Two-layer Thermally Driven Turbulence: Mechanisms for Interface Breakup

It is commonly accepted that the breakup criteria of drops or bubbles in turbulence is governed by surface tension and inertia. However, also {\it{buoyancy}} can play an important role at breakup. In order to better understand this role, here we numerically study Rayleigh-B\'enard convection for two immiscible fluid layers, in order to identify the effects of buoyancy on interface breakup. We explore the parameter space spanned by the Weber number $5\leq We \leq 5000$ (the ratio of inertia to surface tension) and the density ratio between the two fluids $0.001 \leq \Lambda \leq 1$, at fixed Rayleigh number $Ra=10^8$ and Prandtl number $Pr=1$. At low $We$, the interface undulates due to plumes. When $We$ is larger than a critical value, the interface eventually breaks up. Depending on $\Lambda$, two breakup types are observed: The first type occurs at small $\Lambda \ll 1$ (e.g. air-water systems) when local filament thicknesses exceed the Hinze length scale. The second, strikingly different, type occurs at large $\Lambda$ with roughly $0.5<\Lambda \le 1$ (e.g. oil-water systems): The layers undergo a periodic overturning caused by buoyancy overwhelming surface tension. For both types the breakup criteria can be derived from force balance arguments and show good agreement with the numerical results.


Introduction
Liquid can break up or fragment in multiphase turbulence (Deane & Stokes 2002;Villermaux 2007;Wang et al. 2019;Villermaux 2020). This physical phenomenon is very important for raindrops (Villermaux & Bossa 2009;Josserand & Zaleski 2003), for ocean waves and the resulting spray (Veron 2015), and even for the transmission of virus-laden droplets during coughing or sneezing (Bourouiba 2020;Chong et al. 2020). Once the physical mechanisms governing this important phenomenon are un-derstood, one can deduce quantitative criteria for the breakup to occur. In turbulence, for drops or bubbles the breakup criteria can be deduced from the balance of inertial and surface tension forces, as developed in the Kolmogorov-Hinze theory (Kolmogorov 1949;Hinze 1955). This theory well predicts the breakup criteria in experimental and numerical results in various flow systems, e.g., homogeneous isotropic turbulence (Martínez-Bazán et al. 1999;Perlekar et al. 2012;MuKolmogorov-Hinzeerjee et al. 2019), shear flows (Rosti et al. 2019), pipe flows (Hesketh et al. 1991) and ocean waves (Deane & Stokes 2002;Deike et al. 2016).
Whilst the classical Kolmogorov-Hinze theory considers only surface tension and inertial forces, in many multiphase turbulent flows also buoyancy can play an important role. Examples of multiphase buoyant turbulence include the hotspots and superswells in Earth's mantle (Davaille 1999;Tackley 2000) and even flows during sneezing and exhalation (Bourouiba et al. 2014). In such flows, the breakup of the interface between the fluids is the key phenomenon. Yet, the exact mechanisms that drives interface breakup when buoyancy is crucial is unknown.
The objective of the present work is to shed light on this mechanism. As examples for turbulent flow where buoyancy is important and at the same time can easily be tuned, we take thermal convection, namely Rayleigh-Bénard (RB) convection (Ahlers et al. 2009;Lohse & Xia 2010;Chillà & Schumacher 2012) of two immiscible fluids. We numerically investigate the breakup mechanisms of the interface between the two immiscible fluids. The immiscible fluids are first arranged in two layers according to their densities and then heated from below and cooled from above. Most previous studies of such twolayer RB convection were conducted in the non-turbulent regime (Nataf et al. 1988;Prakash & Koster 1994;Busse & Petry 2009;Diwakar et al. 2014). Experimental studies in the turbulent regime were reported by Xie & Xia (2013), who focused on the flow structures in each layer and the coupling modes between the flows in the two layers, including viscous coupling and thermal coupling. Besides the classical rectangular/cylindrical configuration, the two-layer RB convection in spherical-shell geometry was also numerically studied by Yoshida & Hamano (2016). However, these previous studies only considered the case for strong surface tension, where the interface between the layers does not break up. In this study we will for the first time explore the case with interface breakup, which happens when surface tension is sufficiently small.
The control parameters of two-layer RB convection are the density ratio Λ between two fluids and the Weber number W e, which is the ratio of inertia to surface tension. We will keep the Prandtl number P r (a material property) and the Rayleigh number Ra (the dimensionless temperature difference between the plates) fixed, at values allowing for considerable turbulence. Our main result will be the phase diagram in the parameter space (W e, Λ), in which we identify the non-breakup and breakup regimes. At increasing W e, we observe two distinct types of interface breakup. At small Λ ≪ 1, the mechanism is well-described by the Kolmogorov-Hinze theory. However, at large 0.5 < Λ 1, the breakup is dominated by a balance between buoyancy and surface tension forces, leading to a periodic overturning-type breakup.
The organization of this paper is as follows. The numerical methodology is introduced in Section 2. Then in Section 3, we validate our code by studying droplet fragmentation in thermal turbulence and favourably compare the results to the Kolmogorov-Hinze theory. The main results on the interface breakup in two-layer RB turbulence are presented in Section 4, including a discussion of the first and second types of interface breakup in Section 4.1 and Section 4.2, respectively, and the analysis of the critical Weber number for interface breakup in Section 4.3. We finalise and further discuss our findings in Section 5.

