Mass Transfer in Turbulent Flow ChEn 6603 References: S.. Pope. Turbulent Flows. Cambridge University Press, New York, 2000. D. C. Wilcox. Turbulence Modeling for CFD. DCW Industries, La Caada CA, 2000. H. Tennekes and J. L. Lumley. A First Course in Turbulence. MIT Press, Cambridge, MA, 1972. R. O. Fox. Computational Models for Turbulent Reacting Flows. Cambridge University Press, 2003. 1
Mixing in reacting flow (DNS) Methane pool fire ~ 6 cm Rayleigh-Taylor instability (DNS calculation) Photograph of Jupiter from Voyager ~15,000 Miles (2 Earth diameters) William H. Cabot and Andrew W. Cook Nature Physics 2, 562-568 (2006) 2
Origins of Turbulence Energy balance perspective. Consider steady, isothermal, fully developed turbulent flow in a horizontal pipe Increasing pressure drop does not increase flow rate proportionally. Why? Where is the energy going? How? Work done by pressure forces balanced by work done by viscous forces Energy provided at large scales, dissipated at small scales. Length scales reduce to meet demand of energy balance. Smaller length scales steeper gradients more dissipation. Kinetic energy equation: p ρk n t + (ρkv) = v τ v p + ρ ω i v f i 0 = v p + v τ pressure work What assumptions? viscous dissipation i=1 3
Velocity Length Scales L - largest length scale (m) η - smallest length scale (m) U - velocity at L-scale (m/s) ν - kinematic viscosity (m 2 /s) = µ/ρ ε - kinetic energy dissipation rate (m 2 /s 2 s -1 ) Most kinetic energy is contained in large length scales (L). It is dissipated primarily at smallest (Kolmogorov) length scales (η) by molecular viscosity (ν). Can we form a length scale from ε and ν? η ν 3 1/4 UU L/U kinetic energy integral or large time scale Note: ε doesn t depend on ν. ν just determines the smallest length scale in the flow. η L η ν 3 1/4 3/4 L 1/4 ν U 3/4 LU = Re 3/4 ν Key result! Tells us how length-scales separate! 4
Scalar Length Scales Sc > 1 mixing paint - l <η - scalar only feels straining from smallest velocity scales. (mass diffuses slower than momentum) 2 Dt D 1/2 ν η l - smallest scalar length scale (atchelor scale) 1/4 D 1/2 ν 1/4 ν 3 1/2 D = Sc 1/2 ν Form a time scale from the Kolmogorov time scale (i.e. from ν and ε). 1/4 Sc ν D Sc < 1 l>η - at l, there are still velocity fluctuations, but the scalar field is uniform η (mass diffuses faster than momentum) D 3 1/4 1/4 D ν 3 3/4 D = Sc 3/4 ν Relevant parameters are D, ε. (ν only dominant near η). 1/4 L = L η η Re 3/4 Sc 1/2 L = L η η Re 3/4 Sc 3/4 Gases: Sc 1, Liquids: Sc ~ 10 3. 5
Sc http://sensitivelight.com/smoke2 6
Solution Options Increased Modeling Direct Numerical Simulation (DNS) Resolve all time/length scales by solving the governing equations directly. Restricted to small problems. Cost scales as Re 3 for turbulence alone! (Species with Sc>1, and/or complex chemistry could further increase cost) (L/η Re 3/4, 3D, time) Large Eddy Simulation (LES) Resolve large spatial & temporal scales Model small (unresolved) time/space scales Reynolds-Averaged Navier Stokes (RANS) Time-averaged. Describes only mean features of the flow. Model all effects of the flow field Useful only for some classes of problems (usually interfaces like walls) Commonly done in heat transfer & mass transfer (also for some problems involving aerodynamics ) 7
Time-Averaging (RANS) Constant density, viscosity: Definition of time-average: Continuity: v = 0 v = (v v) 1 t ρ p + ν 2 v 1 t0 +T φ lim φ(t) dt T T vdt = 0 0 vdt = 0 0 t 0 v = 0 Momentum: 0 v dt = t 0 = 0 0 vv dt 1 p + ν 2 v ρ vv dt 1 p + ν 2 v ρ index (Einstein) notation: 0 v i v j + 1 ρ v i v j dt = v i v j. p + ν v i x i 8
The Closure Problem φ φ φ Fluctuating component v i v j = ( v i + v i )( v j + v j ) = v i v j + v i v j + v i v j + v i v j φ = 0 φϕ = 0 φ = φ ( v i v j )+ 1 ρ v i v j = v i v j + v i v j p + ν v i + x i vi x v j j =0 Model this term using a gradient diffusion model. vi x v j µ t j ρ x i v j ( v i v j )+ 1 ρ p x i + 1 ρ (µ + µ t) v j =0 For large Re, µt µ (molecular viscosity is negligible). 9
Time-Averaged Species Equations constant properties & density... ω i v + 1 ρ j i = s i /ρ ( ω i v) + 1 ρ j i + ωi v = s i /ρ A very difficult problem... ( ω i v) + 1 ρ ji + j i,turb = s i /ρ MODEL for turbulent species diffusive flux: j i,turb = ρd turb ω i Sc turb = ν turb D turb = µ turb ρd turb D turb - turbulent diffusivity (for mass flux relative to mass avg. velocity) µturb - eddy viscosity Typically, Scturb is specified. @ Large Re, ji,turb ji (molecular diffusion is negligible) ( ω i v) + 1 ρ j i,turb = s i /ρ Multicomponent effects are irrelevant at sufficiently high Re. 10
Spatial Averaging (LES) φ φ(x)g(x)dx G(x) - filter kernel function removes high wavenumber components of ϕ. φ = φ φ = = 0 Courtesy R.J. McDermott φϕ = 0 Courtesy R. J. McDermott Filter governing equations. (similar procedure as for RANS, but a little more complicated). Write models for unclosed terms. Solve filtered equations (for filtered variables). Provides time-varying solutions at a coarse level. 11
Variable Density ρω i t Favre-averaging (RANS) Favre-filtering (LES) = ρω i v j i + s i. φ ρφ ρ ρ φ = ρφ ρ ω i t = ρω i v j i + s i = ρ ω i ṽ ji + j i,turb + si Leads to many additional complications, most of which are typically ignored... LES: if example: ji =? n ρdik o ω k k=1 Δ = filter width, η ρ n k=1 D o ik ω k then j i,turb j i 12