Institute for Applied Sustainable Science, Engineering & Technology Roland Schmehl Flow Problem Analysis in Oil & Gas Industry Conference Rotterdam, January 2 Deformation and Secondary Atomization of Droplets in Technical Two-Phase Flows Slide
Outline Introduction Basics Empirical description Normal Mode Analysis Nonlinear deformation analysis Potential Theory Breakup model Motion of deformed droplets Validation of deformation models Modeling of droplet breakup Validation of breakup models Summary Slide 2
Introduction Droplet breakup in premix zone Spherical droplet Subsequent breakup Dispersion Bag breakup Bag formation Droplet deformation Sheet breakup mm Air assisted pressure swirl atomizer Liquid: Tetradecane D = 6µm We 5 Slide 3
Basics Deformation and breakup phenomena Roland Schmehl Low relative velocities Moderate to high relative velocities (top to bottom) Shape oscillations Bag breakup Bag-plume breakup Forced deformations Plume-sheet breakup Sheet-thinning breakup Image source: Wiegand (987) Image source & terminology: Guildenbecher (29) Slide 4
Basics Forces acting on and in the droplet Surface tension Inertial forces x 2 O d x 3 x v rel Aerodynamic forces Viscous forces Slide 5
Basics Model mechanisms Moderate velocities High velocities Extremly high velocities Transverse deformation by aero- dynamic pressure distribution Superposed separation of liquid by aerodynamic shear forces Superposed hydrodynamic instability of front surface Slide 6
Basics Dimensional analysis Non-dimensional numbers We = ρ v2 rel D σ Weber number t v t t σ t µ Re def = v rel D µ d ρρd Reynolds number of deformational flow On = µ d ρd D σ d Characteristic times Ohnesorge number = We Re def ρd D v rel t = ρ tσ D 3 = ρ d σ t µ = µ d ρ v 2 rel t v = D v rel Pressure distribution Inertia forces Surface tension Inertia forces Pressure distribution Dissipation Flow around droplet Slide 7
Basics Influence of aerodynamic loading 8 6 We 4 2 Droplet in shock tube flow Droplet in free fall Droplet in premix module Droplet in rocket engine preflow 5 5 2 25 3 35 4 45 5 55 6 65 7 T σ = t/t σ Slide 8
Basics Classification of numerical models Empirical description Correlations and similarity laws for description of droplet deformation. + Arbitrary deformations + Simple implementation Limited to specific loading scenarios No dynamic response to flow variations > Limited use for modeling Simple mechanistic models Deformation kinematics reduced to single degree of freedom: Droplet shapes approximated by spheroids. + Specific small & large deformations + Description of nonlinear effects + Low computational effort Fitting of empirical constants required > Suitable for modeling Normal mode analysis Modal discretization of aerodynamic pressure distribution, kinematics and dynamics of deformation. + Small but arbitrary deformations + Low computational effort Neglects nonlinear effects Neglects effect of aerodynamic shear > Suitable for modeling Direct numerical simulation Spatial and temporal discretization of the Navier-Stokes equations in both phases. + Arbitrary deformation + Includes all forces Extremely high computational effort Small scale processes problematic > Can not be used for modeling Slide 9
Empirical description Load-based classification: On-We diagram 3 shear breakup Re def, = 2 transitional breakup We bag-plume breakup bag breakup (oscillatory breakup) shape oscillations deformation > 2% - deformation -2% deformation 5-% Krzeczkowski (98) Hsiang & Faeth (995) deformation < 5% ρ d /ρ=58 2 Re = 24 6-4 -3-2 - 2 3 On Slide