Methodology: Cahn-Hilliard approach coupled to finite difference scheme
In this study, we performed the simulations in a two-dimensional (2D) rectangular domain with aspect ratio Γ = 2 (width divided by height). Although 2D RB convection is different from 3D one, it still captures many essential features thereof (van der Poel et al. 2013). The direct numerical simulations solver for the Navier-Stokes equations is a secondorder finite-difference open source solver (Verzicco & Orlandi 1996;van der Poel et al. 2015), namely AFiD, which has been well validated and used to study various turbulent flows (Stevens et al. 2018;Zhu et al. 2018b;Blass et al. 2020;Wang et al. 2020a,b). To simulate multiphase turbulent flows, we combine AFiD with the phase-field method (Jacqmin 1999;Ding et al. 2007;Liu & Ding 2015), which has also been successfully applied to the interfacial Zhu et al. 2017;Chen et al. 2018Chen et al. , 2020 and turbulent flows (Soligo et al. 2019a;Roccon et al. 2019).
We consider two immiscible fluid layers of the same volume placed in the domain, named fluid H for the heavier fluid initially at the bottom and fluid L for the lighter fluid initially at the top. The mathematically sharp interface between two fluids is modeled by a diffuse interface with finite thickness, and can be represented by contours of the volume fraction C of fluid H. The corresponding volume fraction of fluid L is 1 − C. The evolution of C is governed by the Cahn-Hilliard equations, where u is the flow velocity, and ψ = C 3 − 1.5C 2 + 0.5C − Cn 2 ∇ 2 C is the chemical potential. The Cahn number Cn = 0.75h/D, where h is the mesh size and D the domain height, and the Péclet number P e = 0.9/Cn are set the same as in Liu et al. (2017); Li et al. (2020). To overcome the mass loss in the phase-field method, a correction method as proposed by Wang et al. (2015) is used. This correction method resembles that used in Soligo et al. (2019b) and exhibits good performance (see Section 3). The motion of the fluids is governed by the Navier-Stokes equation, heat transfer equation and continuity, here given in non-dimensionalized form. We have used the material properties of fluid H, the domain height D, the temperature difference ∆ between plates, and the freefall velocity U = √ αgD∆ to make these equations dimensionless, where α is the thermal expansion coefficient of fluid H and g the gravitational acceleration. Then we define ρ = C + Λ(1 − C) as the dimensionless density, P the dimensionless pressure, µ the dimensionless dynamic viscosity, C p the dimensionless specific heat capacity, k the dimensionless thermal conductivity, F st = 6 √ 2ψ∇C/(Cn We) the dimensionless surface tension force, θ the dimensionless temperature, j the unit vector in vertical direction. The superscript T stands for the transpose. Note that ρ, µ, C p and k in general vary in space.
In thermal flows, the density also depends on the temperature, so the dimensional . Further considering the Oberbeck-Boussinesq approximation in the Navier-Stokes equation (2.2), we have the dimensionless density ρ in the inertia term, ρθ, as the buoyancy term, and ρ/F r as the gravity term, which cannot be ignored as in the single phase simulation, due to the different densities of the fluids. Furthermore, we only consider the case without phase transition. The properties of fluid H and fluid L are set as follows: Their density ratio is Λ = ρ L /ρ H 1. Except for the density ρ, all other properties of the two fluids are the same. The other dimensionless parameters are W e = ρ H U 2 D/σ, Ra = αgD 3 ∆/(νκ), P r = ν/κ, the Froude number F r = U 2 /(gD) (the ratio of inertia to gravity). Here σ is the surface tension coefficient, ν = µ/ρ the kinematic viscosity, and κ = k/(ρC p ) the thermal diffusivity. We fix Ra = 10 8 , P r = 1, F r = 1 and Γ = 2 (these values chosen to both ensure flows in the turbulent regime and simplify the simulations), and only vary W e from 5 to 5000 and Λ from 0.001 (e.g. air-water system) to 1 (e.g. oil-water system).
The boundary conditions at the top and bottom plates are set as ∂C/∂j = 0, j·∇ψ = 0, no-slip velocities and fixed temperature θ = 0 (top) and 1 (bottom). Periodic conditions are used in the horizontal direction. All the simulations begin with the same initial velocity and temperature fields, which originates from a well developed turbulent flow at W e = 5 and Λ = 1. Uniform grids with 1000 × 500 gridpoints are used, which are sufficient for Ra = 10 8 and P r = 1, consistent with the grid resolution checks in Zhang et al. (2017). The details of the discretizations can be found in Ding et al. (2007); Verzicco & Orlandi (1996); M. S. Dodd (2014).

