W HAT IS EHD? Introduction EHD without cross-flow Modal - PowerPoint PPT Presentation

Introduction S TABILITY OF PLANAR SHEAR FLOW EHD without cross-flow Modal IN THE PRESENCE OF ELECTROCONVECTION Non-modal EHD with cross-flow Modal F. Martinelli 1 , M.Quadrio 1 , 2 & P .Schmid 1 Non-modal Conclusions 1 LadHyx, École Polytechnique (F) 2 Dip. Ing. Aerospaziale, Politecnico di Milano (I) Ottawa, July 29th, 2011

O UTLINE 1 I NTRODUCTION Introduction EHD without cross-flow 2 EHD WITHOUT CROSS - FLOW Modal Non-modal Modal EHD with Non-modal cross-flow Modal Non-modal Conclusions 3 EHD WITH CROSS - FLOW Modal Non-modal 4 C ONCLUSIONS

O UTLINE 1 I NTRODUCTION Introduction EHD without cross-flow 2 EHD WITHOUT CROSS - FLOW Modal Non-modal Modal EHD with Non-modal cross-flow Modal Non-modal Conclusions 3 EHD WITH CROSS - FLOW Modal Non-modal 4 C ONCLUSIONS

W HAT IS EHD? Introduction EHD without cross-flow Modal Dielectric fluid Non-modal EHD with Negligible magnetic effects cross-flow Modal Charge injection at the boundary Non-modal Conclusions Fully coupled problem owing to Coulomb force

W HAT IS ELECTROCONVECTION ? R EVIEW BY P.A TTEN , IEEE T RANS ., 1996 collector Introduction EHD without cross-flow Φ Modal Non-modal 0 y , v EHD with cross-flow ? Modal Non-modal ? Conclusions x , u liquid with z , w charged particles injector Planar indefinite geometry (periodic box) Unipolar autonomous injection "Analogous" to Rayleigh-Bénard thermal convection

W HAT IS KNOWN ABOUT ELECTROCONVECTION ? R ESULTS FOR LINEAR STABILITY DATE BACK TO ’70-’80 no cross-flow cross-flow Introduction EHD without cross-flow asymptotic Modal ? Non-modal stability EHD with cross-flow Modal Non-modal Conclusions non-modal ? ? stability

E QUATIONS T WO - WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD ∇ 2 Φ = − q ε Introduction EHD without cross-flow Modal Non-modal EHD with cross-flow Modal Non-modal Conclusions Quasi-electrostatic limit of Maxwell equations

E QUATIONS T WO - WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD ∇ 2 Φ = − q ε Introduction EHD without ∂ q cross-flow ∂ t + ∇ · ( q V + qK E − D ∇ q ) = 0 Modal Non-modal EHD with cross-flow Modal Non-modal Conclusions Conservation of charge density q

E QUATIONS T WO - WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD ∇ 2 Φ = − q ε Introduction EHD without ∂ q cross-flow ∂ t + ∇ · ( q V + qK E − D ∇ q ) = 0 Modal Non-modal EHD with cross-flow ∂ V ∂ t +( V · ∇ ) V = − 1 ρ ∇ P + ν ∇ 2 V + F e Modal Non-modal Conclusions Electric force is F e = q E (no dielectric force since ε is uniform)

E QUATIONS T WO - WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD ∇ 2 Φ = − q ε Introduction EHD without ∂ q cross-flow ∂ t + ∇ · ( q V + qK E − D ∇ q ) = 0 Modal Non-modal EHD with cross-flow ∂ V ∂ t +( V · ∇ ) V = − 1 ρ ∇ P + ν ∇ 2 V + F e Modal Non-modal Conclusions ∇ · V = 0 Incompressibility

D IMENSIONLESS PARAMETERS Introduction Reference length, potential and velocity are h , Φ 0 and EHD without K Φ 0 / h cross-flow Modal Non-modal Taylor number T (forcing par., fluid properties + Φ 0 ) EHD with cross-flow Ionic mobility M (fluid properties) Modal Non-modal Charge diffusivity Fe (fluid properties + Φ 0 ) Conclusions Moreover: Charge injection coefficient C (boundary condition only) Reynolds number Re (in base flow)

