# Some aspects of spray formation and development in the near field of two-phase coaxial injectors

Moongeun Hong, Alain Cartellier, Emil Hopfinger, Jean-Philippe Matas

LEGI, Laboratoire des Écoulements Géophysiques et Industriels, BP 53, 38041 Grenoble cedex 9, France.

## 1. Introduction

Coaxial injection is commonly employed in various industrial processes and notably in combustion engines. In most reactors used in aeronautic, for example, sheets or films of fuel are sheared and stripped by air streams. In LOX/H2 cryotechnic rocket engines, gaseous hydrogen at high velocity is used to atomize the core of liquid oxygen. Such injection devices have proved efficient and reliable, but there is still room for improvements. Indeed, various aspects of their functioning are still poorly understood. On the one hand, the primary atomization mechanisms responsible for the initial drop size distribution are still subject to controversy (Lasheras & Hopfinger 2000, Hopfinger 2001, Villermaux & Marmottant 2004). Improved understanding in that area would help implement efficient control procedures, and would better ascertain the extrapolation of laboratory experiments based on substitution fluids to actual engine conditions. Note also that aside drop sizes, little information is available concerning the initial drop flux and its spatial distribution. On the other hand, concerning spray formation, our understanding of the two-phase mixing process downstream the injection remains largely speculative and the predictive capacity of current numerical simulations is consequently limited. In the perspective of model improvement, the relative importance of coalescence and break-up mechanisms on the spray development as well as the influence of lateral dispersion need to be better understood. In particular, drop breakup due to the mean relative velocity (shear break-up) or due to turbulence have been invoked to explain the observed spatial evolution of drop sizes (Lasheras et al. 1998) but recent proposals indicate that collisions can be crucial in the spray development (Vallet et al. 2001). In addition, the role of collisions is expected to evolve along the spray since the mean drop size has been shown to decrease at moderate distance but to increase further downstream (Lasheras et al. 1998). However, among the available data, the near injection zone has scarcely been examined in detail, except for the evolution of the dispersed phase fraction (Carreau et al. 1997).

In order to build refined databases able to back up numerical simulations, we have carried out dedicated laboratory experiments using non-reacting, substitution fluids under ambient conditions. The experimental facilities are briefly described in Section 2. Emphasis is then put on the two-phase mixing process at short distances from the injection. Section 3 is devoted to the spray characteristics in a section located at the end of the liquid intact length. It also includes some considerations on the primary atomization mechanisms. The spatial evolution of the spray further downstream is analyzed in Section 4 with emphasis put on the relative role of break-up and coalescence mechanisms.

## 2. Experimental set-up and measuring techniques

Experiments have been performed with water and air under ambient conditions. A first experimental set-up consists of a planar mixing layer, with a liquid stream in the bottom channel and an air stream in the upper channel. Dimensions are H<sub>G</sub>=10mm, H<sub>L</sub>=10 mm over a 100 mm width (in that case the y=0 plane is a rigid wall). The second experiment is the coaxial injector
previously used by Raynal (1997, 1998) and by Marmottant (2001), its dimensions are H<sub>G</sub>=1.7mm, D<sub>L</sub>= 2H<sub>L</sub>=7.6mm (Fig.1). In both experiments, strongly convergent sections (contraction ratio 10 for the planar configuration and above 6 for the coaxial geometry) are imposed on both fluids just before the exit section. The splitter plate between the two streams is 0.1 - 0.2 mm thick at its extremity. The mean liquid velocity U<sub>L</sub> was varied between 0.1 and 1m/s, while the mean gas velocity U<sub>G</sub> ranged up to about 100m/s. Both experiments were carefully qualified by previous contributors (Raynal 1997, Marmottant 2001). In particular, the gas boundary layers at the exit section have been shown to be laminar. The corresponding vorticity thickness, evaluated as δ = Max(U) / Max(∂U/∂y) was found to evolve as:

$$ \delta / H_G = c_\delta Re^{-1/2} \qquad (1) $$

where Re=U<sub>G</sub> H<sub>G</sub> / v<sub>G</sub>. In the planar configuration, the coefficient c<sub>δ</sub> equals 6.0 (Raynal 1998). In the coaxial configuration, c<sub>δ</sub> ranges between 3.5 when measured 0.5mm inside the injector (Raynal 1998), and 5.3 (Raynal 1998) or 5.6 (Marmottant 2001) when evaluated slightly outside the injector. Note that, for the two geometry and for the velocity range considered, the vorticity layer in the gas is set by the boundary layer and not by any wake effect from the splitter plate.

<figure><img src="image_1.png" /><figcaption>Fig.1: Sketch of the gas-liquid mixing layer. For coaxial geometry, the bottom wall is absent and the x axis coincides with the axis of symmetry.</figcaption></figure>

In order to access the dispersed phase characteristics, the phase detection optical probe technique previously developed for bubbly flows (Barrau et al. 1999) has been extended to sprays. The drop velocity (more precisely the velocity component along the probe axis) is inferred from the dewetting time, and, combined with the liquid residence time, one gets the chord pierced along the fluid inclusion. Introducing some assumptions on the flow structure (Cartellier 1999), it is possible to deduce the joint size-velocity product density f(x, d, v), the number density n and its flux φ, the local liquid phase fraction α and the interfacial area density Γ. The details of this system as well as some qualifications of its performances are available elsewhere (Hong et al. 2004). Typically, the uncertainty on the above mentioned quantities is less than 15% for sprays mainly consisting of inclusions larger than about 10μm.

