Report

Laboratoire Environnement, Géomécanique & Ouvrages Soutenance de thèse Transport, dépôt et relargage de particules inertielles dans une fracture à rugosité périodique T. Nizkaya Directeur de thèse: M. Buès Co-directeur de thèse: J.-R. Angilella, LAEGO, Université de Lorraine Ecole doctorale RP2E 1er Octobre 2012 Nancy, Lorraine 1 Particle-laden flows Photo: NASA's Goddard Space Flight Center Particles: air and water pollutants, dust, sprays and aerosols, etc… 2 Particle-laden flows through fractures Hydrogeology: Flows through fractures often carry particles (sediments, organic debris etc.). How to model particle-laden flows? 3 Two models of particles Tracer particles: Inertial particles: point particles finite size, density advected by the fluid different from fluid. (+ brownian motion) Example: dye in water Example: sand in the air 4 Two models of particles Tracer particles: Inertial particles: point particles finite size, density advected by the fluid different from fluid. (+ brownian motion) Advection-diffusion equations for particle concentration. Example: sand in the air 5 Two models of particles Tracer particles: Inertial particles: point particles finite size, density advected by the fluid different from fluid. (+ brownian motion) Particle inertia is important. Advection-diffusion Even weakly-inertial particles equations for particle are concentration. very different from tracers! 6 Clustering of inertial particles Inertial particles tend to cluster in certain zones of the flow. rain initiation Wilkinson & Mehlig (2006) planet formation Barge & Sommeria (1995) aerosol engineering Fernandez de la Mora (1996) Particles in fractures: clustering can lead to redistribution of particles across the fracture? 7 Clustering of inertial particles Inertial particles tend to cluster in certain zones of the flow. rain initiation Wilkinson & Mehlig (2006) planet formation Barge & Sommeria (1995) aerosol engineering Fernandez de la Mora (1996) In periodic flows particle focus to a single trajectory: Robinson (1955), Maxey&Corrsin (1986), etc. 8 Goal of the thesis Theoretical study of focusing effect on particle transport in a fracture with periodic corrugations. Water + particles 0 homogeneos distribution «focusing» 9 Outline of the talk I. Single-phase flow in a model fracture II. Focusing of inertial particles in the fracture III. Influence of lift force on particle focusing IV.Conclusion and perspectives 10 I. Single-phase flow in a thin fracture. 11 I. Single-phase flow in a thin fracture. Goal: Obtain an explicit fluid velocity field for arbitrary fracture shapes Method: Asymptotic expansions 12 Simplified model of a fracture Z = () = () X Model fracture: a thin 2D channel with «slow» corrugation. Typical corrugation length L0 >> typical aperture H0. Small parameter: = ≪ 13 Single-phase flow in fracture Single-phase flow in fracture: 2D, incompressible, stationary Navier-Stokes equations: = = () (, ) = , = () Streamfunction: , = × Non-dimensional variables: , = ; = = ; ; = , = Reynolds number: = () ≪ = 14 Equations of inertial lubrication theory Navier-Stokes equations in non-dimensional variables: Boundary conditions: No slip at the walls Hasegawa and Izuchi (1983) Borisov (1982), etc. 15 Equations of inertial lubrication theory Navier-Stokes equations in non-dimensional variables Boundary conditions: ≪ ≪ No slip at the walls Small parameter ε ⇒ perturbative method 16 Generalization of previous works Crosnier (2002) Hasegawa and Izuchi (1983) } Present thesis: full parametrization of the fracture geometry. Borisov (1982) 17 The cross-channel variable: = () (, ) → (, ) h(x) h(x) = = = () = () = − Cross-channel variable − () : = ℎ() 2 − 1 () ℎ() = half-aperture of the channel 2 1 + 2 () () = middle-line profile 2 18 Asymptotic solution of 2nd order 0th : 1st: 2nd: 19 Asymptotic solution of 2nd order 0th : «local cubic law» 1st: 2nd: inertial corrections viscous correction 3rd… etc. 20 Numerical verification: mirror-symmetric = . --- LCL flow, 2nd order asymptotics, numerical simulation 21 Numerical verification: flat top wall = . --- LCL flow, 2nd order asymptotics, numerical simulation 22 Application: corrections to Darcy’s law Flow rate depends on pressure drop: = − - curve Q ∞ Darcy’s law Inertial corrections: analytical expression? Small flow rates Larger flow rates 23 Corrections to Darcy’s law Pressure drop (from 2nd order asymptotic solution): 2 P 12 Q 2 2 Q 1 K1 K 3 3 L Hh No quadratic term! In accordance with Lo Jacono et al. (2005) and many others. 24 Corrections to Darcy’s law Pressure drop (from 2nd order asymptotic solution): 2 P 12 Q 2 2 Q 1 K1 K 3 3 L Hh Geometrical factors: () = () () 25 Corrections to Darcy’s law Pressure drop (from 2nd order asymptotic solution): 2 P 12 Q 2 2 Q 1 K1 K 3 3 L Hh Geometrical factors: () = () () Slope of the linear law depends on both aperture and shape of the middle line. 26 Corrections to Darcy’s law Pressure drop (from 2nd order asymptotic solution): 2 P 12 Q 2 2 Q 1 K1 K 3 3 L Hh Geometrical factors: () = () () Cubic correction only depends on aperture variation. 27 Numerical verification Pressure drop vs Reynolds number Darcy’s law numerics (mirror-symmetric channel) our asymptotic solution numerics (channel with flat top wall) 28 II. Transport of particles in the periodic fracture 29 Periodic channel corrugation period «focusing» Particles: small, non-brownian, non-interacting, passive. Flow: asymptotic solution (leading order) 30 Particle motion equations Re p ( ) aVs 1 = − − Particle dynamics: mp dV p dt FH m p g from Stokes equations around the particle Maxey-Riley equations Gatignol (1983) Maxey and Riley (1983) 31 Particle motion equations Maxey-Riley equations: mp dV p drag force added mass Basset’s memory term dt fluid pressure gradient + gravity mf DU f (m p m f ) g Dt a2 6a U f V p U f 6 2 m f dV p D a U f U f 2 dt Dt 10 t a2 1 d U f V p U f ds 6 t s ds 0 32 Typical long-time behaviors (numerics - LCL flow, no gravity) Heavy particles Q Light particles Q Heavy particles can focus to a single trajectory (or not!) depending on channel geometry. Q 33 Typical long-time behaviors (numerics - LCL flow, with gravity) Light particles Heavy particles Low Q High Q Focusing persists in presence of gravity, if the flow rate Q is high enough (permanent suspension) 34 Goal: Find conditions for particle focusing depending on channel geometry and flow rate. Method: Poincaré map + asymptotic motion equations for weakly-inertial particles 35 Simplified Maxey-Riley equations Particle response time: 2 Re H 9R a H0 2 Density contrast: 2 f R 2 p f 36 Simplified Maxey-Riley equations Particle response time: 2 Re H a 9R H 0 For weakly-inertial particles: Density contrast: 2 f R 2 p f 1 x p u f ( x p ) v1( x p ) O 3 / 2 fluid velocity 2 Maxey (1987) particle inertia + weight from Maxey-Riley equations 37 Poincaré map for weakly-inertial particles + = rescaled cross-channel variable z = ( ) after k periods +1 = + ( ) from simplified Maxey-Riley 1 equations 38 Poincaré map for weakly-inertial particles + Poincaré map: +1 = + ( ) Stable fixed point: ∞ = 0; −1 < ′ ∞ < 0. Particles converge to the streamline Focusing! , = ∞ 39 Analytical expression for the Poincaré map Poincaré map for the LCL flow: 2 3R 9 2 f ( ) 1 ~ J J 1 G h h z 2 ' ( ) 8 Fluid/particle density ratio heavier 3R 1 than fluid 2 3R lighter 1 than fluid 2 Channel geometry Gravity number J h h' h 2 J h '2 2 h '1 2 g z L0 1 Gz 2 U0 Fr h 40 Analytical expression for the Poincaré map Poincaré map for the LCL flow: 2 3R 9 2 f ( ) 1 ~ J J 1 G h h z 2 ' ( ) 8 Fluid/particle density ratio heavier 3R 1 than fluid 2 3R lighter 1 than fluid 2 Channel geometry Gravity number J h h' h 2 J h '2 ∞ = 0; −1 < ′ ∞ < 0. 2 h '1 2 g z L0 1 Gz 2 U0 Fr h Attractor position ∞ 41 Focusing/sedimentation diagram Rescaled gravity: 8 GZ z 9 h' 2 h Corrugation asymmetry factor: '2 h '1 2 ( ) (analytical expression) cr z z 2 h 2 h' h J h / J h 42 Focusing/sedimentation diagram Light particles ( ) cr z Heavy particles A Case A: = 0, = −0.217 J h / J h 43 Focusing/sedimentation diagram ( ) cr z z B Case B: = 0, = −0.5 Jh / J h 44 Focusing/sedimentation diagram ( ) cr z C z Case C: = −3, = 0 Jh / J h 45 Focusing/sedimentation diagram z ( ) cr z D Case D: = −3, = −0.5 Jh / J h 46 Other applications of Poincaré map Using the Poincaré map we can calculate: • Percentage of deposited particles • Maximal deposition length • Focusing rate Verified numerically Ok 47 Influence of channel geometry on transport properties 48 Shape factors of the channel Aperture-weighted norm: ah 2 1 l l = () () = () 2 a ( x)dx 0 h3 ( x) () = () Shape factors: h 3 1h 2 J h h' h 2 «apparent» aperture aperture variation J ' h 2 J h '2 h '1 2 2 h middle line difference between corrugation wall corrugations 49 Single phase flow: geometry influence Pressure drop curve: 2 P 12 Q 2 2 Q 1 K1 K 3 3 L Hh Slope of the linear law: H 03 H 3 h 3 h Inertial correction: h 3 Shape factors: 1h 2 J h h' h J ' h 2 2 J h '2 h '1 2 Weak dependence on channel shape! 2 h 50 Particle transport: geometry influence Particle Poincaré map: P ( ) f ( ) 3R / 2 1 ~ ' ( ) 2 9 2 J h Jh 1 Gz P ( ) 8 3 Shape factors: 1h 2 Particle behavior depends h strongly on the difference J h h' h J ' h 2 2 J h '2 h '1 2 in wall corrugations! 2 h 51 Particle transport: geometry influence flat J flat J h h J hflat J hmirror mirror J flat J h h Example: channel with flat top wall and mirror-symmetric channel. Equivalent for single phase flow but different for particles. J mirror 0 h 52 IV. The effect of lift force on particle focusing 53 Motion equations with lift U f (X ) Particle motion equations: mp dV p dt mf DU f Dt 6a (U f m f dV p 2 dt (m p m f ) g Vp ) DU f + Lift force Dt FL VP Lift in simple shear flow (Saffman, 1956) Lift appears when particle leads or lags the fluid. No formula for lift in a general flow… «Generalization» of Saffman’s lift: 2 1/ 2 FL 6.46a ( f ) U f 1/ 2 ((U f V p ) 0 ) 54 Lift force induced by gravity Gravity in the direction of the flow (vertical channel): Heavy particles lead Light particles lag pushed to the walls pushed to the center Effect opposite to focusing! Poincaré map with lift calculated analytically 55 Lift force induced by gravity G Two attracting streamlines (theory) Effect opposite to focusing! Poincaré map with lift shows splitting of the attractor. 56 Lift force induced by particle inertia No gravity FL lead lag FL k 1 Lift-induced chaos at finite response times Particles lead or lag because of their proper inertia. The direction of lift changes many times. 57 Lift force induced by particle inertia No gravity FL lead lag FL k 1 Lift-induced chaos at finite response times Particles lead or lag because of their proper inertia. The direction of lift changes many times. Effect on focusing? Poincaré map does not work here… 58 Lift-induced chaos at finite response times Period doubling cascade Chaos! k (response time) Feigenbaum constants: 59 IV. Conclusion 60 Conclusions: single-phase flow • A new asymptotic solution of Navier-Stokes equations is obtained for thin channels. • This solution generalizes previous results to arbitrary wall shapes. • Inertial corrections to Darcy’s law are calculated analytically as functions of channel geometrical parameters. 61 Conclusions: particle transport • Particles transported in a periodic channel can focus to an attracting streamline which depends on channel geometry. • This attractor persists in presence of gravity, if the flow rate is high enough. • The full focusing/sedimentation diagram for particles in periodic channels has been obtained analytically, using Poincaré map technique. 62 Conclusions: lift effect • Lift has been taken into account in form of a classical generalization of Saffman (1965). • In presence of gravity (vertical channel), lift causes attractor splitting: two attracting streamlines are visible. • In the absence of gravity, lift causes a period-doubling cascade leading to chaotic particle dynamics. 63 Perspectives • Particles in a non-periodic (disordered) fracture Do particles still cluster? How to quantify the clustering? • Collisions Does focusing increases collision rates? • Brownian particles with inertia Maxey-Riley equations with noise? • Experimental verification Experimental setup is under construction at LAEGO 64 Perspectives • Particles in a non-periodic (disordered) fracture Do particles still cluster? How to quantify the clustering? • Collisions Does focusing increases collision rates? • Brownian particles with inertia Maxey-Riley equations with noise? • Experimental verification Experimental setup is under construction at LAEGO. Thank you for attention! 65