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Turbulent mixing of oil droplets in a round water jet

Turbulent mixing of oil droplets in a round water jet

If two

If two droplets collide, during the interaction time, t int , the fluid film between the droplets mustvanish, during the drainage time, t drain . The probability of coalescence in case of a collisiondepends on the ratio of these time scales:P coal = e − t draint int (2.61)It is beyond the scope of this project to give a complete quantitative analysis of this coalescenceprocess so this probability is simply set to 1. In order to predict whether coalescence might playa role of importance, the collision frequency of droplets per unit volume, n c ω col , within the microscale of size η k is compared to the reciprocal time the droplets are in this micro structure, 1/τ k .Then, if 1/τ k ≫ n c ω col ηk 3 , the effects of collisions are neglected. This statement can be rewrittenas:c 2 (η k /d d ) 3 ≪ 1 (2.62)2.2.4 Concentration development in a turbulent jetIn this section, the oil droplets are assumed to behave as a passive scalar in a turbulent flow fieldwithout diffusion. After Reynolds decomposition of the oil concentration (c = c+c ′ ), the averagedtransport equation reads:∂ū¯c∂z + 1 ∂r¯v¯c= − ∂u′ c ′r ∂r ∂z− 1 ∂rv ′ c ′(2.63)r ∂rThe mean concentration scales with c s , while the turbulent transport terms u ′ c ′ and v ′ c ′ scalewith UC. It will be verified experimentally that the ratio of these scales is determined by:UC l)= O((2.64)u s c s LApplying the boundary-layer approximation (2.5) and the continuity equation (2.3), the equationof transport can consequently be written as:u ∂c∂z + v ∂c∂r = −1 ∂rv ′ c ′r ∂r(2.65)The term v ′ c ′ can be interpreted as the turbulent transport of scalar concentration in the r-direction (turbulent flux).Like the velocity profiles, the concentration profiles are assumed to be self-similar as well. Moreover,it is assumed that the typical length scale of the concentration profile, l c , scales with thewidth of the velocity profile: l c = γl. Consequently, the concentration profile can be expressed asa function of η as well.c = c s h(η)v ′ c ′ = UCk(η) (2.66)in which c s it the concentration at the centreline of the jet. Now, the conservation of solute matterflux, Q, is considered:Q = 2π∫ ∞0∫ ∞rū¯cdr = 2πu s c s l 2 ηf(η)h(η)dη = Q 0 = c 0 Φ jet (2.67)in which Q 0 is the initial solute matter flux defined by the jet flow, Φ jet , and the initial concentration,c 0 . Using u s l = constant, it follows that, to ensure self-similarity, also c s l must be constant.Thus, the development of the centreline concentration as a function of z becomes:c s = c 2l = c 2α(z − z 0 )016(2.68)

Substituting (2.66) in the equation of transport (2.65) and using (1/c s )dc s /dz = −(1/l)dl/dz, itfollows that:Uc s dCu s dη F (η)h(η) = d ηk(η) (2.69)dηA self-similar solution is only possible if:Uc sCu s= c 3 = constant (2.70)Integrating (2.69) to η, gives the following relation between the self-similar concentration andvelocity profiles. The boundary condition F(0)=0 is used to determine the integration constant.F (η)h(η) = 1 c 3ηk(η) (2.71)In order to find a solution for the concentration profile h(η) it is necessary to formulate a closurehypothesis for the turbulent flux, v ′ c ′ . Again, K-theory is applied [Hinze, 1975]:v ′ c ′ ∂c≡ UCk(η) = −K c∂r = −K cc s dh(η) (2.72)l dηIn analogy with the eddy-viscosity, K, the eddy-diffusivity, K c , scales with the macro structure ofthe turbulence: K c = κUL. The unknown transport coefficient, κ, that is introduced, determinesthe ratio between matter transport and momentum transport.Substituting k(η) in the equation of transport (2.69) and taking K c = κu s Bαl (see section 2.1.5),it follows that:F (η)h(η) = −κBη d h(η) (2.73)dηUsing the expression for F (η) derived in section 2.1.3, the solution for c s h(η) is given by:c ≡ c s h(η) = c s1(1 + η 2 /8B) 2/κ (2.74)Note, that if κ = 1, the concentration profile and the velocity profile collapse. The larger thistransport coefficient, the wider the concentration profile becomes. In figure 2.7a the concentrationprofile is plotted for two values of κ.From the expression for the concentration profile it is possible to derive the spreading rate of theconcentration, α c , as a function of κ. The characteristic length scale, l c , is defined as the widthfor which the concentration is equal to 1 e c s. Say η = γ = l c /l, then the solution for h(η) reads:1e = 1(2.75)(1 + γ 2 /8B) 2/κThe relation between the spreading coefficient, γ, and the transport coefficient, κ, becomes:√κ = 2ln(1 + γ 2 /8B) or γ = 8B(e −1/2κ − 1) (2.76)The spreading coefficient, γ must be determined experimentally. It is known from previous experimentsthat γ > 1, so the scalar transport is faster than the momentum transfer. In terms of κ,κ ≈ 1.4 [Hinze, 1975]. This could be explained by the fact that the fluid carries the oil droplets,while its kinetic energy is dissipated and momentum is lost whereas the scalar quantity is always17

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