### 3. Spray formation and boundary conditions

Various approaches have been considered to simulate the spray formation and its spatial evolution. In particular, direct simulations are well adapted to investigate the primary atomization process (Scardovelli & Zaleski 1999), although 3D computations are still scarce because of technical constraints (Lafaurie et al. 1998). Concerning the spray itself, two-fluid averaged descriptions are generally preferred because they are less numerically cumbersome, especially when coupled with combustion kinetics. In order to capture the change in drop size, extra equations need to be considered. These are either the full pdf f(x, d, v) used in place of the dispersed phase conservation equations, or some moments of f - such as the interfacial area density - used in addition of the above mentioned conservation equations. However, such descriptions require
boundary conditions in a suitably selected cross-section as close as possible to the injector output. In that scope, various characteristics of the spray have been experimentally determined at a distance corresponding to the liquid intact length where the primary atomization mechanisms have already stripped the bulk liquid. In the following, we examine the evolution of various dispersed phase characteristics with the gas velocity at a fixed dynamic pressure ratio namely $M = \rho_G U_G^2 / (\rho_L U_L^2) = 16$. In parallel, we propose and discuss the validity of simple models to understand the origin of the main observed features.

## 3-1 Drop size

One of the main unknowns in atomization is the size of the droplets. For coaxial injection, recent investigations have shown that at least two mechanisms play a role in the interface destabilization. Along the longitudinal direction, long waves are formed while in the transverse direction, ligaments merge, grow and then break into droplets. The axial structure is due to a Kelvin-Helmholtz instability, but its wavelength $\lambda_{//}$ is controlled by the vorticity thickness $\delta$ in the more energetic gas phase. Namely,

$$ \lambda_{//} = C_{//} \delta (\rho_L/\rho_G)^{1/2} \qquad (2) $$

with $C_{//} \approx 1$ to $1.7$ in the coaxial geometry (Raynal 1998, Marmottant 2001) and $C_{//} \approx 4$ in the planar geometry (Raynal 1997). Along the transverse direction, and similarly to Joseph et al.'s (1999) analysis of drop breakup, we and Varga (2002) have proposed that the ligaments arise from a Rayleigh-Taylor instability due to the axial acceleration of the wave crests. The resulting most amplified wavelength is (Hong et al. 2002):

$$ \lambda_{RT} = 2\pi (3 \gamma C_{//}/C_d)^{1/2} \delta (\rho_L/\rho_G)^{1/4} (\rho_G (U_G-U_c))^2 \delta/\sigma^{-1/2} \qquad (3) $$

where $C_d$ is the drag coefficient associated with the hydrodynamic force exerted by the gas on the wave, and where it is assumed that the characteristic thickness of the wave subject to destabilization is a fraction $\gamma$ of the axial period $\lambda_{//}$. The dependencies of $\lambda_{RT}$ with the phasic velocities and with the surface tension $\sigma$ have been confirmed by experiments performed on various injectors (Hong et al. 2002). In addition, the comparison of eq.(3) with direct measurements of $\lambda_{RT}$ indicates that the portion of the wave crest subject to acceleration is thinner in the planar geometry ($\gamma \approx 0.04$, assuming $C_d \approx 1$) than in the coaxial configuration ($\gamma \approx 0.2$, also assuming $C_d \approx 1$). Finally, and as pointed out by Marmottant 2001, the mean drop size appears to be proportional to $\lambda_{RT}$. More precisely, experiments with water have shown that in the planar geometry, $d_{32} \approx 0.10 \lambda_{RT}$ while $d_{32} \approx 0.27 \lambda_{RT}$ in the coaxial geometry (Hong et al. 2002).

