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46) called inner scale or small scale, gives the lowest scale for existence of eddies. This length scale is also called in the literature as Taylor micro-scale of turbulence. Below this scale eddies dissipate their mechanical energy into heat. 47) where Vl ′ is the mean characteristic velocity of pulsation of a large eddy with size l e ,l . 49) where cη′ is an empirical constant. This formula is called Kolmogorov–Pandtl expression, Kolmogorov (1942), Prandtl (1945). In analogy to the definition expression for the dissipation of the turbulent kinetic energy in isotropic turbulence given by Eq.

2. 3. 3. 92 , which is different to the constant for single-phase flow. 3. Lee et al. 3. 272 . We will provide more information of the source terms in the next chapters. Nomenclature Latin c vm cd cL cp is the mass concentration of the inert component i in the velocity field l, dimensionless virtual mass force coefficient, dimensionless drag force coefficient, dimensionless lift force coefficient, dimensionless specific heat at constant pressure, J / ( kgK ) cη viscosity coefficient, dimensionless Cil cη are the modeling constants for the conservation equation of the energy dissipation, dimensionless model coefficients in k-eps model D Dl Dhy diffusivity, m 2 / s particle size in field l, m hydraulic diameter (4 times cross-sectional area / perimeter), m Dil := ν l Scil , coefficient of molecular diffusion for species i into the field l, cε 1 , cε 2 , cε 3 m2 / s Nomenclature 49 Dilt := ν lt / Scilt coefficient of turbulent diffusion, m 2 / s Dil* := Dil + Dilt , effective diffusion coefficient, m 2 / s DCil right-hand side of the non-conservative conservation equation for the in- ( ert component, kg / sm3 d e Gk ,l ) total differential specific internal energy, J/kg production of turbulent kinetic energy due to bubble relocation in chang- Pk ing pressure field per unit mass of the filed l, W/kg (m²/s³) acceleration due to gravity, m / s 2 specific enthalpy, J/kg latent heat of evaporation, J/kg unit matrix, dimensionless kinetic energy of turbulent pulsation, m 2 / s 2 irreversibly dissipated power from the viscous forces due to deformation of the local volume and time average velocities in the space, W/kg l = 1: partial pressure inside the velocity field l l = 2,3: pressure of the velocity field l pressure, Pa surface averaged difference between the pressure at the surface dσ and the bulk pressure of c, effective interfacial stagnation pressure difference in the continuum, Pa surface averaged difference between the pressure at the surface wσ and the bulk pressure of c, effective wall-continuum stagnation pressure difference in the continuum, Pa production of the turbulent kinetic energy per unit mass, W/kg Pk ,l in ν lt, ss Pk ,l which is the production of the turbulent kinetic energy per unit Pkw,l mass of the velocity field l due to deformation of the velocity field l, W/kg production of turbulent kinetic energy per unit mass of the field l due to Pk μ ,l friction with the wall, W/kg production of turbulent kinetic energy per unit mass of the field l due to g h Δh I k P pli p Δpcdσ Δpcwσ * Pε w,l friction evaporation or condensation, W/kg production of the dissipation of the turbulent kinetic energy per unit mass, W/kg production of the dissipation of the turbulent kinetic energy per unit mass Pkw,c of the field l due to friction with the wall, W/kg generation of turbulent kinetic energy per unit mass of the continuum, Pε W/kg 50 2 Introduction to turbulence of multi-phase flows Pε w,c “production” of dissipation of the turbulent kinetic energy per unit mass Prll of the continuum, W/kg := ρl c plν ll / λll , molecular Prandtl number, dimensionless PrTt ,l := ρl c plν lt / λlt , turbulent Prandtl number, dimensionless Prkt q&σ′′′l turbulent Prandtl number describing diffusion of the turbulent kinetic energy, dimensionless turbulent Prandtl number describing diffusion of the dissipation of the turbulent kinetic energy, dimensionless thermal energy introduced into the velocity field l per unit volume of the flow, W/m³ l = 1,2,3.

That is why we write the equation for the rate of dissipation of the kinetic energy of isotropic turbulence in analogy with the derivation for single phase flows intuitive, without strong proof ∂ (α ρ ε γ ) + ∇. ⎡⎣α c ρ c ( Vcε c − ν cε ∇ε c ) γ ⎤⎦ ∂τ c c c v ⎧ε ⎫ = α c ρ cγ v ⎨ c ⎡ cε 1 ν ct Pk ,c + Pkw,c + Pkwζ ,c − cε 2ε c + cε 3Gk ,c ⎤ ⎬ . 60) 46 2 Introduction to turbulence of multi-phase flows Here k ε c is characteristic time scale of the dissipation. Note that Lopez de Bertodano (1992) proposed especially for the dissipation of the turbulence generated by the particles, cε 3Gk ,c , to use different characteristic time scale, namely 2 cvm Dd .

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