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Turbulent mixing of oil droplets in a round water jet

Turbulent mixing of oil droplets in a round water jet

Consequently, a

Consequently, a self-similar solution of the equation of motion is only possible if:u 2 s dlU 2 = constant (2.14)dzThe ratio u s /U is assumed to be constant in the self-similar region, as only a single velocity scaleis needed to describe the flow field. If the constant in (2.14) is incorporated in the ratio u s /U, itfollows that:dldz = U 2u 2 = constant (2.15)sThis is consistent with relation 2.6 that must be verified experimentally. Combined with u s l = c 1 ,the following scaling relations for the width and centreline velocity of the jet may be applied:l = U 2u 2 (z − z 0 ) = α(z − z 0 ) (2.16)su s =c 1α(z − z 0 )(2.17)in which α is the spreading rate of the jet (dl/dz) and z 0 is the z-coordinate from which theself-similar part of the jet virtually originates. In figure 2.2 a schematic drawing of the self-similarturbulent jet is drawn.Figure 2.2: Schematic drawing of a self-similar jetThe position downstream for which the self-similar region is reached, is about 70 jet diameters[Wygnanski and Fiedler, 1969]. There has been an extensive debate on the question whether theself-similarity is universal, implying that the spreading rate, α, has the same value in every jetexperiment. However, some recent numerical studies show a strong dependence of the velocityprofile on initial conditions [Boersma, 1998] which agrees with the analysis of [George, 1989].George proposes a scaling of the flow variables in which a dependence on the initial conditions iskept in order to get universal self-similarity. Therefore, the conclusion that the spreading rate isnot a universal constant seems reasonable. When taking l as the width of the jet for which theaxial velocity is 1 e u s, [Panchapakesan and Lumley, 1993] found a spreading rate of 0.115. [Aanen,2002] found a spreading rate of 0.109.6

2.1.3 Velocity profilesIt is not possible to solve the equation of motion (2.7) and derive a velocity profile, u s f(η), withouta closure hypothesis for the Reynolds shear stress (see section 2.1.1). Nowadays, it is very wellpossible to perform a direct numerical simulation to a jet with low Reynolds number [Boersma,1998]. The axial velocity profile found with the simulations of Boersma are Gaussian and agreewith experimental observations. In this section, two possible closures are presented: K-theory andthe Prandtl mixing length hypothesis. At first, the analytical solution of the spatial developmentof the jet is presented according to K-theory (Boussinesq-hypothesis). This theory makes use ofthe analogy with a molecular model for shear stresses (τ) [Nieuwstadt, 1998]:τρ = ν ∂U∂y(2.18)In a gas, the kinematic viscosity, ν, is equal to αaλ, in which α is a constant, a is the velocity ofthe molecules and λ is the free path length. Instead of the viscosity caused by colliding molecules,K-theory assigns an eddy-viscosity (K) to colliding eddies:−u ′ v ′ = K ∂u∂r(2.19)The eddy-viscosity is determined by the typical length (L) and velocity (U) scales of the turbulence:K ∼ UL (2.20)As the molecular shear stresses are determined by local quantities (λ and a), the analogy used bythe Boussinesq hypothesis is possibly valid, only if the turbulence scales are local quantities aswell. Unfortunately, there is no physical ground for this assumption as the turbulence is normallynot local, but scales with the length scale of the flow geometry instead (L ∼ L). Nevertheless,it turns out that for a jet geometry L ≪ L (see section 2.1.5) and K-theory is applicable in thisspecific case.Continuing from (2.19), an expression for the function g(η) in terms of F is found:−u ′ v ′ ≡ U 2 g(η) = K ∂u∂r = K u s ∂( F′ )l ∂η ηTo ensure self-similarity, it is necessary that:Ku sU 2 l⇒g(η) = Ku sU 2 l∂∂η( F′ )η(2.21)= B = constant (2.22)In order to come to a solution of the equation of motion (2.13), it is first integrated to η, yielding:− F F ′η= ηg + A (2.23)The integration constant A is equal to zero, as g and F ′ /η → 0 for η → 0. Substitution ofthe expressions for g(η) (2.21) and B (2.22) in the integrated equation of motion (2.23), leads toanother integrable equation with the following solution:− 1 2 F 2 = B{(ηF ′ ) − 2F } + C (2.24)in which C = 0 because F = 0 for η = 0 as the jet axis is a streamline (ψ must be constant, u s l 27

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