A second important aspect for drop production concerns the size distribution. From the analysis of ligament breakup, Marmottant has found that the size distributions are exponential and that they solely depend on the mean diameter $d_{10}$, namely $P(d/d_{10})=\exp(-d/d_{10})$. This result is relevant for the drops that have just been formed. In order to examine whether such a distribution remains unaltered further downstream, drop size measurements have been performed at the end of the liquid intact length ($x=3H_L \approx L$, $y=H_L$) where the primary atomization is expected to be completed. When normalized by $d_{10}$, all size distributions collapse irrespective of the gas velocity or of the injector geometry (Fig.2), at least when $U_G$ is not too close to the onset of atomization. Also, the shape of the pdf remains unchanged further downstream (up to $x = 18H_L$). Only does the upper size limit evolve with $U_G$ and with the position. The existence of such a "quasi-universal" shape indicates that some mechanisms are present which ensure a rapid equilibrium in the size distribution, and that it is sufficient to follow the evolution of a single moment to describe such sprays. In that sense, simulations making use of an evolution equation for a selected moment such as the interfacial area density (see for example Vallet et al. 2001, Barbeau et al. 2001) appear to be fully pertinent.
The pdfs shown on Fig.2 clearly indicate a departure from an exponential distribution. In particular, small sizes are slightly depleted while larger sizes are more probable than in the case of an exponential distribution. This trend is also apparent from the various moments: the ratio $d_{20}/d_{10}^2$ is close to 2 as for the exponential law, but $d_{30}/d_{10}^3$ is about 8-9 and $d_{32}/d_{10}$ is about 4, instead of respectively 6 and 3 for the exponential distribution. Therefore, when compared with Marmottant's findings, it seems that the size pdf changes along the liquid intact length, and shifts from an exponential to a more complex form. Such an evolution may be due to coalescence in the mixing layer, a region where the number density is fairly high (see section 3-2), and where both the size dependent increase of the mean drop velocity and the presence of large scale coherent structures favor collisions. Yet, the existence of such a mechanism cannot be unambiguously proven from the present results. Indeed, a close examination of the measured pdfs just above the interface and at various distances from the injector does not show any neat increase of the probability for large sizes. Note however that in this region, the raw data do probably include the detection of some ligaments so that the largest observed structures may not correspond to detached droplets. Finally, there is a possibility that the drop size distribution does not change at all between the loci of ligament breakup and the end of the liquid intact length. Indeed, Fig.2 shows that the exponential behavior holds over 2 decades and a half counted along the ordinate: this is also the extent of Marmottant's data, and the deviations from the exponential decay are recorded for the rarest events corresponding to the largest sizes. A closer examination of the spray is therefore required to decide whether the large sizes originate from ligament break-up or from coalescence. At any rate, boundary conditions must account for the actual pdf.

<figure><img src="image_2.png" /><figcaption>Fig.2: Normalized size pdf measured at x=3H_L≈L, y=H_L for various gas velocities and two geometry (M=16).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig.3: Transverse profiles at x=3H_L≈L of the d_{10} and the d_{32} for the planar (U_G=60m/s) and the coaxial (U_G=70m/s) geometry at M=16.</figcaption></figure>

So far, drop sizes have been examined at a fixed position (x=3H_L≈L, y=H_L), and it is worthwhile to examine how they evolve along the y (transverse or radial) direction. Profiles of mean drop diameters in the cross-section x=3H_L are given in Fig.3 for both geometries at comparable gas velocities and fixed M=16. No measurements have been performed below y/H_L=0.5 in the planar experiment because of the presence of a liquid film on the bottom wall. Clearly, both the arithmetic mean $d_{10}$ and the Sauter mean $d_{32}$ diameters neatly decrease when moving towards the outer boundary of the mixing layer. For the coaxial injector, very large liquid lumps are detected in the vicinity of the axis: they arise from an oscillating liquid tongue formed by a helical instability.
Their rupture is not achieved until about 6-7 $H_L$, and produces droplets 5 to 10 times larger than those due to the Rayleigh-Taylor scenario discussed above (see also Hong et al. 2002). Hence, for coaxial injectors, if boundary conditions are to be given at the $x=L$ cross-section, the break-up of the central liquid tongue must be accounted for by the code. This statement is enforced by the fact that the central region $y/H_L<0.5$ corresponds to the main fraction of the liquid flux (see Fig.5).

Going back to the radial profile of drop size, it must be underlined that the trend shown on fig. 3 is opposite to the one reported by Lasheras et al. (1998) who observed the largest sizes at the jet periphery. Their experiments, performed at $M=300$, correspond to the so-called recirculating regime (Villermaux 1997). This is probably the reason why they observed such a strongly amplified helical instability (see their Figures 2 and 3), which, in its turn, is responsible for the off axis presence of the largest drops at 30 diameters downstream the injector exit.

### 3-2 Flux of drops and related quantities

The specification of the number of droplets injected per unit area is also an important parameter for two-phase flow simulations. In the perspective of exploiting models inspired from kinetic theory, which deal with space variables relative to inclusions' center and which have the capability to handle situations involving break-up and/or coalescence, we give the results in terms of the number density $n$ (#/m$^3$) and its flux $\phi$ (#/m$^2$ s).

Another important parameter needed for the description of the spray is the local volumetric flux $J_L$. By definition, $J_L$ equals the entrainment velocity $U_e$. The latter is set by the continuity of the turbulent stress at the interface (Villermaux 1997, Raynal 1998). Namely:

$$ \rho_L U_e^2 = c_1 \rho_G u_G'^2 \quad (4) $$

where $u_G'$ denotes the turbulent fluctuation in the gas. In the case of homogeneous mixing layers, $c_1$ is about 0.25, and $u_G'$ is assumed to be linear in the velocity difference ($U_G-U_L$), with a proportionality constant close to 0.17. Since the local volumetric flux $J_L$ is also given by $(\pi/6)$ $d_{30}$ $\phi$ when assuming uncorrelated sizes and velocities, one gets:

$$ \phi = (6/\pi) U_e / d_{30} = (0.5/\pi) (\rho_G/\rho_L)^{1/2} (U_G-U_L) / d_{30} \quad (5) $$

Computing $\phi$ using the above equations with the various associated constants provides a very reasonable estimate when compared with measurements. For the coaxial injectors, the agreement also holds at different radial locations such that $y/H_L \ge 1$.

