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

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

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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, cole

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Introduction EHD without cross-flow

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STABILITY OF PLANAR SHEAR FLOW

IN THE PRESENCE OF ELECTROCONVECTION

  • F. Martinelli1, M.Quadrio1,2 & P

.Schmid1

1LadHyx, École Polytechnique (F)

  • 2Dip. Ing. Aerospaziale, Politecnico di Milano (I)

Ottawa, July 29th, 2011

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Introduction EHD without cross-flow

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EHD with cross-flow

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OUTLINE

1 INTRODUCTION 2 EHD WITHOUT CROSS-FLOW

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3 EHD WITH CROSS-FLOW

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4 CONCLUSIONS

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Introduction EHD without cross-flow

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EHD with cross-flow

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Conclusions

OUTLINE

1 INTRODUCTION 2 EHD WITHOUT CROSS-FLOW

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3 EHD WITH CROSS-FLOW

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4 CONCLUSIONS

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Introduction EHD without cross-flow

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WHAT IS EHD?

Dielectric fluid Negligible magnetic effects Charge injection at the boundary Fully coupled problem owing to Coulomb force

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WHAT IS ELECTROCONVECTION?

REVIEW BY P.ATTEN, IEEE TRANS., 1996

? ? Φ

x,u y,v z,w

collector injector liquid with charged particles Planar indefinite geometry (periodic box) Unipolar autonomous injection "Analogous" to Rayleigh-Bénard thermal convection

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WHAT IS KNOWN ABOUT ELECTROCONVECTION?

RESULTS FOR LINEAR STABILITY DATE BACK TO ’70-’80

? ? ?

asymptotic non-modal cross-flow no cross-flow stability stability

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EQUATIONS

TWO-WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD

∇2Φ = −q ε Quasi-electrostatic limit of Maxwell equations

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EQUATIONS

TWO-WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD

∇2Φ = −q ε ∂q ∂t +∇·(qV+qKE−D∇q) = 0 Conservation of charge density q

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EQUATIONS

TWO-WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD

∇2Φ = −q ε ∂q ∂t +∇·(qV+qKE−D∇q) = 0 ∂V ∂t +(V·∇)V = − 1 ρ ∇P+ν∇2V+Fe Electric force is Fe = qE (no dielectric force since ε is uniform)

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EQUATIONS

TWO-WAY COUPLING BETWEEN KINETIC AND ELECTRIC FIELD

∇2Φ = −q ε ∂q ∂t +∇·(qV+qKE−D∇q) = 0 ∂V ∂t +(V·∇)V = − 1 ρ ∇P+ν∇2V+Fe ∇·V = 0 Incompressibility

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DIMENSIONLESS PARAMETERS

Reference length, potential and velocity are h, Φ0 and KΦ0/h Taylor number T (forcing par., fluid properties + Φ0) Ionic mobility M (fluid properties) Charge diffusivity Fe (fluid properties + Φ0) Moreover: Charge injection coefficient C (boundary condition only) Reynolds number Re (in base flow)

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FORMULATION, NUMERICS

v-η-Φ formulation Fourier transform in x,z directions Small perturbations, linearization y discretization with N Chebyshev polynomials

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OUTLINE

1 INTRODUCTION 2 EHD WITHOUT CROSS-FLOW

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3 EHD WITH CROSS-FLOW

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4 CONCLUSIONS

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STATE OF THE ART

P.ATTEN 1996

Charge diffusion assumed to be negligible, Fe → ∞ Instability for κ ≈ 2.5 and T = Tc ≈ 161 Discrepancy between numerical Tc and experimental Tc ≈ 100

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NEUTRAL CURVE

DIFFUSION MATTERS!

T κ Neutral curves. N=250, M=100, C=50 155 156 157 158 159 160 161 162 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Fe=104 Fe=105 Fe=106 Fe=107

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"OPTIMAL" Fe

EXPLAINS DIFFERENCE BETWEEN EXPERIMENTAL AND NUMERICAL Tc?

Fe Optimal charge diffusivity. N=100, M=100, C=50, κ=2.5 100 110 120 130 140 150 100 200 300 400 500 600 700 800 900 1000

Tc

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DEFINITION OF ENERGY

Total energy of the system split into mechanical and electric contributions E = Em +Ee = 1 2(u2 +v2 +w2)+ 1 2εE·E Transient growth function defined as G(t) = max E (t) E (0) = max

x0=0

x(t)2 E x2

0 E

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MAP OF Gmax

MILD TRANSIENT GROWTH

T κ Gmax curves for Fe=200, N=150, M=10, C=50 20 40 60 80 100 120 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 5

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Introduction EHD without cross-flow