Empirical description Load-based classification: We-WeRe.5 diagram Atmospheric data: Hinze (955), Krzeczkowski (98), Hsiang & Faeth (995), Vieille (995), Dai & Faeth (2), Schmelz (22). Low pressure data: Zerf (998). High pressure data: Vieille (998). 4 3 2 bag breakup bag-plume breakup transitional breakup shear breakup On g = 2 WeRe.5 3 p. MPa p. MPa p>. MPa 2 3 4 5 6 7 8 9 We 4 Slide
Empirical description Temporal stages Data source: Krzeczkowski (98), Dai & Faeth (2) 8 7 6 bag bag-plume plume-shear shear breakup breakup T b T= t/t 5 4 ring breakup plume plume/core droplet complex larger fragments 3 2 bag bag breakup plume bump plume breakup peripheral bags separation at equator ligament formation at equator T i T hmin transverse distortion + flattening 2 3 4 5 6 7 8 9 We Slide 2
Empirical description Global secondary droplet properties Root-normal distribution: f (x) = ( ) 2 x µ 2σ 2πx exp 2 σ with D.5 D 32 =.2, µ=., σ=.22..8.8.6.6 V/V.4.2 bag breakup F(x) We =5 V/V.4.2 bag-plume breakup F(x) We =25 2 3 x= D/D.5 2 3 x= D/D.5 Data sources: Hsiang & Faeth (992) and Chou & Faeth (998). Slide 3
Empirical description Global secondary droplet properties Sauter diameter: D 32 D = 6.2 On.5 We.25 = Re.5 def, On<.,We < 3..8.8.6.6 V/V.4.2 transitional breakup F(x) F red (x) We =4 We =7 2 3 x= D/D.5 V/V.4 shear breakup F(x) F red (x).2 We =25 We =25 We =375 2 3 x= D/D.5 Data sources: Hsiang & Faeth (992) and Chou & Faeth (998). Slide 4
Empirical description Scondary droplet properties - differentiated by origin.8.8 V/V.6.4 ring plume bag core D32/D.6.4 ring plume core global.2.2 2 3 4 5 6 We 2 3 4 5 6 We Data source: Dai & Faeth (2) Slide 5
Normal Mode Analysis Langrangian description of flow kinematics v v rel u U P r O d r u d r d u d O g t Slide 6
Normal Mode Analysis Linear normal mode decomposition x 2 a,v rel x 3 δ θ r n= n= n=2 n=3 n=4 n=5 Decomposition of an arbitrary axisymmetric droplet shape into orthogonal deformation modes Slide 7
Normal Mode Analysis Derivation of dynamic deformation equations Linearized Navier-Stokes equations Series expansion (v = ψ) r 2 r (r2 v r )+ r sinθ θ (v θ sinθ)= ρ d v r t ρ d v θ t = p [ r +µ d 2 v r 2 r 2v r 2 r 2 sinθ ] θ (v θ sinθ) +ρ d a cosθ = p [ r θ +µ d 2 v θ + 2 ] v r r 2 r θ r 2 sinθ v θ ρ d a sinθ p s p = ρv2 rel 2 p = ρ dv 2 rel 2 δ r = R C n P n (cosθ) n= ( r n β n Pn (cosθ) R) n= ( r n α n Pn (cosθ) R) n= Deformation equations (n 2) formulated on nondimensional time scale T σ = t/t σ d 2 α n dt 2 σ d 2 α n dt 2 σ + 8n(n )On dα n dt σ + 8n(n )(n+2)α n = 2nC n We (Hinze 948) + 8(2n+)(n )On dα n dt σ + 8n(n )(n+2)α n = 2nC n We (Isshiki 959) Viscous term different in both theories! Slide 8
Normal Mode Analysis Pressure distribution on spherical surface.5.5 Re=5, Tomboulides und Orszag (2) Re=, Tomboulides und Orszag (2) Re=5, Bagchi et al. (2) Re= 4, Constantinescu und Squires (2) Re=.62 5, Achenbach (972) potential flow ps p ρv 2 rel /2 -.5 - -.5 3 6 9 2 5 8 θ [ ] Slide 9
Normal Mode Analysis Modal representation of pressure boundary condition Cn.2.8.6.4 C 2 C 3 C 4 C 5 transition lam.-turb. wake unsteady wake asymmetric flow separation rigid sphere, in uniform flow.2 -.2-2 3 4 5 Re Slide 2
Normal Mode Analysis Stationary deformation Water droplet in vertical air flow (Pruppacher et al. 97) D [mm]: 8. 7.35 5.8 5.3 3.45 2.7 v rel [m/s]: 9.2 9.2 9.7 9.3 8.46 7.7 We:..2 8. 7.3 4. 2.6 Re: 4723 434 343 35 873 334 Slide 2
Normal Mode Analysis Deformation from shock loading Water droplet in horizontal shock tube flow: D = mm, On=3.38 3 We: 5 2 Re def : 296 662 25 Re: 49 99 7 v rel : 7.7 m/s 7.2 m/s 26.68 m/s Slide 22
Nonlinear deformation analysis Motivation Linear analysis: First order theory Forces and displacements at undeformed droplet Nonlinear analysis: Second order theory Forces and displacements dependend on deformation Nonlinear phenomena: Mode coupling, excitation of higher modes Oscillation dynamics depends on amplitude (frequency and period) Nonlinear resonance effects Hydrodynamic instabilities Slide 23
