ 1 scale, shows harmonics for Ug 12 m sÀ 1 ¼ and Ul 0.26 m sÀ ; (b) Downstream variation¼ of the spectrum of the amplitude 1 of the instability: Ug 12 m sÀ , 1 ¼ U 0.26 m sÀ . The spectrum is com- l ¼ puted every 2dg,uptox 65dg:theam- plitude of the maximum¼ increases with downstream distance.

the same. This is different from the behaviour observed by (ai 0). It can be seen that the agreement with the velocity Fuster et al.19 on their numerical simulation of a similar deficit¼ prediction is very good. We now compare on Fig. 17 liquid-gas configuration, where the frequency decreases with our experimental results to the experimental results of Ray- the downstream distance. nal et al.7 (crosses) and Ben Rayana9,13 (squares). As men- This frequency is measured for various liquid and gas tioned in the introduction, these frequencies are both larger velocities. Below the smallest gas velocity investigated than the classical inviscid prediction without a velocity defi- 1 (Ug 12 m sÀ ) the amplitude of the perturbation is too small cit, shown as a dotted line. Our prediction derived from the ¼ 1 given our optical resolution. Beyond Ug 30 m sÀ , measure- analysis with a velocity deficit (solid line) is in a rather good ment is made difficult by the break-up of¼ the waves; in addi- agreement with Raynal data, except for large values of M. tion, we took care to measure the growth rate of the instability This could be due to the change in topology when M conjointly with the frequency for each set of conditions (the becomes large: due to the strong atomization the length L of 1 method will be detailed below), and for Ug > 30 m sÀ the intact liquid core shortens: L 6Hg=pM, as proposed by the rapid saturation of the waves impairs the measurement of Raynal.12 Hence the interface becomes steeper and the quasi the growth rate. Figure 15 shows that the frequency increases parallel flow assumption is no longer valid.ffiffiffiffiffi Ben Rayana data with both the gas velocity and the liquid velocity. (obtained for a fixed M 16) is also closer to our prediction The analysis of Sec. II has predicted a dimensionless than to the classical one,¼ but it is still at about twice the pre- 1=2 frequency varying with MÀ ,ifM is not small. We there- dicted value. Ben Rayana data (all for M 16) were ¼ fore plot on Fig. 16 the variations of fdg=Ug as a function of obtained by visualization, through counting of the number of 1=2 MÀ : the data of Fig. 15 collapse on a single curve. The waves over a given duration, while both Raynal and our dotted line shows the prediction derived for M > 1 for the ve- method are the spectral methods, which identify the maxi- locity profile of Fig. 1(b), while the solid line represents the mum of a spectrum derived from the interface location. prediction when a full velocity deficit is taken into account

1=2 FIG. 16. (Color online) Dimensionless frequency as a function of MÀ .The 1=2 dotted line is f r=2p 1 p2MÀ (asymptotic prediction for profile of ¼ð Þð þ Þ 5p 1=2 FIG. 15. (Color online) Experimental frequency as a function of the liquid Fig. 1(b) and the solid line is f r=2p 1 2 2MÀ (asymptotic predic- 1 1 ¼ðffiffiffi Þð þ Þ velocity Ul, for different Ug: *: Ug 12 m sÀ ; h: Ug 17 m sÀ ; : tionforprofileofFig.1(c). Symbols correspond to different values of Ug, *: 1 1 ¼ ¼  1 1 ffiffiffi 1 1 U 22 m sÀ ; : U 27 m sÀ . U 12 m sÀ ; h: U 17 m sÀ ; : U 22 m sÀ ; : U 27 m sÀ g ¼  g ¼ g ¼ g ¼  g ¼  g ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-9 Experimental and analytical study Phys. Fluids 23, 094112 (2011)