The number density $n$ and $\phi$ are related by $\phi = \langle V \rangle n$, where the mean drop velocity $\langle V \rangle$ is defined by $\langle V \rangle(x)=\int dR \int dV V f(x,R,V) / \int dR \int dV f(x,R,V)$. Let us recall that by definition $J_L = \alpha V_{PL}$ so that $V_{PL} = \langle Vd^3 \rangle / \langle d^3 \rangle$, and therefore, $\langle V \rangle$ and $V_{PL}$ are identical for uncorrelated sizes and velocities, but they differ if size and velocity are correlated.

Experimental results show that the liquid fraction $\alpha$ at a fixed location does not vary much with the gas velocity. A direct consequence is that the interfacial area density $\Gamma$, which at first order equals $6 \alpha / d_{32}$, is mainly governed by the size of droplets, and not by their concentration (again at a given location).

The experimental behaviours of $\alpha$, $\phi$ and $\Gamma$ can be studied as a function of the gas velocity $U_G$, and these behaviours can be compared to the predictions of simple models. If the comparison is made for the ratio $V_{PL}/U_G$, it can then be seen that experimentally the relative velocity scaled by $U_G$ does not significantly evolve with $U_G$ at fixed $M$. Let us examine what predicts the simplest approach that assumes the gas velocity field to be uniform (equals to $U_G$). The droplet dynamics then obeys $dV_d/dt = f(U_G-V_d)/\tau$ where $\tau=(\rho_L/\rho_G)d^2/(18v_G)$ is the characteristic response time of a drop of size $d$, and where the friction factor $f$ is given by $1+Re_p^{1/6}/6$, $Re_p=(U_G-V_d)d/v_G$ being the particulate Reynolds number. Since $Re_p$ is about 200±40 at early times, $f$ can be taken as constant,
and in the limit $V_d / U_G \ll 1$, the velocity $V_d$ of a drop of diameter $d$ at a distance $x$ from the injection should be:

$$V_d(x,d) / U_G = V_d(x=0,d) / U_G + (2 f x / \tau U_G)^{1/2} \propto x^{1/2} / (d U_G^{1/2}) \quad (6)$$

where the ejection velocity $V_d$ ($x=0,d$) is of the order of $U_e$. The $x^{1/2}$ behavior predicted by eq.(6) agrees with observations: indeed, as shown Fig.4 the mean velocities $\langle V \rangle$ and $V_{PL}$ as well as the mean drop velocities for given size intervals grow like $x^{0.5 \pm 0.1}$, and that holds at various radial positions. However, the above prediction has two weaknesses. First, from eq.(6), one expects $V_d/U_G$ to evolve with $U_G$. Such a behavior should also hold for the mean phasic velocity since $V_{PL}/U_G = \langle (V_d/U_G) d^3 \rangle / \langle d^3 \rangle \propto x^{1/2} / (d_{32} U_G^{1/2})$. In section 3.1, $d_{32}$ has been shown to vary like $U_G^{-5/4}$, so that $V_{PL}/U_G$ should grow like $U_G^{4/3}$. This is not consistent with the observations. Second, on a more quantitative ground, eq.(6) systematically overestimates the velocity magnitude for all range of sizes considered: for example, the velocity predicted for a 200µm drop at $x=10 H_L$ is 0.4 $U_G$ while the measured velocities never exceed 0.23-0.24 $U_G$. In an attempt to correct this defect, account can be made for the gas flow expansion. Although in better quantitative agreement than the simplest model, the predictions are then still unable to capture the evolutions in the drop velocity, especially at small distances from the injection (not shown here). Consequently, and notably within a few diameters downstream the injector, the actual gas velocity decay is much more pronounced than the one expected from the expansion. To further confirm such a statement, the momentum exchange between phases can be estimated from the number density and an average drag force deduced from our experimental results. This term is then shown to overwhelm the other losses of momentum for the gas phase (including its expansion), and that holds up to $x/H_L$ about 8-10. Therefore, it is clear that a 2D two-fluid approach incorporating the momentum exchanges between phases is required to fully understand the observed velocity behaviors. This conclusion also holds for the dispersed phase fraction. Indeed, eq.(6) combined with the Zuber & Findlay cinematic model which gives $\alpha$ as a function of $U_G$ predicts that $\alpha$ should decay as $x^{-0.5 \text{ down to } -0.7}$. In the experiments, $\alpha$ drops like $x^{-1.3 \pm 0.1}$ at low heights in the planar geometry and on the axis in the coaxial geometry: such a depletion which is much stronger than the one expected from the droplet acceleration alone, is clearly due to the lateral liquid phase transport.