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EHD with cross-flow

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OUTLINE

1 INTRODUCTION 2 EHD WITHOUT CROSS-FLOW

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3 EHD WITH CROSS-FLOW

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4 CONCLUSIONS

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NEUTRAL CURVE

SQUIRE THEOREM STILL APPLIES: β = 0

Re α Neutral curves for Fe=200,C=50 1000 2000 3000 4000 5000 6000 7000 8000 0.2 0.4 0.6 0.8 1 1.2 1.4 M=10, T=2000 M=10, T=4000 M=5, T=2000

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MOST UNSTABLE HYDRODYNAMIC MODE

Re = 7000, α = 1

−200 −100 −300 −250 −200 −150 −100 −50 real(ω) imag(ω) Spectrum −1 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 y Velocity R I −1 1 −1 −0.5 0.5 1 1.5 x 10

−3

y Potential R I

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MOST UNSTABLE ELECTRIC MODE

Re = 100, α = 1

−40 −20 −15 −10 −5 5 10 real(ω) imag(ω) Spectrum −1 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 y Velocity R I −1 1 −0.02 −0.01 0.01 0.02 0.03 0.04 y Potential R I

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TRANSIENT GROWTH AT β = 0

Re α Gmax contours for Fe=200,M=10C=50T=2000 1000 2000 3000 4000 5000 6000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 50 100 150

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OPTIMAL INPUT FOR β = 0

ORR MECHANISM. α = 1, β = 0, Re = 1000

x y 1 2 3 4 5 6 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0.5 1 1.5 x 10

−3

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OPTIMAL OUTPUT FOR β = 0

ORR MECHANISM

x y 1 2 3 4 5 6 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025

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DOES EHD ENHANCE TRANSIENT GROWTH?

LOOKING AT KINETIC ENERGY ALONE, β = 0

Re α Maximum amplification of kinetic energy − M=10, T=2000, Fe=200, C=50 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000

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Introduction EHD without cross-flow

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EHD with cross-flow

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OUTLINE

1 INTRODUCTION 2 EHD WITHOUT CROSS-FLOW

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3 EHD WITH CROSS-FLOW

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4 CONCLUSIONS

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CONCLUSIONS

Electroconvection (stability) revisited Role of diffusion Non-modal effects (esp. with cross-flow) Non-linear effects? EHD as a extremely-low-power flow control device?

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DIMENSIONLESS NUMBERS

Reference length, potential, velocity, time and pressure are: h, Φ0, KΦ0/h, h2/KΦ0 and ρK2Φ2

0/h2

M = 1 K ε ρ T = εΦ0 µK Fe = KΦ0 D C = q0h2 εΦ0 K is ionic mobility, ρ and µ fluid density and dynamic viscosity, D is charge diffusivity, ε fluid (uniform) fluid permittivity.

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DIMENSIONLESS EQUATIONS

U(y) and Φ(y) are the base velocity and potential profiles ∂ ˆ ∆ˆ v ∂t = − jαU ˆ ∆ˆ v+ jαU′′ ˆ v+M2 Φ′′′κ2 ˆ ψ −Φ′κ2 ˆ ∆ ˆ ψ

  • + M2

T ˆ ∆ˆ ∆ˆ v ∂ ˆ η ∂t = − jβU′ ˆ v− jαU ˆ η + M2 T ˆ ∆ ˆ η ∂ ˆ ∆ ˆ ψ ∂t = Φ

′ ∂ ˆ

∆ ˆ ψ ∂y +Φ

′′′ ∂ ˆ

ψ ∂y +2Φ

′′ ˆ

∆ ˆ ψ − jαU ˆ ∆ ˆ ψ −Φ

′′′ ˆ

v+ 1 Fe ˆ ∆ˆ ∆ ˆ ψ,

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EXAMPLE OF FLUID PROPERTIES

DATA FOR PYRALENE 1460

K = 3.2E −9 D = 8.2E −11 ε = 5.224E −11 µ = 0.01 ρ = 1.41E3 M = 60 T = 1.6325Φ0 Fe = 0.6Φ0

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APPLICATIONS

Ion-drag pumping EHD turbulent mixing EHD heat transfer augmentation

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TRANSIENT GROWTH AT α = 0

Re β Gmax contours for Fe=200,M=10,C=50,T=2000 200 400 600 800 1000 1200 1400 1600 1800 2000 0.05 0.1 0.15 0.2 0.25 0.3 0.35 100 200 300 400 500 600 700 800 900 1000

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OPTIMAL INPUT FOR α = 0

LIFTUP MECHANISM. α = 0, β = 0.2, Re = 1000

z y 5 10 15 20 25 30 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −3 −2 −1 1 2 3 x 10

−3

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OPTIMAL OUTPUT FOR α = 0

LIFTUP MECHANISM

z y 5 10 15 20 25 30 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −2 −1 1 2 x 10

−4