F ORMULATION , NUMERICS Introduction EHD without cross-flow Modal v - η - Φ formulation Non-modal EHD with Fourier transform in x , z directions cross-flow Modal Small perturbations, linearization Non-modal Conclusions y discretization with N Chebyshev polynomials

O UTLINE 1 I NTRODUCTION Introduction EHD without cross-flow 2 EHD WITHOUT CROSS - FLOW Modal Non-modal Modal EHD with Non-modal cross-flow Modal Non-modal Conclusions 3 EHD WITH CROSS - FLOW Modal Non-modal 4 C ONCLUSIONS

S TATE OF THE ART P.A TTEN 1996 Introduction EHD without cross-flow Modal Charge diffusion assumed to be negligible, Fe → ∞ Non-modal EHD with Instability for κ ≈ 2 . 5 and T = T c ≈ 161 cross-flow Modal Discrepancy between numerical T c and experimental Non-modal Conclusions T c ≈ 100

N EUTRAL CURVE D IFFUSION MATTERS ! Neutral curves. N=250, M=100, C=50 3 2.9 Introduction EHD without 2.8 cross-flow Modal 2.7 Non-modal EHD with 2.6 cross-flow Modal Non-modal 2.5 κ Conclusions 2.4 Fe=10 4 2.3 Fe=10 5 2.2 Fe=10 6 2.1 Fe=10 7 2 155 156 157 158 159 160 161 162 T

"O PTIMAL " Fe E XPLAINS DIFFERENCE BETWEEN EXPERIMENTAL AND NUMERICAL T c ? κ =2.5 Optimal charge diffusivity. N=100, M=100, C=50, 1000 900 Introduction EHD without 800 cross-flow Modal 700 Non-modal EHD with 600 cross-flow Modal Non-modal Fe 500 Conclusions 400 300 200 100 100 110 120 130 140 150 T c

D EFINITION OF ENERGY Introduction Total energy of the system split into mechanical and EHD without cross-flow electric contributions Modal Non-modal E = E m + E e = 1 2 ( u 2 + v 2 + w 2 )+ 1 EHD with 2 ε E · E cross-flow Modal Non-modal Transient growth function defined as Conclusions � x ( t ) 2 � E G ( t ) = max E ( t ) E ( 0 ) = max � x 2 x 0 � = 0 0 � E

M AP OF G max M ILD TRANSIENT GROWTH G max curves for Fe=200, N=150, M=10, C=50 3 5 Introduction 4.5 EHD without cross-flow 4 Modal Non-modal 2.5 EHD with 3.5 cross-flow Modal Non-modal 3 κ Conclusions 2.5 2 2 1.5 1.5 1 20 40 60 80 100 120 T

O UTLINE 1 I NTRODUCTION Introduction EHD without cross-flow 2 EHD WITHOUT CROSS - FLOW Modal Non-modal Modal EHD with Non-modal cross-flow Modal Non-modal Conclusions 3 EHD WITH CROSS - FLOW Modal Non-modal 4 C ONCLUSIONS

N EUTRAL CURVE S QUIRE THEOREM STILL APPLIES : β = 0 Neutral curves for Fe=200,C=50 M=10, T=2000 1.4 M=10, T=4000 Introduction M=5, T=2000 EHD without 1.2 cross-flow Modal Non-modal 1 EHD with cross-flow Modal 0.8 Non-modal α Conclusions 0.6 0.4 0.2 1000 2000 3000 4000 5000 6000 7000 8000 Re

