We formulate the droplet entrainment detached from a thin liquid film sheared by a turbulent gas in a circular pipe. In a time-averaged sense, the film has a Couette flow with a mean velocity of um. Then, a roll wave of wavelength λ and phase velocity uc is formed destabilized through Kelvin–Helmholtz instability, followed by a ripple wave of wavelength λp due to Rayleigh–Taylor instability, wherein the vorticity thickness of the gas stream is consistently a characteristic length scale. Superposing the two types of waves in axial and transverse directions produces conical cusps as the root of ligaments, from which droplets are torn off. The droplet entrainment rate is derived as λpλucum, validated by recent experimental results.

Circular pipe flow, in which a fast gas flowing along the central channel accompanies a liquid film slowly propagating on the wall, has been an important issue in various situations such as in film-cooling techniques,1,2 the drying of cleaning solutions,3 icing on aircrafts,4,5 and mechanical erosion.6 The liquid–gas interface is no longer smooth by the shear stress due to the large velocity gap between the gas and the liquid, leading to liquid film destabilization and droplet entrainment. The wavy film directly enhances the heat/mass transfer across the interface, and the entrainment influences turbulent intensities in the gas stream.7 Past studies assessed the sequential scenario,8–11 as schematically illustrated in Fig. 1, that the gas flows with velocity ug and vorticity thickness δ over the smooth film on the wall, soon drives the film with a mean velocity of um and thickness of h, initiates a roll wave of wavelength λ and amplitude a by Kelvin–Helmholtz (KH) instability, accelerates wave crests producing transverse ripple waves of wavelength λp by Rayleigh–Taylor (RT) instability, stretches ligaments, and finally entrains fragmented droplets. Simultaneously, bubbles burst and droplets impact on the film,8,12 and the waves intricately merge10 and break.13 The complex wavy patterns are also called disturbance waves.5,6,14,15

FIG. 1.

Schematic events of thin liquid film dynamics on a channel wall subjected to fast gas stream.

FIG. 1.

Schematic events of thin liquid film dynamics on a channel wall subjected to fast gas stream.

Close modal

Recent progress includes quantitative measurement for unsteady film characters, film thickness, advective and phase velocities, wavelength and frequency of instability waves, and entrainment rate, defined by the flow rate ratio of entrained droplets to initial liquid, for the liquid–gas flow inside pipes6,16 and on flat plates.5,10,12 Another approach, theoretical framework, has been devoted to formulating those gas-sheared liquid film phenomena; however, the incorporated empirical correlations restrict the prediction applicability.1,14,17 For instance, Lane et al.17 proposed two mathematical models for calculating the entrained flow rate from the roll wave, switching them according to flow conditions. The development of a physical model is still under progress.1,14,15,17

Here, in this paper, we theoretically express the comprehensive thin liquid film dynamics sheared by turbulent gas, considering KH and RT instabilities, eventually achieving the straightforward formulation of the entrainment rate. The validity is convinced by the experimental results of Yarygin et al.16 We also discuss the underlying mechanism of the wavy film formation and entrainment.

Inside the pipe with diameter D, the liquid film wets the entire inner surface. Defining the axial direction as coordinate x and liquid Reynolds number as Rel = ρlumh/ηl, wherein ρl is the liquid density and ηl is the viscosity coefficient, the liquid film is fully covered by the velocity boundary layer at x/h > Rel, with a typical area of x ≥ 10−3 m at Rel = 10 and h = 10−4 m. The time-averaged liquid volume flow rate Q is given as

Q=πDhum.
(1)

The shear stress of τ by the turbulent gas flow acting on a smooth pipe or on the film in the present analysis is established by Blasius,18–20 

τρgug2=0.040Reg1/4,
(2)

where ρg is the gas density and Reg = ρgugD/ηg is the Reynolds number with viscosity coefficient ηg. Assuming Couette flow for the film, in which the velocity equals zero on the wall and 2um at the liquid–gas interface with a linear profile,8 the balance of shear stress across the interface is deduced,

ηl2umh=τ.
(3)

From Eqs. (1)(3), we obtain the time-averaged liquid velocity and thickness in the normalized form, respectively,

umQ/(πD2)=0.14ηgηlρlρgRel1/2Reg7/8,
(4)
hD=7.1ηlηgρgρlRel1/2Reg7/8.
(5)

Hereafter, the theoretical results of the present formulation are excluded for any parameters fitting to the experimental results. Figure 2 demonstrates the validity of Eqs. (4) and (5). Working fluids are air for the gas stream and ethanol for the liquid film. The film flow is laminar at Rel = 8.8, corresponding to Q = 0.42 μm3/s. As Reg increases, the gas stream accelerates the liquid film [Fig. 2(a)] following umReg7/8, inversely thinning the thickness [Fig. 2(b)], convincing strong dependency of the film flow on the shearing gas. The film front velocity coincides with the main film body upstream. The overestimation of h calculated by Eq. (5) is attributed to non-implementation of the entrainment effect, thinning h. The agreement between the theoretical and experimental results evidences that the Couette flow pattern formed inside the thin liquid film is a reasonable assumption in a time-averaged sense. Here, surface tension plays no role, and gravity is insignificant at Reg > 104 because the Froude number, um/(hg)0.5, is much larger than unity (g = 9.8 m/s2).