Droplet fragmentation in turbulent flow
We have verified our code against the existing theory from the literature. In this section RB convection with drops are simulated. Initially, the temperature field has a linear profile, the velocity field is set to 0, and a big drop of fluid H with a diameter of 0.5D is placed at the center of the domain. The plates are superhydrophobic for fluid H, i.e. C = 0 is used on both plates in the verification cases. We set Λ = 1 and W e from 2000 to 16000, and use uniform grids with 2000 × 1000 gridpoints. Note that the value of W e is large because it is a system Weber number, and the local Weber number of drops calculated from simulations is of the order of 1. where Smax is measured in the same way as in Hinze (1955), that is the diameter of the equivalent drop occupying 95% of the total dispersed area.

We
From the snapshot with the advecting drops in figure 1, we observe the large scale circulation, which is well known from single phase convection (Krishnamurti & Howard 1981;Xi et al. 2004;Zhu et al. 2018a;Xie et al. 2018). Figure 2 where S Hinze is the Hinze length scale and ǫ the energy dissipation rate of the turbulent flow. In the Kolmogorov-Hinze theory, one assumption is that the local Weber number defined by the size and velocity of the drops adjusts such that it is W e local ∼ O(1). Indeed, in our simulations the local drop size adjusts correspondingly, so this assumption is fulfilled. The second assumption is that the flow exhibits inertial subrange scaling in the region of wave lengths comparable to the size of the largest drops. The spatial location where this assumption holds in RB convection is in the bulk region of convection (Lohse & Xia 2010). Therefore, the Kolmogorov-Hinze theory can indeed also be reasonably applied to 2D RB convection. At the same time the results show that the code and the approach give consistent results. Besides, figure 2(a) shows the temporal evolution of mass error E mass = |M t − M 0 |/M 0 , where M t is the mass of fluid H at time t and and M 0 the initial mass. The results indicate good mass conservation.

Interfacial breakup in two-layer turbulent thermal convection
In two-layer RB convection with an initially smooth interface between the two fluids (Xie & Xia 2013), the densities of the fluids depend on both Λ and the local temperature. At small Λ, for example, the air-water system with Λ = 0.001, fluid H (water) is always heavier than fluid L (air) even if fluid H, as the bottom layer, is hotter. In contrast, at large Λ, e.g., an oil-water system with Λ ≈ 1, fluid H (water) is hotter than fluid L (oil), so it can be lighter. So depending on the value of Λ two distinct types of flow phenomena emerge.