al (Fig. 6), but what is unexpected is that the experimental points end up closer to the asymptotic law than to the exact numerical solution of the dispersion relation. Figure 18(b) next shows the same ratio fexp=fth computed for the experi- mental points of Raynal et al.7 and Ben Rayana.9 The data points of Raynal et al. (crosses) lie along a curve going from a ratio fexp=fth 0.7 to a ratio 2, across the several orders of magnitude of¼ M spanned by their data. Though for a given value of M there is a strong dispersion in the frequency ratio, this global trend suggests that M has a strong influence on the departure to the prediction of the linear stability analysis. As mentioned in the discussion of Fig. 17, this could be due to the parallel flow assumption not being valid at large M, in particular when the liquid intact length L becomes shorter than the wavelength. The same ratio fexp=fth is shown for the data of Ben Rayana, all for a fixed M 16: the ratio of fre- ¼ FIG. 17. Comparison of data sets for the dimensionless frequency as a func- quencies is larger for these measurements than for other data 1=2 7,12 h 9,13 tion of MÀ : x results of Raynal ; results of Ben Rayana ; results sets, in line with the difference observed on Fig. 17. As men- 1=2  of the present study. The dotted line is f r=2p 1 p2MÀ (asymp- tioned in the preceding paragraph, this could be due to a bias totic prediction for profile of Fig. ¼ð1(b) andÞð þ the solidÞ line is 5p 1=2 ffiffiffi introduced by the counting method used by Ben Rayana. f r=2p 1 2 2MÀ (asymptotic prediction for profile of Fig. 1(c). ¼ð Þð þ Þ We also carried out measurements of the spatial growth ffiffiffi Spectra for large M are usually quite noisy and display rate ki of the instability, using the following method: we strong harmonics: this may explain why the frequency meas- superpose each captured interface; for a given downstream ured by Ben Rayana et al.9 as an average frequency is signif- location, we make a histogram of the interface positions icantly larger than the maximum frequency retained by (typically over several hundred periods of the instability), spectral methods. and exclude the lowest and highest 1% events: this procedure In order to be more precise on the comparison of the lin- aims at removing single events (a much larger isolated wave ear stability analysis frequency and the experimental fre- for example) which might occur over the course of the mea- quency, we plot on Fig. 18(a) the ratio of the experimental to surement. The remaining width of the histogram is then predicted frequencies, as a function of M. The predicted fre- taken to be the amplitude A of the waves. Figure 19(a) illus- quency is here the solution of the spatial inviscid stability trates this procedure: the solid line shows the position of the analysis, taking into account a full velocity deficit in the liq- interface for a given time t; the histograms below were made uid phase, as well as gravity and surface tension. Though at the locations shown by the three dot dashed lines; the this ratio is in average close to one (which is not the case if dashed lines on the histograms show the limit value retained the velocity deficit is not taken into account, as evidenced by for the amplitude after the exclusion of the lowest and high- Fig. 16), the points are quite scattered: the ratio varies est 1%; the dashed line wrapping the interface corresponds between 0.8 and 1.2 for the conditions of our present experi- to this same limits once reported on the mixing layer. The ment. Note that for low M the theory tends to overestimate amplitude is plotted as a function of downstream position, 1 the experimental frequency (ratio f =f as low as 0.8), a for a fixed Ul 0.37 m sÀ and for the different Ug investi- exp th ¼ behaviour not observed on Fig. 16 where the experimental gated, see Fig. 19(b). It can be seen that there is a region of frequency was compared to the asymptotic prediction: this exponential growth (enhanced by the dashed line on the difference is expected since the asymptotic law of section B graph), whose extent decreases when Ug is increased: after a departs from the full solution precisely for low M, i.e. large limit position, the growth becomes slower, indicating a

FIG. 18. (Color online) (a) Ratio of the experi- mental frequency and the predicted frequency, as a function of M. Symbols correspond to dif- 1 ferent values of Ug, *: Ug 12 m sÀ ; 1 ¼ 1 h: Ug 17 m sÀ ; : Ug 22 m sÀ ;*: ¼ 1 Â ¼ Ug 27 m sÀ . (b) Same plot with the data of Raynal¼ (crosses) and Ben Rayana (diamonds).