<figure><img src="image_4.png" /><figcaption>Fig.4 Evolutions of the phasic and the mean drop velocities for two different size intervals with the downstream distance at various heights (planar geometry, $U_G=60m/s$, $M=16$). For $y=H_L$, $\langle V \rangle$ represents the mean velocity assuming correlated size and velocity.</figcaption></figure>

Also, as one moves toward large heights or away from the axis, the decay of $\alpha$ with $x$ becomes less and less steep, again because of the transverse flux of droplets induced by their interactions with the mixing layer. These interactions also affect the spatial distribution of the mean velocities because of the size segregation induced by inertia sensitive dispersion by the mixing layer (this is notably
manifest from the mean size profile of Fig.3). Further progress must therefore be grounded on
refined comparisons with numerical simulations.

3.3 Boundary conditions at the x=L cross-section

<figure><img src="image_5.png" /><figcaption>Fig.5 Transverse profiles at x=3H_L≈L of the local liquid volumetric flux J_L made dimensionless by U_L and the mean phasic velocity V_PL made dimensionless by U_G for the planar (U_G=60m/s) and for the coaxial (squares: U_G=50m/s, triangles: U_G=70m/s) geometry at M = 16.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig.6 Transverse profiles at x=3H_L≈L of the liquid phase fraction for the planar (U_G=60m/s) and for the coaxial geometry at M=16.</figcaption></figure>

The previous sections have started to stress the spatial non-uniformity of the two-phase co-
current jets. To get more insight on their structure, and also to provide a complete set of boundary
condition, the profiles of dispersed phase variables have been determined at a distance x=L. The
size distribution has already been presented (see Fig.3). The local liquid volumetric flux J_L and the
mean phasic velocity VPL are shown Fig.5. These distributions are very similar for the planar and
for the coaxial experiments. In both cases, the volumetric flux decreases in the y direction, and the
main part of the incoming liquid flux is confined at low heights. In the planar configuration, about
90% of the flow rate enters the liquid film formed on the wall, and in the coaxial geometry, the
region y/H_L≤0.5 collects 80-85% of the flux. Contrary to J_L, the phasic velocity monotonously
increases when moving off axis: this behavior is related with the significant diminution of the mean
drop sizes with y presented Fig.3. The inverted trends between J_L and VPL are due to the behavior of
the dispersed phase fraction α that experiences a striking exponential decrease along the transverse
coordinate as shown Fig.6. Note that, near the axis of the coaxial jet, α is close to but not equal to
unity: it is about 0.9 for U_G=70m/s and 0.7 for U_G=50m/s.

As the previous variables, the number density fluxes exhibit similar distributions in both
geometries (Fig.7). However, they present a maximum around y/H<sub>L</sub>=1 while the liquid volumetric
fluxes J<sub>L</sub> are monotonous. This change originates from the lateral size segregation: the largest sizes
located at low y damp the number density flux in that region. Contrary to others quantities, the
number densities (Fig.7) and consequently the interfacial area densities (Fig.8) exhibit profound
differences between the two injectors. In the coaxial geometry, n and Γ go to a minimum near the
axis because of the helical instability that forms large lumps of liquid instead of numerous fine
droplets. But for the planar geometry, n and Γ drastically increase as ones approaches low heights.
In addition, when considering the position y/H<sub>L</sub>=1, the magnitudes of the quantities n, φ or Γ
obtained in both geometries are somewhat similar. But for lower y positions, they become
drastically different: this is notably so at y/H<sub>L</sub>=0.5, the initial boundary between the liquid and gas
streams. Again, these differences arise from the presence of the helical instability in the coaxial
injector, while in the planar configuration, the atomization mechanism discussed in Section 3.1 is
unhampered and the droplets are both thinner and much more numerous.

<figure><img src="image_7.png" /><figcaption>Fig.7 : Transverse profiles at x=3H_L≈L of the number density and of the number density flux for the planar and for the coaxial geometry at M=16 (Note the multiplying factors in the legend).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig.8: Transverse profiles at x=3H_L≈L of the interfacial area density for both geometries at M=16.</figcaption></figure>

**4. Spatial evolution of the spray:**

The spatial evolution of the spray further downstream the injector has been investigated for
fixed conditions, namely U<sub>G</sub>=60m/s, M=16. Extensive results are available for the planar geometry
up to x/H<sub>L</sub>= 10: they are presented and discussed in Section 4.1. Based on flux conservation for
various size classes, the role of break-up/coalescence phenomena in the early development of the
spray is then investigated in Section 4.2.

4-1 Spatial structure of the spray

To illustrate some global features of the spray development, iso-contour of various dispersed
phase characteristics are presented Fig.9. In all these plots, the limits of the gas potential cone and
of the external mixing layer (with stagnant air) as expected in an unbounded single-phase flow have
been indicated as dashed lines. The region y/H<sub>L</sub><0.6 has not been scrutinized because of the
presence of a liquid film along the bottom wall, and because of the deposition process that takes
place here.

