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Turbulent mixing of oil droplets in a round water jet

Turbulent mixing of oil droplets in a round water jet

2.1.1 Boundary-layer

2.1.1 Boundary-layer equationsStarting point for the derivation of the governing equations of a turbulent jet flow is the Reynoldsdecomposition of the flow variables, which divides an instantaneous variable into a mean part anda fluctuating part. For example, for the instantaneous axial velocity:u = u + u ′ (2.2)The ensemble averaging operator, denoted with an overbar, fulfils the so called Reynolds conditions[Nieuwstadt, 1998]. Applying this decomposition and neglecting the molecular terms (Re ≫1), the averaged Navier-Stokes equations for a stationary axisymmetric geometry in cylindricalcoordinates become:∂u∂z + 1 ∂rvr ∂r = 0 (2.3)∂ūū∂z + 1 ∂rū¯vr ∂r= − 1 ∂pρ ∂z − ∂u′ u ′− 1 ∂ru ′ v ′∂z r ∂r∂ū¯v∂z + 1 ∂r¯v¯v= − 1 ∂pr ∂r ρ ∂r − 1 ∂rv ′ v ′− ∂u′ v ′r ∂r ∂zin which u is the mean velocity in the axial direction, z, v the mean velocity in the radial direction,r and p the mean pressure. Equation 2.3 describes the conservation of mass (continuity) andequations 2.4 the conservation of momentum.The jet flow develops as a function of the downstream (axial) direction, z, with characteristiclength scale L. This development is very slow in comparison to the development in the radialdirection, r, with characteristic length scale l. Thus, the following boundary-layer approximationmay be applied:∂∂z ≈ 1 L ≪ ∂ ∂r ≈ 1 (2.5)lFurthermore, the mean axial velocity, u, is scaled by a characteristic velocity scale u s and themean radial velocity is scaled according to 2.3 by u s l/L. The turbulent terms are scaled by themacro scale U: u ′ i u′ j ∼ U 2 . It is hypothesized that the ratio between these two scales is determinedby:U 2u 2 s( l= OL)From the Navier-Stokes equations (2.4), an approximation of the equation for conservation ofmomentum in the z-direction can be derived by applying this scaling and thereafter neglecting theterms with a relatively small order of magnitude:(2.4)(2.6)u ∂u∂z + v ∂u∂r = −1 ∂ru ′ v ′r ∂r(2.7)This equation and the continuity equation 2.3, are the so-called boundary-layer equations [Nieuwstadt,1998]. Note that it is assumed that the pressure outside the turbulent area is constant(dp 0 /dz = 0). The term u ′ v ′ is the Reynolds shear stress, which can be interpreted as the transportin the r-direction of momentum in the z-direction. As it concerns an extra unknown variable,an extra relation is needed to solve this equation analytically. This is called a closure-problem[Nieuwstadt, 1998]. In the next sections a possible solution of the equation of motion is formulated.First, a mathematical description of the flow field is introduced assuming self-similarity (2.1.2).Thereafter, the velocity profiles are derived using K-theory as a closure hypothesis (2.1.3).4

2.1.2 Self-similarityOne essential assumption in the theory describing a turbulent jet is that during the developmentof the flow in the downstream direction, the turbulence maintains its structure. In other words:the jet is self-similar. By definition: if all velocities are reduced by one velocity scale and alldimensions by one length scale, the flow patterns expressed in the reduced quantities becomeidentical [Hinze, 1975]. As a consequence, the following mathematical description of the flow fieldcan be applied.( y)u = u s fl( y)−u ′ v ′ = U 2 g(2.8)lAs a turbulent jet is an axisymmetric geometry, this description is transformed to cylindricalcoordinates. Therefore, a stream function, ψ, is introduced which fulfils the continuity equation(2.3):u = 1 ∂ψr ∂rv = − 1 ∂ψ(2.9)r ∂zThe condition of self-similarity applied to the stream function, ψ, leads to the following descriptionof the flow field:( r)ψ = u s l 2 Fl( r)−u ′ v ′ = U 2 g(2.10)lSubstitution of ψ in (2.9), yields for the velocities:u = u sF ′ (η)η= u s f(η)∂l(v = u s F ′ − F )∂z η(2.11)in which η = r/l. Normalising f(η) with f(0) = 1, makes u s the velocity at the centreline of thejet. In the derivation of v the assumption is made that u s l = constant, which will be verifiedbelow.In this description of the flow field, two scales appear: the typical velocity (u s ) and length (l) scale.These scales are a function of z. To investigate their z-dependence, the equation of conservationof momentum flux I is considered:I = 2π∫ ∞0∫ ∞u 2 rdr = 2πu 2 sl 2 ηf 2 (η)dη = I 0 (2.12)in which I 0 is the momentum flux at the jet tube. In order to have a constant momentum flux, itis obvious that the product u s l must be constant. Substituting the expressions for the velocities(2.11) in the equation of motion (2.7) and applying (1/u s )du s /dz = −(1/l)dl/dz, which followsfrom u s l = constant, the equation of motion reads:0− u2 s dl{( F′ ) 2 F d( F′ )}+U 2 = 1 d(ηg) (2.13)dz η η dη η η dη5

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