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-10 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011)

FIG. 19. (Color online) (a) Illustration of the growth rate measurement (Ug 12 m=s, Ul 0.37 m=s): a histogram of the interface positions at a given down- stream position is made; the amplitude of the instability is deduced from the¼ width of the histogram¼ (dashed line); (b)Variation of the dimensionless amplitude 1 A=d of the waves, as a function of downstream distance. From right to left, U 12, 17, 22, and 27 m sÀ . The region of exponential growth (enhanced by the g g ¼ dashed line) is drastically reduced as Ug is increased.

possible saturation. It is interesting to note that the region of growth rate kith deduced from the spatial stability analysis exponential growth is preceded by another region where the (numerical resolution of the dispersion relation, with a full growth rate steadily decreases: the extent of this latter region velocity deficit, gravity perpendicular to the interface and decreases when Ug is increased. For each gas velocity, the surface tension included). Figure 21 shows the variations of region of exponential growth appears to be reached when the the ratio kiexp=kith as a function of the gas velocity. It can be amplitude of the waves reaches a threshold value of approxi- seen that this ratio is around two for the lowest gas velocity mately 0.7dg. The nature of this first region remains unclear: investigated, but when Ug is increased it increases up to a ra- our guess is that it could be due to a wake caused by the tio eight between the experimental and predicted growth splitter plate, and at any rate be caused by the finite thickness rate. These variations are quite similar to the variations of of this plate (e 150 lm at its end). the experimental ki shown on Fig. 20(b): this is because the We now present¼ on Fig. 20 the values of the dimension- gas velocity has very little influence on the spatial growth less spatial growth rate of the instability measured in the rate predicted by the stability analysis. Figure 21 shows that zone of exponential growth, as a function of the gas velocity the measured growth rate is much larger than the predicted Ug and as a function of M. It can be seen that ki increases one: while it manages to capture the frequency of the insta- steadily with Ug, (Fig. 20(b)) while it is extremely scattered bility, the inviscid stability analysis strongly underestimates when plotted as a function of M (Fig. 20(a)). This indicates its growth rate. This could be due to the strong spatial varia- that while M controls the variations of the dimensionless fre- tions induced by the instability: PIV measurements of the quency, it does not control the variations of the dimension- gas velocity field carried out on an analogous but axisym- 2 less spatial growth rate. The latter increases roughly as Ug. metric coaxial jet set-up showed that when the amplitude of This experimental growth rate can be compared to the spatial

FIG. 20. (Color online) (a) Dimensionless measured growth rate as a func- 1 1 1 tion of M, 8: Ug 12 m sÀ ; h: Ug 17 m sÀ ;x:Ug 22 m sÀ ; : FIG. 21. (Color online) Ratio of the experimental and predicted spatial 1 ¼ ¼ ¼  Ug 27 m sÀ ; (b) Dimensionless measured growth rate as a function of growth rate, as a function of Ug. Symbols correspond to different values of ¼ 1 1 1 1 1 1 Ug;*:Ul 0.26 m sÀ ; ^: Ul 0.31 m sÀ ; !: Ul 0.37 m sÀ ; ~: Ul,*:Ul 0.26 m sÀ ; ^: Ul 0.31 m sÀ ; !: Ul 0.37 m sÀ ; ~: ¼ 1 ¼1 1 ¼ ¼ 1 ¼1 1 ¼ U 0.50 m sÀ ; /: U 0.76 m sÀ ; .: U 0.95 m sÀ . U 0.50 m sÀ ; /: U 0.76 m sÀ ; .: U 0.95 m sÀ . l ¼ l ¼ l ¼ l ¼ l ¼ l ¼

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-11 Experimental and analytical study Phys. Fluids 23, 094112 (2011)

FIG. 22. (Color online) PIV visualiza- tion of the gas flow around a wave, for an annular gas flow (Hg 1.5 mm) around a liquid jet (radius ¼R 4 mm), 1 ¼ 1 Ul 0.5 m sÀ , Ug 20 m sÀ . The white¼ dashed line enhances¼ the limit of the liquid jet downstream the wave: the gas jet is detached from the liquid.