<figure><img src="image_9.png" /><figcaption>Fig.9: Iso-contours of various dispersed phase variables (planar mixing layer, U<sub>G</sub>=60m/s, M=1.6, here D = 2 H<sub>L</sub>):</figcaption></figure>

a) α, b) V<sub>PL</sub>/U<sub>G</sub>, c) φ, d) Γ

The examination of the dispersed phase fraction α in Fig.9-a shows that most of the liquid remains concentrated at low heights, typically up to y/H_L<1 - 1.2. This weak lateral spreading of the two-phase zone also clearly appears on the spatial distribution of the interfacial area density Γ shown Fig.9-d. In particular, Γ iso-contours have an initial spread angle of order of 2-3°, while, further downstream they tend to become almost parallel to the x axis: these trends are quite consistent with the LOX/H2 flame boundary recently identified by OH chemi-luminescence (Juniper 2001). Although the mass flux is essentially located at low heights at all downstream distances (not shown), the dispersed phase velocity VPL is steadily increasing with x (fig 9-b). Interestingly, the velocity maximum is reached at moderate heights, say for y/H_L between 0.7 and 1.5 at x/H_L above 7-8, that correspond to the external, and not the inner, mixing layer. This region is essentially occupied by small droplets (d<sub>32</sub> is less than 100µm to be compared with 400µm at low heights), which are indeed those experiencing the strongest acceleration. Yet, this is not the only reason. Indeed, the drop velocity depends on the gas velocity distribution. Although the latter has not been measured, its axial decrease is probably much less pronounced at moderate than at low heights because of the huge diminution (by a 10-100 factor) of the number density and therefore of the momentum exchange between phases.

The region located just above the liquid intact zone (indicated by a continuous line) deserves some comments. It must first be stressed that the data gathered in that area should be interpreted with caution because the probe technique does not distinguish between attached ligaments and droplets. In addition, let us recall that the measurements of the size distribution, the number density and its flux are based on the assumption that all inclusions are spherical. When moving slightly to
higher positions φ reaches its maximum (red zone in Fig. 9-c) and the mean size strongly decreases (not shown), indicating that the ligaments have already been broken apart. In that zone of strong drop production, the number density can reach a large magnitude, of the order of $3 \times 10^{11} \#/m^3$. Such values, combined with the large relative velocity, induce drastic loss of momentum in the gas, as evoked in section 3.2. Downstream the liquid intact length, α (like n) is steadily decreasing as

<figure><img src="image_10.png" /><figcaption>Fig. 10: a -Radial distributions of the volumetric flux of droplets (U<sub>G</sub>=90m/s), and b - droplet sizes (U<sub>G</sub>=50m/s, x=9 H<sub>L</sub> ≈ 3 L) in the coaxial injector system at M=16.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 11: Variation of the dispersed phase fraction with y/D, for different downstream positions.</figcaption></figure>
expected for a dispersed flow whose relative velocity decays, and which experiences lateral dispersion (the role of coalescence will be discussed in section 4.3). The number density flux $\phi$ also experiences a continuous decay, but it is much slower than that of $n$ and $\alpha$ because of the longitudinal increase in the velocity.

For the coaxial injector, the whole flux profile is available. As in the planar geometry, the main fraction of the flux remains in the vicinity of the axis ($y/H_L<1$) as shown on Fig.10a, and that holds even at large downstream distances from the injector. In addition, as shown on Fig.10b, the largest inclusions are located on the axis, while their size strongly decreases with the radial distance (up to a factor 5).

<figure><img src="image_12.png" /><figcaption>Fig.12: Iso-contours of the number density flux for different class sizes : a) $d < 90$ µm, b) $90$ µm $< d < 180$ µm, c) $180$ µm $< d < 270$ µm</figcaption></figure>

## 4.2 Dispersion

To investigate more precisely the dispersion mechanisms, we show on figure 11 results of the local dispersed phase fraction profiles for different dowstream distances $x$. For all $x$, the trend is a close to exponential decrease of $\alpha$ with $y$, but it can be seen that the decrease is less and less steep
when one moves to larger x. This corresponds to the larger extension of the spray as one goes
towards downstream positions.

The behaviours observed in different size classes are shown on figure 12. Isocontours of the number density flux are represented for three size classes. It can be seen that the dispersion is much stronger in the case of the smallest particles (d < 90 µm) than for the other particles, which is consistent with the expected behaviour.

4-3 Coalescence

The above results emphasize the role of the large drops emitted along the axis. However,
various mechanisms do play a role in the development of the spray and notably on the spatial
organization of drop sizes. Aside from secondary breakup due to turbulence or shear and
coalescence (Lasheras et al. 1998), it has been recently suggested that drop-drop collisions can be
responsible for efficient breakup especially at short downstream distances (Vallet et al 2001). The
latter proposal has been developed for droplets smaller than the Kolmogorov length scale, with their
relative velocity controlled by the turbulence of the gas. However, according to the extent of the
measured size distributions, it is likely that the differential inertia between droplets and thus their
differential acceleration by the main gas stream plays a major role in this process. To test these
aspects, one cannot merely rely on the examination of the size variation with spatial coordinates,
the reason being that the size distributions are affected by the lateral dispersion and by the relative
velocities (which control the volume fraction for each size class). Budgets of volumetric fluxes per
size class have been considered instead. These fluxes have been integrated over the radial
coordinate in order to get rid of dispersion mechanisms. Two such results are given in Fig.11 where
the flux change is evaluated between the downstream distances x= 6 and 9 H<sub>L</sub>.

In the planar configuration, the loss of droplets smaller than 120 µm and the gain in the
larger size classes clearly demonstrates that coalescence is dominant (these results are significant
since the total flux was recovered within about 2.5%). This finding is confirmed by the joint size-
velocity pdf (not shown). Indeed, a much larger probability was obtained for collision Weber
numbers (based on the relative velocity between drops) in the range 15-30, that approximately
corresponds to a coalescence dominated regime, than for Weber numbers above 30 that correspond
to break-up (see for ex. Post & Abraham 2002). Both mechanisms are, therefore, effective, but the
net effect is toward coalescence. It is interesting to note that the d<sub>10</sub> is nearly constant (Fig.12b).
Such a plot can be mistakenly interpreted as a frozen size distribution while it results from
differential accelerations of drops combined with coalescence.

<figure><img src="image_13.png" /><figcaption>Fig.11: Volumetric fluxes losses and gains per size classes between the cross-sections x/H<sub>L</sub>=6 and x/H<sub>L</sub>=9: a) planar (U<sub>G</sub>=60 m/s, M=16), b) coaxial injector (U<sub>G</sub>= 70 m/s, M=16).</figcaption></figure>
For the coaxial geometry, the smaller size classes disappear, again indicating that coalescence is active. Meanwhile, the largest classes also disappear due to the breakup of the large liquid inclusions located near the axis. Although the flux conservation was not as good as in the planar system (the global flux is recovered to within 5-10%), the net result is a strengthening of intermediate size classes. Depending on the radial location considered, the $d_{10}$ either decreases (near the axis at small distance) because of strong breakup, or increases (for $y/H_L \ge 1$) because of coalescence and dispersion (Fig. 12a).

The main result is that, in both geometries, the smallest drops disappear by coalescence within very short distances from the injection. Such a behavior has not yet been reported. It is at odds with the expectations of Vallet et al. (2001) who predict a strong breakup in the near region in the case of very small droplets whose motions are driven by the turbulence. On the experimental side, Lasheras et al (1998) report a decrease of the Sauter mean diameter between 10 and 20 injector diameters but do not provide any data at shorter distances. Aside from differences in the injector geometry and flow conditions, a comparison with their experiments is therefore quite speculative.

Generally speaking, direct comparisons between experiments remain delicate because at least three ingredients are expected to play key roles in the spray development:

* the size distribution resulting from the primary atomization, whose origins, as shown in section 2, are not fully understood,

* the structure of the gas velocity field that governs the acceleration of droplets. In particular, the extent of the gas potential cone compared with the liquid intact length is expected to crucially affect the collision Weber number,

* the possible presence of a helical instability mode, that alters not only the size distribution but also modifies the spatial distribution of the initial fluxes emitted from the primary atomization zone. To progress further in the understanding of spray development, it would be necessary to vary the geometry in a more systematic manner, and also to back-up the experiments by adequate spray modelling (and the corresponding simulations). Since, as shown above, the relative velocities between size classes are required, two-fluids models including population balance equations seem quite appropriate. In addition, the variables used in such approaches coincide with those delivered by the metrology we have developed.

<figure><img src="image_14.png" /><figcaption>Fig. 12: Downstream evolution of the $d_{10}$ at various locations with respect to the axis for M=16, a - coaxial (U<sub>G</sub>=70m/s), b - planar (U<sub>G</sub>=60m/s).</figcaption></figure>
**5. Conclusion**

We have studied the evolution of a two-phase mixing layer, in plane or coaxial geometries. Our results provide a data base for the comparison with numerical simulations. However, there is still a lack of measurement for the gas phase velocity, and efforts should be made in that direction for future experiments.

Due to the presence of a solid boundary, the planar configuration studied here is not fully representative of coaxial injectors used. However, the presence of this wall freezes the equivalent of the helical instability observed in axisymmetrical geometry, (Squire like instability), and thus it is more adapted to code benchmarking, in particular with respect to the simulation of coalescence due to collisions.

The extrapolation of our results to liquids used in engine injectors can be tricky or simply not valid. This is especially so for cryogenic fluids that experience a transition from subcritical to supercritical thermodynamic conditions (Chehroudi et al 2002).

**References:**

* Barbeau B., Jay S., Lacas F., Candel S., Simulation numérique de la combustion dans les injecteurs coaxiaux, in *Combustion dans les moteurs fusées - Actes du Colloque de Synthèse du Groupe de Recherche CNES / CNRS / ONERA / SNECMA*, 26-28 Juin Toulouse, Cépaduès-Editions pp.207-217 (2001).

* Barrau E., Rivière N., Poupot Ch., Cartellier A., 1999, Single and double optical probes in air-water two-phase flows: real time signal processing and sensors performances. *Int. J. Multiphase Flow*, **25**, 229-256.

* Carreau J.L., Porcheron E., Le-Visage D., Prevost L., Roger F., Liquid core characterization of coaxial liquid oxygen/inert gas jets, *Int. J. of Fluid Mech. Research*, **24**, no.4-6, 498-507 (1997).