Nonlinear deformation analysis Kinematics and basic equations x 2 x 2 y y s x 3 x v rel x x 3 ζ= : ζ=± : ζ x 3 s =, v s= v s,max x 3 s =, v s= Dynamic equilibrium of mechanical energy contributions ρ d 2 d v 2 dv+σ ds dt V dt = S p s v nds V ΦdV = F(y,ẏ,ÿ) = Slide 24
Potential Theory Breakup model Energy contributions Potential energy of surface: polynomial approximation σ ds dt = σ ds dy dy dt, S ds dy = 9.98y 3 3.34y 2 + 33.94y 3.58,.5< y<,.67y 3 4.y 2 + 9.2y 5.67, y<2.3. Kinetic energy: viscous potential flow ρ d 2 d v 2 dv = 8 dt V 5 πρ dr 5 ( + 2 ) d 2 y 2 3 y 6 dt 2 y 7 ( ) 2 dy dt dy dt Viscous dissipation: viscous potential flow V ΦdV = 2µ d V ( v2 x 2 ) 2 ( ) 2 dv = 6πR 3 dy µ d y dt Slide 25
Potential Theory Breakup model Work performed by aerodynamic forces Velocity potential on surface: stationary external flow ( ψ s = 2 2 e2 artanhe e 2 e v rel x 3, γ = 2 γ 2 e 2 ( ) e 2 arcsine e, y<, ), y>. Pressure distribution on surface p s p ρ/2v 2 rel ( ) 2 x3 = p max, s ( ) 2 x3 = s ζ 2 ( y 6 )ζ 2, s : Surface coordinate ζ : Non-dimensional axial coordinate Total work performed by aerodynamic pressure forces S p s v nds= ρv2 rel λ dy + 2 2πR3 p max y dt, λ 4ζ 2 + 3ζ 4 = ( y 6 )ζ 2 dζ= 2 e2 e 4 2 e2 e 4 [ (3 e 2 ) artanhe e ] 3 [ 3 2e 2 arcsin e 3 e 2 e ], y<, y> Slide 26
Potential Theory Breakup model Nonlinearity of aerodynamic load term 2.9.8 λ, λ/y, C2λ/y.7.6.5.4.3 λ, exact λ /y, exact.2 C 2 = 2/3 p max, potential theory C 2 = 2/3 p max, CFD-simulation. C 2 λ /y, with C 2 from CFD-simulation C 2 λ /y, polynomial approximation.5.5.5 2 y.5 C2 Slide 27
Potential Theory Breakup model Comparison of linear and nonlinear models Taylor Analogy Breakup (TAB) model d 2 y dt 2 σ + 4On dy dt σ + 64(y )=2C 2 We Potential Theory Breakup (PTB) model ( + 2 ) d 2 y 3 y 6 dtσ 2 2 ( ) 2 dy y 7 + 4On dy dt σ y 2 + 2 ds dt σ S dy = 5 C 2 λ We 4 y Slide 28
Motion of deformed droplets Empirical description Equation of motion m d du d dt = π D2 8 ρ c D v rel v rel + m d g 2y h E= h 2y Exposed cross section of droplet π D 2 =π D 2 y2 Aerodynamic drag coefficient c D = f c D,sphere + ( f )c D,disc c D,sphere =.36+5.48Re.573 + 24 Re, Re 4 c D,disk =.+ 64 πre f= E 2 cd 3 2.5 2.5.5 E-based interpolation Spheroid E=.5, exp. 2 3 4 Re E.25.5.75 Slide 29
Motion of deformed droplets Droplets falling into horizontal free jet flow Wiegand (987), experiment Normal Mode Analysis Slide 3
Motion of deformed droplets Computed motion and deformation -.2 -.4 -.6 -.8 y [m] -. -.2 -.4 Wiegand (987) Versuch 7-3W -.6 Slide 3
Validation of deformation models Effect of viscous damping on free shape oscillations.8.6 On a:.77 b:.77 c:.77 Becker et al. (994) NLTAB3 NM.4 a E.2 c b.8.6 2 4 6 8 2 2 2T σ Aspect ratio: E= +α 2 2 α, Nomal Mode Analysis 2 E= y 3, Spheroid-based models Slide 32
Validation of deformation models Effect of increasing amplitude on free shape oscillations.2 ω/ω.98.96.94.92 NLTAB3 model: On=.89 On=.39 On=.46 On=.63 Tσ/ Tσ,. On=.89 On=.39 On=.46 On=.63 prolate spheroid oblate spheroid.9..2.3.4 α 2,.9..2.3.4 α 2, Frequency shift Asymmetry of oscillation period Slide 33
Validation of deformation models Stationary deformation of droplets in free fall.9.8 experiment correlation NM TAB C 2 = 4/3 9 8 E.7.6.7. 4/3 C 2 ud, 7 6 experiment c D,sphere c D (Re,We) NM model NLTAB3 mod., C 2 = 4/3.5 2 3 4 5 6 7 8 D [mm] 5 2 3 4 5 6 7 8 D [mm] v rel prescribed coupled with droplet deformation Slide 34