the waves exceeds dg, the vorticity thickness becomes signif- velocity deficit: a zero velocity at the interface has been icantly reduced at the crest of the waves: downstream the assumed in the stability results presented here. When the ex- crest the gas jet then detaches from the liquid, creating low perimental growth rate of the instability becomes larger it velocity recirculations in the following trough of the wave. causes steeper spatial variations, accelerating the failure of A typical velocity field obtained on this coaxial set-up is the stability analysis: the ratio fexp=fth increases, and the ex- shown on Fig. 22. The variations of the amplitude of the perimental growth rate strongly departs from its prediction. waves shown on Fig. 19(b) show that the distance after which the amplitude reaches dg varies from x 10dg for IV. CONCLUSION 1 1  U 12 m sÀ ,tox d for U 27 m sÀ : the typical pic- g g g We have extended the inviscid stability analysis of a liq- ture¼ of Fig. 22 is therefore expected¼ to occur well before the uid stream destabilized by a parallel gas stream, in order to end of the first wavelength, even for relatively low gas veloc- take into account key features of the velocity profile: namely ities. These strong spatial variations in the experiment are the liquid velocity deficit caused by the splitter plate at the likely to play a part in the departure from the predictions of liquid-gas interface, and the finite thickness of the gas the stability analysis. stream. We find that the velocity deficit leads to a significant Following the idea that the departure from the growth increase in the predicted frequency of the most unstable rate predicted by the linear stability analysis is caused by the mode. It also predicts a stronger influence of the liquid ve- large amplitude of the waves, we plot on Fig. 23 the ratio of locity on this frequency, and it removes the inviscid liquid the measured to predicted frequency, as a function of the ex- mode (scaling with d ) not observed in the experiments but perimental spatial growth rate. The predicted frequency, as l predicted by previous analyses.3 The effect of a finite gas in Fig. 18, is the numerical solution of the dispersion relation stream H is also to increase the frequency of the instability: including surface tension, gravity and a full velocity deficit g this effect becomes relevant for H < 0.1 k. Note however (a 0). It can be seen that the data points of Fig. 18(a) are g i that the scaling of the most unstable mode remains essen- now¼ significantly less scattered: this is because among a se- tially controlled by the vorticity thickness of the fast gas ries corresponding to a given U (given symbol), an increas- g phase. ing k corresponds in average to an increasing f =f . The i exp th We have carried out experimental measurements of the fact that for low growth rates the ratio f =f is smaller than exp th frequency and of the growth rate of the instability. The one (as low as 0.8) could be due to an overestimation of the results show a good agreement of the measured frequency with the frequency predicted by the inviscid stability analy- sis, provided that the liquid velocity deficit at the interface is taken into account. The experimental growth rate is on the contrary significantly larger than the predicted growth rate. We attribute this to the strong spatial variations observed in the experiment, and in particular to the impact of the large amplitude of the waves on the gas flow. The present analysis is inviscid: recent studies including viscosity in the stability analysis of a similar two phase mix- ing layer (Boeck and Zaleski,15 Fuster et al.19), but starting from a smooth version of the base flow of Fig. 1(a), predict a different most unstable mode for our experimental condi- tions, not scaling with dg as is observed in the experiment. It would be interesting to extend these viscous analyses to a FIG. 23. (Color online) Ratio of the experimental and predicted frequency smooth version of the velocity profile of Fig. 1(c), to deter- as a function of the experimental growth rate. mine how the interface velocity affects the viscous mode,