First type of interface breakup occurring for small Λ ≪ 1
At small Λ, fluid H forms the bottom layer and fluid L the top one and large scale circulations are observed in each layer between the interface and the respective plate, as seen in figure 3(a). With increasing W e, i.e. decreasing effects of surface tension compared to inertia, the interface becomes more unstable. At low W e (figure 3a), the interface only slightly deforms because the surface tension is large enough to resist the inertia, such that the convection rolls are well-ordered. The temperature profile is similar to that obtained from two-layer RB convection experiments (Davaille 1999). As W e increases (figure 3b), the interface undulates due to the plumes. Each crest and trough on the interface is caused by a rising, or respectively settling plume in the heavier fluid H. In this situation, inertia is resisted by gravity together with surface tension. When W e keeps increasing (figure 3c), the interface eventually breaks up and drops detach from the interface between two layers.
This "first type of interface breakup" (as we call it) occurs at small Λ. The process of the breakup begins from a settling plume in fluid H thanks to which the interface is pulled downwards, leading to a filament (or trough) on the interface (see figure 4a). If the filament length S f ila (defined in figure 4a) grows larger than the Hinze length scale S Hinze , the filament will pinch off from the interface. Within the Kolmogrov-Hinze theory (Hinze 1955), the Hinze length scale S Hinze in (3.1) is determined by the energy dissipation rate ǫ of the turbulent flow. As seen from figure 3(c), more drops of fluid L exist in fluid H than of fluid H in fluid L. This finding resembles the breakup of the ocean waves (Deane & Stokes 2002), leading to more bubbles in water than drops in air. The reason is that S Hinze is smaller in fluid H than in fluid L as ρ H > ρ L .

4.2.
Second type of interface breakup occurring for large 0.5 < Λ 1 We now come to the large Λ ≈ 1 case: Since fluid H carries hotter fluid than fluid L, due to thermal expansion it can become lighter than fluid L, inverting the original density contrast at equal temperature. In this situation, buoyancy drives fluid H upwards and fluid L downwards. This leads to wave crests and troughs, as shown in figure 5. If W e is low (figure 5a), the surface tension can maintain a stable interface, though it is wobbling. However, if W e increases (figure 5b), the wobbling wave on the interface can amplify more and more until it finally touches the upper and/or lower plate and breaks up. We call this type of interface breakup the "second type of breakup".
For this second type of interface breakup, the breakup process is strikingly different from the first type. A periodic overturning is observed both in the fluid dynamics and in the heat transfer (see figure 5c and  strong enhancement of the Nusselt number N u, in the shown case 5 times of N u without breakup, as shown in figure 6. Then, fluid H on the top gets cooler and thus heavier, while fluid L at the bottom warmer and thus lighter. Once again, the breakup occurs after t 3 and the fluids swap their positions (see figure 5c). Since fluid L is lighter than fluid H at the same temperature, it is easier to rise and touch the cold plate. Thus, with fluid L as the bottom layer, the temperature of the bottom layer during the breakup is lower than that when fluid H is the bottom layer. This is also reflected in the heat transfer. The peak of N u after t 3 is smaller, only 3 times of N u without breakup, and the preparation time for breakup from t 2 to t 3 is shorter than that from t 4 to t 5 .