Validation of deformation models Maximum transverse distortion of droplets in shock tube flow ymax 2.2 2.8.6.4 TAB, C 2 = 2/3 TAB, C 2 =.5 TAB, C 2 = 4/3 NM, D = mm NLTAB3 C 2 =.5 PTB C 2 =.5 PTB C 2 λ /y, Polynom Hsiang & Faeth (992) Exp. Hsiang & Faeth (995) Exp. Temkin & Kim (98) Exp. Dai & Faeth (2) Exp. Haywood et al. (994) CFD Leppinen et al. (996) CFD Hase (22) CFD y M y c.2.5 2 3 4 5 7 9 3 5 7 9 2 We Slide 35
Validation of deformation models Deformation of droplet in shock tube flow VOF method NM model.5. VOF method, Hase (24) NM model TAB model output times S/S.5 4.92 We 9.83..2.3.4.5.6.7.8.9. t[s] Slide 36
Validation of deformation models Droplet falling through horizontal jet flow at We =.5 -. -.2 Test 7-6W c D,sphere c D (Re,We) NLTAB3, y =. NLTAB3, y =.3 NLTAB3, y =.5 y.4.2 y =.3 y =. yd -.3 -.4 -.5.8..2 t.2 y =.3: c D We.5.4 cd We -.6 We =.5 -.7..2.3.4.5 x d Trajectories in laminar core flow.8.6..2 t Trajectory data NLTAB3 model.3.2 Slide 37
Validation of deformation models Droplet falling through horizontal jet flow at We = 3.3 -. -.2 Test -W c D,sphere c D (Re,We) NLTAB3, y =. NLTAB3, y =.3 NM, y =. NM, y =.3 y.4.2 y =.3 y =. yd -.3.8.5..5 t -.4 -.5.7 y =.3: c D We 4 3 cd.6 We -.6 2 We = 3.3.5 -.7..2.3.4.5 x d.5..5 t Trajectories in laminar core flow Trajectory data NLTAB3 model Slide 38
Validation of deformation models Droplet falling through horizontal jet flow at We =.8 We =.8 2.8 y =.3 y =. -..6 y.4 -.2.2 yd -.3.8.2.4.6 t -.4 Test 7-3W c D,sphere -.5 c D (Re,We) NLTAB3, y =. NLTAB3, y =.3 -.6 NM, y =. NM, y =.3 NM, y =.5 -.7..2.3.4.5 x d cd.9.8.7.6 y =.3: c D We.5.2.4.6 t 2 8 6 4 We Trajectories in laminar core flow Trajectory data NLTAB3 model Slide 39
Modeling of droplet breakup Breakup criterion based on critical deformation y 2.8 2.6 2.4 2.2 2.8.6.4 Leppinen et al. (996), We = 2, LLT-C Kim (977), We = 2.6 Dai & Faeth (2), We = 5 Dai & Faeth (2), We = 2 y, NLTAB3 We y, empirical y max, PTB 2 y max, NLTAB3 5 y max, TAB 3 disk- bagbulging dy dt = expansion dy dt = 3.2 y M y c.2 2.8.5.5 2 2.5 T= t/t Slide 4
Modeling of droplet breakup Simulation of the On-We diagram 2 PTB, c D = c D (Re) PTB, c D = c D (Re, A) Hsiang & Faeth (992), Exp. Hsiang & Faeth (995), Exp. shear breakup We y=.8 = 69 multimode breakup We y=.8 = 3 We y max.9.7 stability limit y max =.8.2..5 critical damping -3-2 - 2 On Slide 4
Modeling of droplet breakup Setup of modeling framework 8 7 6 5 Empirical models T 4 3 2 Dynamic boundary layer models Dynamic deformation models 2 3 4 5 6 7 8 9 We Slide 42
Validation of droplet breakup Size distribution computed from differentiated model We = 5 (bag breakup) We = 25 (shear breakup).8.8.6.6 V/V.4.2 experiment experiment root-normal computed V/V.4.2 experiment root-normal computed 2 3 x= D/D.5 2 3 x= D/D.5 Data sources: Hsiang & Faeth (992) and Chou & Faeth (998). Slide 43
Validation of droplet breakup Single droplet falling in horizontal free jet Air & Ethanol We=68.5, On=.76, Re=4362 Software: OpenFOAM & Ladrop Dispersion model switched off Experimental data: Guildenbecher (29) Slide 44
Validation of droplet breakup Simulation using load-based breakup criterion 2 2 2 2 Slide 45
Validation of droplet breakup Simulation using deformation-based breakup criterion 2 2 2 2 Slide 46
Validation of droplet breakup Comparison of load- and deformation-based simulations breakup for We>We,c breakup for y>y max 2 2 2 2 2 2 2 2 Slide 47
Summary Analytical models for description of linear and nonlinear deformation dynamics Simple mechanistic approach for coupling of droplet deformation and motion Empirical stability criteria, classification and kinematics of breakup process Systematic validation and assessment of models based on fundamental test cases Future: Test modelling framework within practical Euler-Lagrange simulations Slide 48