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions 094112-12 Matas, Marty, and Cartellier Phys. Fluids 23, 094112 (2011) and to see in particular if the most unstable modes in both 8P. Marmottant and E. Villermaux, “On spray formation,” J. Fluid. Mech. the viscous and inviscid cases turn out to be similar when the 498, 73 (2004). 9F. Ben Rayana, A. Cartellier, and E. J. Hopfinger, “Assisted atomization deficit is included. of a liquid layer: Investigation of the parameters affecting the mean drop In most industrial applications of coaxial injectors the size prediction,” (paper ICLASS06-190), in Proceedings of the interna- geometry of the flow is axisymmetric, and not planar. The tional conference on Liquid Atomization and Spray Systems (ICLASS), 27 same inviscid mode, driven by the gas vorticity thickness, August–1 September, Kyoto, Japan, ISBN 4-9902774-1-4 (Academic Pub- lication and Printings Co. Ltd., Osaka, Japan, 2006). has been observed on experiments carried out in this geome- 10E. Villermaux, “Mixing and spray formation in coaxial jets,” J. Propul. 8 try by Marmottant and Villermaux , but with discrepancies Power 14, 807 (1998). between the predicted frequency and the experimental fre- 11J. M. Gordillo, M. Pe´rez-Saborid, and A. M. Gan˜a´n-Calvo, “Linear stabil- 448 quency. We believe that the inclusion of the finite thickness ity of co-flowing liquid-gas jets,” J. Fluid. Mech. , 23 (2001). 12L. Raynal, “Instabilite´ et entraıˆnement a` l’interface d’une couche de Hg of the gas stream and of the cylindrical geometry in the me´lange liquide-gaz”, Ph.D. dissertation, Universite´J. Fourier Grenoble I, theory could significantly affect the results of the predictions France, 1997. 13 for this geometry. F. Ben Rayana, “Contribution a` l’e´tude des instabilite´s interfaciales liquide-gaz en atomisation assiste´e et tailles de gouttes”, Ph.D. disserta- tion, INP Grenoble, France, 2007. 14P. E. Dimotakis, “Two-dimensional shear-layer entrainment,” AIAA J. 24, 1A.H. Lefebvre, Atomization and Sprays (Hemisphere, New York, 1989). 1791 (1986). 2J. Lasheras and E. J. Hopfinger, “Liquid jet instability and atomization in a 15T. Boeck and S. Zaleski, “Viscous versus inviscid instability of two-phase coaxial gas stream,” Annu. Rev. Fluid Mech. 32, 275 (2000). mixing layers with continuous velocity profile,” Phys. Fluids 17, 032106 3J. Eggers and E. Villermaux, “Physics of liquid jets,” Rep.Prog.Phys. 71, (2005). 036601 (2008). 16E. Hinch, “A note on the mechanism of the instability at the interface 4L. Rayleigh, “On the stability, or instability, of certain fluid motions,” between two fluids,” J. Fluid Mech. 144, 463 (1984). Proc. London Math. Soc. 11, 57 (1879). 17S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, 5G. A. Lawrence, F. K. Browand, and L. G. Redekopp, “The stability of a New York, 1981). sheared density interface,” Phys. Fluids A 3(10), 2360 (1991). 18M. Gaster, “A note on the relation between temporally-increasing and 6H. Eroglu and N. Chigier, “Wave characteristics of liquid jets from airblast spatially-increasing disturbances in hydrodynamic stability,” J. Fluid. coaxial atomizers,” Atomization Sprays 1, 349 (1991). Mech. 14, 222 (1962). 7L. Raynal, E. Villermaux, J. Lasheras, and E. J. Hopfinger, “Primary insta- 19D. Fuster, A. Bague´, T. Boeck, L. Le Moyne, A. Leboissetier, S. Popinet, bility in liquid gas shear layers,” in 11th Symposium on Turbulent Shear P. Ray, T. R. Scardovelli, and S. Zaleski, “Simulation of primary atomiza- Flows, 7–10 September 1997, Grenoble, France, Vol. 3, pp. 27.1–27.5, tion with an octree adaptive mesh refinement and VOF method,” Int. J. OCLC 40626641, INP-CNRS-UJF. Multiphase Flow 35, 550 (2009).

Downloaded 04 Nov 2011 to 194.254.66.83. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions