- Text
- Velocity,
- Turbulent,
- Concentration,
- Droplets,
- Equation,
- Axial,
- Radial,
- Profiles,
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- Mixing

Turbulent mixing of oil droplets in a round water jet

2.1.1 Boundary-layer equationsStart**in**g po**in**t for the derivation **of** the govern**in**g equations **of** a turbulent **jet** flow is the Reynoldsdecomposition **of** the flow variables, which divides an **in**stantaneous variable **in**to a mean part anda fluctuat**in**g part. For example, for the **in**stantaneous axial velocity:u = u + u ′ (2.2)The ensemble averag**in**g operator, denoted with an overbar, fulfils the so called Reynolds conditions[Nieuwstadt, 1998]. Apply**in**g this decomposition and neglect**in**g the molecular terms (Re ≫1), the averaged Navier-Stokes equations for a stationary axisymmetric geometry **in** cyl**in**dricalcoord**in**ates 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 ∂z**in** 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 (cont**in**uity) 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 follow**in**g 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 accord**in**g 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 determ**in**edby:U 2u 2 s( l= OL)From the Navier-Stokes equations (2.4), an approximation **of** the equation for conservation **of**momentum **in** the z-direction can be derived by apply**in**g this scal**in**g and thereafter neglect**in**g 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 cont**in**uity 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 **in**terpreted as the transport**in** 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 **in**troduced assum**in**g self-similarity (2.1.2).Thereafter, the velocity pr**of**iles are derived us**in**g K-theory as a closure hypothesis (2.1.3).4

2.1.2 Self-similarityOne essential assumption **in** the theory describ**in**g a turbulent **jet** is that dur**in**g the development**of** the flow **in** the downstream direction, the turbulence ma**in**ta**in**s its structure. In other words:the **jet** is self-similar. By def**in**ition: if all velocities are reduced by one velocity scale and alldimensions by one length scale, the flow patterns expressed **in** the reduced quantities becomeidentical [H**in**ze, 1975]. As a consequence, the follow**in**g 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 cyl**in**dricalcoord**in**ates. Therefore, a stream function, ψ, is **in**troduced which fulfils the cont**in**uity 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 follow**in**g description**of** 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. Normalis**in**g f(η) with f(0) = 1, makes u s the velocity at the centrel**in**e **of** the**jet**. 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 **in**vestigate their z-dependence, the equation **of** conservation**of** 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. Substitut**in**g the expressions for the velocities(2.11) **in** the equation **of** motion (2.7) and apply**in**g (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

- Page 1: Turbulent mixing of oil dropletsin
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agreement (γ = 1.2). The turbulent

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Appendix ASingle-phase jet experime

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1measurementsGaussian fittheoretica

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0.2measurementsAanen0.15V rms/U c0.

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15measured valuesself−similarity

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0.15−r∂U/∂z∂(rV)/∂rresidu

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taken into account. Also, a theoret

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10510095z [mm]90858075−30 −20

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90PIVWareDaVis8580U c[mm/s]75706590

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0.2PIVWareDaVis0.15V rms/U c0.10.05

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List of Figures2.1 A turbulent jet,

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List of SymbolsRomana - velocity of

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Greekα - spreading rate of the jet

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ReferencesAanen, L. (2002), Measure