M OST UNSTABLE HYDRODYNAMIC MODE Re = 7000, α = 1 Spectrum Potential Velocity −3 x 10 −50 0.4 1.5 R R I I Introduction 0.2 EHD without −100 1 cross-flow Modal 0 Non-modal EHD with cross-flow −150 0.5 −0.2 Modal imag( ω ) Non-modal Conclusions −0.4 −200 0 −0.6 −250 −0.5 −0.8 −300 −1 −1 −200 −100 0 −1 0 1 −1 0 1 real( ω ) y y

M OST UNSTABLE ELECTRIC MODE Re = 100, α = 1 Spectrum Velocity Potential 10 0.4 0.04 R I Introduction 0.2 0.03 EHD without 5 cross-flow Modal 0 Non-modal 0.02 EHD with cross-flow 0 −0.2 Modal imag( ω ) R Non-modal 0.01 I Conclusions −0.4 −5 0 −0.6 −10 −0.01 −0.8 −15 −1 −0.02 −40 −20 0 −1 0 1 −1 0 1 real( ω ) y y

T RANSIENT GROWTH AT β = 0 G max contours for Fe=200,M=10C=50T=2000 150 1.1 Introduction EHD without 1 cross-flow Modal 0.9 Non-modal 100 EHD with 0.8 cross-flow Modal 0.7 Non-modal α 0.6 Conclusions 0.5 50 0.4 0.3 0.2 0.1 0 1000 2000 3000 4000 5000 6000 Re

O PTIMAL INPUT FOR β = 0 O RR MECHANISM . α = 1, β = 0, Re = 1000 −3 x 10 1 1.5 0.8 Introduction 1 EHD without 0.6 cross-flow Modal Non-modal 0.4 0.5 EHD with cross-flow 0.2 Modal Non-modal 0 0 y Conclusions −0.2 −0.5 −0.4 −0.6 −1 −0.8 −1 −1.5 0 1 2 3 4 5 6 x

O PTIMAL OUTPUT FOR β = 0 O RR MECHANISM 1 0.025 0.8 0.02 Introduction EHD without 0.6 0.015 cross-flow Modal Non-modal 0.4 0.01 EHD with cross-flow 0.2 0.005 Modal Non-modal 0 0 y Conclusions −0.2 −0.005 −0.4 −0.01 −0.6 −0.015 −0.8 −0.02 −1 −0.025 0 1 2 3 4 5 6 x

D OES EHD ENHANCE TRANSIENT GROWTH ? L OOKING AT KINETIC ENERGY ALONE , β = 0 Maximum amplification of kinetic energy − M=10, T=2000, Fe=200, C=50 1 6000 5500 0.9 Introduction EHD without 5000 0.8 cross-flow Modal 4500 Non-modal 0.7 EHD with 4000 cross-flow 0.6 Modal 3500 Non-modal α 0.5 Conclusions 3000 0.4 2500 0.3 2000 0.2 1500 0.1 1000 0 500 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Re

O UTLINE 1 I NTRODUCTION Introduction EHD without cross-flow 2 EHD WITHOUT CROSS - FLOW Modal Non-modal Modal EHD with Non-modal cross-flow Modal Non-modal Conclusions 3 EHD WITH CROSS - FLOW Modal Non-modal 4 C ONCLUSIONS

C ONCLUSIONS Introduction EHD without cross-flow Electroconvection (stability) revisited Modal Non-modal Role of diffusion EHD with cross-flow Non-modal effects (esp. with cross-flow) Modal Non-modal Non-linear effects? Conclusions EHD as a extremely-low-power flow control device?

D IMENSIONLESS NUMBERS Reference length, potential, velocity, time and pressure are: h , Φ 0 , K Φ 0 / h , h 2 / K Φ 0 and ρ K 2 Φ 2 0 / h 2 Introduction � ε EHD without M = 1 cross-flow ρ K Modal Non-modal T = ε Φ 0 EHD with cross-flow µ K Modal Non-modal Fe = K Φ 0 Conclusions D C = q 0 h 2 ε Φ 0 K is ionic mobility, ρ and µ fluid density and dynamic viscosity, D is charge diffusivity, ε fluid (uniform) fluid permittivity.