Critical Weber number for interface breakup
The full phase diagram in the parameter space (W e, Λ) -revealing when and what regime occurs -is plotted in figure 7. When W e is larger than a certain critical value W e c , which depends on Λ, the interface breaks up. It is noteworthy that the transition between the non-breakup and breakup regimes show two distinct trends, which correspond to the two above identified types of interface breakup, respectively. The natural question that arises here is then: What sets the critical value W e c at given Λ?
In the first type of interface breakup (small Λ ≪ 1), detaching drops are generated from the initial interface when the filament length S f ila is of the order of the Hinze length scale S Hinze . S f ila is estimated by analyzing the force balance: the sum of gravity and surface tension force counteracts the inertial force, where U H = αg(D/2)(∆/2) and δρ is the density difference from the bottom (fluid H) to the top (fluid L) of the filament. We define δρ = ρ H (T H ) − ρ L (T L ), where T i is the temperature of fluid i. We further found that the value of gravity is one order of magnitude greater than surface tension based on the data of cases near the transition region in the first type. This indicates that the generation of the filament is dominated by gravity and inertia. Therefore we neglect the term σ/S f ila in (4.1). Note however that the * * * * surface tension force still plays an important role to determine S Hinze in (3.1). Combining (4.1), (3.1) and the exact relation ǫ = ν 3 /D 4 (N u − 1)RaP r −2 (Shraiman & Siggia 1990;Ahlers et al. 2009), the dimensionless form of S f ila ∼ S Hinze yields with the non-dimensional temperatures θ i = (T i − T c )/∆ with i being H and L. θ H and θ L are both taken as 0.5 given that the filament is generated near the interface, where the temperature is 0.5. To further simplify (4.2), N u is regarded as constant because the simulation data show that N u varies only within 15% in the non-breakup regime. Given that the N u, Ra, P r and F r are all constant, the criteria for the first type of interface breakup simplifies to In the second type of interface breakup (large Λ ≈ 1), the hot fluid H is lighter than the cold fluid L, so the buoyancy caused by the unstable temperature stratification can overcome the surface tension, leading to waves on the interface. The buoyancy acting on the wave originates from the density difference between the fluid above and below the wobbling interface (see figure 5). The balance is described by where T i is the average temperature in the bulk of fluid i, and the interface deformation is of the order of D because the breakup occurs when the wave amplitude is larger than half of the plate distance D, thus that the interface touches the plates (see figure 5b).
The dimensionless form of (4.4) reads (4.5) From the temperature profile in figure 3, we estimate θ H = 0.75 and θ L = 0.25. Then (4.5) simplifies to (4.6) Figure 7 shows that (4.3) and (4.6) indeed well describe the scaling relations of transitions between the non-breakup and breakup regimes.
In the breakup regimes at large Λ ∼ 1, as W e increases, the periodically overturning of fluid layers gradually becomes chaotic with more and more drops generated from the breakup of the interface. Eventually, the flow pattern is determined by the advection of the small drops (figure 1). There is no clear boundary in the parameter space for this transition and it happens over a considerable range in parameter space. Therefore, in this study, we only focus on when and how the breakup occurs.

Conclusions
In summary, we have numerically shown two distinct types of interface breakup in two-layer RB convection, which result from different dominant forces. At small Λ ≪ 1, a filament is generated on the interface due to the competition of inertial force and buoyancy. The interface breaks up in the form of filament detachment when the local filament thicknesses exceed the Hinze length scale. At large Λ with roughly 0.5 < Λ 1, the periodic overturning-type breakup is caused by buoyancy overwhelming surface tension. From the force balance arguments above, we derive the breakup criteria for two types, respectively. Our approaches show good agreements with the numerical results.
The threshold of regimes in this work is derived from a force balance argument, which is not limited to 2D. For 3D, of course, some expressions need to be modified, for example, surface tension force from F st = σ/R in 2D to F st = σ(1/R 1 + 1/R 2 ) in 3D. Besides, we also note that previous studies also showed that 2D and 3D Rayleigh-Bénard convection have similar features for P r 1, see van der Poel et al. (2013). Therefore, we expect our results from 2D flow could be directly extended to 3D flow.
Our findings clearly demonstrate that interface breakup in multilayer thermally driven turbulence cannot be solely described by the Kolmogorov-Hinze theory, which is only applicable when the lower layer is much denser than the upper layer or when thermal expansion effects do not play a role. Interestingly, when the lower layer is less dense than the upper layer, the system is unstably stratified, leading to the new breakup type described in this paper where buoyancy and surface tension are the dominant forces. It would be interesting to test our predictions of (4.2) and (4.5) for the transitions between the different regimes in a larger range of the control parameters Ra, P r, and F r, which were kept fixed here. More generally, our findings emphasize the role of buoyancy in interfacial breakup. Clearly, buoyancy will also play a prominent role in the interfacial breakup in other turbulent flows, such as Bénard-Marangoni convection, and horizontal and vertical convection. All these phenomena add to the richness of physicochemical hydrodynamics of droplets far from equilibrium (Lohse & Zhang 2020).