The large velocity difference ug/ul ≫ 1 induces KH instability. At the viscous flow condition, the instability is stimulated by the inflectional point created inside the boundary layer, which was originally declared by Lord Rayleigh21 and generalized by Villermaux22 to any velocity profile. The effect of viscosity on the wavelength and growth rate of the instability becomes apparent at Rel ≪ 10; thus, the present analysis of Rel ≈ 10 can neglect the viscous damping.22 Here, the gas stream has a continuous velocity profile with a non-zero boundary layer on the liquid–gas interface (see Fig. 1), represented by vorticity thickness δ (=ηgug/τ) at the circular pipe as20 

δD=25Reg3/4.
(6)
FIG. 2.

Time-averaged liquid film characteristics and experimental results by Yarygin et al.:16D = 10 mm, ρl = 790 kg/m3, ηl = 1.2 mPa s, ρg ≃ 1 kg/m3, ηg = 18 μPa s, and ug is up to 320 m/s (Mach number of unity), proportional to the mass flow rate of 0.4 g/s–20 g/s. (a) Film mean velocity. The closed circle for the velocity of the film main body is obtained from conservation of Q using experimental results of h in (b) for film thickness. (b) The absolute measurement error of h is ±5 μm or h/D = 5 × 10−4.

FIG. 2.

Time-averaged liquid film characteristics and experimental results by Yarygin et al.:16D = 10 mm, ρl = 790 kg/m3, ηl = 1.2 mPa s, ρg ≃ 1 kg/m3, ηg = 18 μPa s, and ug is up to 320 m/s (Mach number of unity), proportional to the mass flow rate of 0.4 g/s–20 g/s. (a) Film mean velocity. The closed circle for the velocity of the film main body is obtained from conservation of Q using experimental results of h in (b) for film thickness. (b) The absolute measurement error of h is ±5 μm or h/D = 5 × 10−4.

Close modal

For the KH instability accompanying a finite-thickness vorticity layer, δ is a characteristic length scale, providing the wavelength of the roll wave independent of capillarity11 as

λδρlρg.
(7)

The roll wave propagates downstream with phase velocity23 

uc=umρl+ugρgρl+ρgugρgρl.
(8)

Neglecting the term of um produces less than 10% discrepancy. Figure 3 compares the results of the experiment16 and Eq. (8). At the constant ρg condition, uc clearly increases being proportional to Reg. We calculate λ from Eq. (7), showing that λ is shorter than 5 mm at Reg = 4 × 104. The experiment16 allocates two conductive sensors along the axial direction with 5 mm distance for the velocity measurement. At a large Reg region, the lack of spatial resolution causes a slight change in velocity trend. The good agreement, however, at least resolved experimental results at Reg ≤ 4 × 104, indicates that the wave characters of Eqs. (7) and (8), originally established for free shear layers, are applicable to the present film dynamics on the wall, without considering the shear stress fluctuations.24 The validity of Eq. (8) for the film flow was also demonstrated by Berna et al.15 

The gas friction is superior to the stabilizing force of surface tension as τσ/λ, defining the surface tension coefficient of σ. Before overlapping the KH instability waves being significant, the amplitude of the roll wave grows to the order of film thickness5 as ah. Then, the gas progressively accelerates the heavy wave crests toward a much lighter gas phase, leading to RT instability.25,26 As a result, the regularly lined roll wave is strongly distorted into the transverse direction being a ripple wave. The wavelength of λp is given as follows:11 

λp=δWeδ1/3ρgρl1/3.
(9)

Here, the Weber number is Weδ = ρgug2δ/σ based on the length scale of δ. The amplitude of the ripple wave enlarges, creating a cusp by the superposed roll wave and ripple wave, as shown in Fig. 4(a). Since the cusp is the origin of ligament downstream producing droplets, the cusp volume approximately coincides with the total droplet volume detached from the cusp. As discussed later, we derive λp < λ; thus, the cross-sectional area of the conical cusp corresponds to λp2 and the height is a equivalent with h. The volume of a single cusp is estimated as vλp2h. As shown in Fig. 4(b), the number of cusps produced is πD/λp along the circumference of the pipe, propagating with a rate of (λ/uc)1 along the axial direction. The entrained volume flow rate Qd is determined by the product of v and all cusp numbers passing in unit time, eventually formulating the entrainment rate by the wavelength ratio and velocity ratio,

QdQπDhλpλ/uc/Q=λpλucum.
(10)
FIG. 3.