* Cartellier A., 1999, Post-treatment for phase detection probes in non-uniform two-phase flows. *Int. J. Multiphase Flow*, **25**, 201-228.

* Gicquel P., Vingert L., Lecourt R., Barat M., Étude expérimentale des sprays cryogéniques en combustion dans des conditions sub et supercritiques, in *Combustion dans les moteurs fusées, - Actes du Colloque de Synthèse du Groupe de Recherche CNES / CNRS / ONERA / SNECMA*, 26-28 Juin Toulouse, Cépaduès-Editions 359-369 (2001).

* Hong M., Cartellier A. & Hopfinger E., 2002, Atomization and mixing in coaxial injection, Proc. 4th Int. Conference on Launcher Technology "Space Launcher Liquid Propulsion", Liège, Belgique, 3-6 Décembre 2002.

* Hong M., Cartellier A. & Hopfinger E., Characterization of phase detection optical probes for the measurement of the dispersed phase parameters in sprays, *Int. J. Multiphase Flow*, **30** 615-648 (2004).

* Hopfinger, E. J., Atomization of a liquid jet by a coaxial gas jet: A critical review, in *Combustion dans les moteurs fusées - Actes du Colloque de Synthèse du Groupe de Recherche CNES / CNRS / ONERA / SNECMA*, 26-28 Juin Toulouse, Cépaduès-Editions pp.33-47 (2001).

* Joseph D. D., Belanger J., Beavers G.S., Breakup of a liquid drop suddenly exposed to a high-speed airstream, *Int. J. Multiphase Flow*, **25**, 1263-1303 (1999).

* Juniper M., Structures et stabilisation de flammes cryotechniques, PhD École Centrale Paris, 2001.

* Lafaurie B., Mantel T., Zaleski S., Direct Numerical simulation of liquid jet atomization, Proc. ICMF'98, Lyon, France, 8-12 Juin (1998).

* Lasheras, J. C., Hopfinger, E. J., Liquid jet instability and atomization in a coaxial gas stream, *Annual Rev. Fluid Mech.* **32**, 275-308 (2000).

* Lasheras J., Villermaux E., Hopfinger E.J., Break-up and atomization of a round water jet by a high-speed annular air jet. *J. Fluid Mech.* vol. 357, 351-379 (1998).

* Marmottant Ph., Atomisation d'un liquide par un courant gazeux, PhD INP Grenoble, 2001.

* Marmottant, Ph., Villermaux, E., Mécanismes d'atomisation primaire dans les jets coaxiaux, in *Combustion dans les moteurs fusées - Actes du colloque de synthèse du Groupe de Recherche CNES / CNRS / ONERA / SNECMA*, Ed. Cépaduès-Éditions, pp. 51-66, 2001.

* Pal S., Moser M.D., H.M. Ryan, Foust M.J., Santoro R.J., Drop size measurement in a LOX/GH2 propellant rocket, Proc. Combustion Fundamentals and Applications, March 15-17, New Orleans, USA (1993).

* Post S., Abraham J., Modeling the outcome of drop-drop collisions in Diesel Sprays, *Int. J. Multiphase Flow* **28**, 997-1019 (2002).

* Raynal, L., Instabilité et entraînement à l'interface d'une couche de mélange liquide-gaz, PhD UJF, Grenoble, 1997.

* Raynal L. Pulvérisation assistée d'un jet liquide par un jet de gaz : étude des mécanismes de la brisure primaire, Rapport Final ref n° 93.811.000.094 DGA, Octobre 1998.
* Rehab H., Villermaux E., Hopfinger E., Flow regimes of large velocity-ratio coaxial jets, *J. Fluid Mech.* **345**, 357-381 (1997).

* Sadr R., Klewicki J.C., An experimental investigation of the near-field flow development in coaxial jets, *Phys. Fluids* **15**(5), 1233-1246, (2003).

* Scardovelli R., Zaleski S., Direct Numerical simulation of free-surface and interfacial flow, *Ann. Rev. Fluid Mech.* **31**, 567-603 (1999).

* Vallet A., Burluka A., Borghi R., Development of an Eulerian model for the "Atomization" of a liquid jet. *Atomization and Spray* **11** (5), 521-544 (2001).

* Varga C., Initial break-up of small diameter liquid jet by a high speed gas stream, PhD. UCSD, (2002).

* Villermaux E., Mixing and spray formation in coaxial jets, Proc. 3rd Int. Symp. on Space Propulsion, Beijing, China, August 11-13, (1997).

* Villermaux E., Marmottant Ph., On spray formation, *J. Fluid Mech.* **498**, 73-112 (2004).

* Zurbach St., Thomas J.-L., Sion M., Défis technologiques et industriels pour la combustion dans les moteurs fusées, in *Combustion dans les moteurs fusées - Actes du Colloque de Synthèse du Groupe de Recherche CNES / CNRS / ONERA / SNECMA*, 26-28 Juin Toulouse, Cépaduès-Éditions pp. 409-427 (2001).