Phase velocity of the roll wave. The same condition as in Fig. 2.

FIG. 3.

Phase velocity of the roll wave. The same condition as in Fig. 2.

Close modal
FIG. 4.

Cusp model as superposing roll waves and ripple waves. The flow direction is from left to right. (a) Single cusp. (b) Top view. The closed circles indicate cusps, arranged in the axial and circumferential directions.

FIG. 4.

Cusp model as superposing roll waves and ripple waves. The flow direction is from left to right. (a) Single cusp. (b) Top view. The closed circles indicate cusps, arranged in the axial and circumferential directions.

Close modal

Figure 5 presents the calculated results of Eq. (10) plotted against the film Weber number, Weh=ρgug2h/σ. At Weh ≥ 10, up to 60% of the initial liquid film is torn off in the experiment,16 wherein the theoretical prediction well reproduces the experimental results. Thus, the validity of the present sequential modeling at every intermediate step achieving Eq. (10) is convinced, including λ and λp due to the two types of instabilities.

FIG. 5.

Entrainment rate. Flow conditions: Q = 0.42 μm3/s, σ = 23 N/m, and ug = 320 m/s. The mass flow rate of gas is 0.4 g/s–20 g/s.16 The experimental data typically at Weh = 10 have a relative error of 15%.

FIG. 5.

Entrainment rate. Flow conditions: Q = 0.42 μm3/s, σ = 23 N/m, and ug = 320 m/s. The mass flow rate of gas is 0.4 g/s–20 g/s.16 The experimental data typically at Weh = 10 have a relative error of 15%.

Close modal

We calculate λ, λp, δ, and h, shown in Fig. 6(a), under the same flow condition. At a surface tension dominant region of Weh < 10, the long wavelength of λ and λp, even longer than D, indicates that the almost flat liquid film is retained and cusps are rarely produced. In contrast, as Weh increases, the wavelengths shorten, producing many cusps on the wavy film, and droplets detach from the cusps, convincing that the cusp formation by film instabilities is an essential factor for droplet entrainment. Those wavelengths are longer than δ and h.

FIG. 6.

Calculated results of length and time scales on the wavy film. The flow condition is the same as in Fig. 5. (a) Normalized length scales by D. (b) Timescales. t1: cusp formation time, t2: cusp breakup time, and t3: wave passing time through the nozzle.

FIG. 6.

Calculated results of length and time scales on the wavy film. The flow condition is the same as in Fig. 5. (a) Normalized length scales by D. (b) Timescales. t1: cusp formation time, t2: cusp breakup time, and t3: wave passing time through the nozzle.

Close modal

The film grows to the amplitude of a/h, order of unity, in a timescale of t1λ/(2πuc) ln(a/h) ≈ λ/(2πuc) through KH instability, and the following cusp fragments in the capillary timescale of t2(ρlλp2h/σ)0.5. The wave passes through the nozzle in t3 = 2D/uc at a nozzle axial length of 2D. Satisfying t1 + t2 < t3 confirms that the entrainment event completes inside the nozzle. Figure 6(b) shows the calculated timescales. As Weh increases, t1 and t2 steeply shorten as the wavelengths are short. At Weh > 6, the condition t1 + t2 < t3 is realized, corresponding to the entrainment rate exceeding 10% in Fig. 5, consistently well validating the present model.

In summary, we deduced the comprehensive formulation describing the liquid film dynamics driven by the co-current fast gas in the pipe. The recent experiment conducted by Yarygin et al.16 confirmed the validity of the present formulation at the linear flow regime that the thin film was of Rel ≈ 10, the amplitude of instability waves was comparable to the film thickness, and the gas velocity was up to the Mach number of unity. The time-averaged film flow and the roll wave were of inertia–viscosity matter, and the ripple wave and the entrainment were identified as the inertia–viscosity–capillarity problem. For both the roll wave and ripple wave stimulated by respective instabilities of KH and RT, the vorticity thickness of the turbulent gas flow on the liquid film was an essential length scale, consistent with the work of Marmottant and Villermaux.11 The entrainment was obviously the result of growing cusps on the wavy film, superimposing the roll and ripple waves together, successfully described by the present physical modeling framework.

As remaining issues, the present simple strategy is extended to a thick film flow with large disturbance waves10 and to predicting the size of the droplet fragment from the ligaments.

Valuable comments were given through discussion by Dr. Daimon, Mr. Fujii, and Mr. Kawatsu in JAXA.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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