Department ofMechanical Engineering

IOWA STATE UNIVERSITYOF SCIENCE AND TECHNOLOGY

An Integral Boundary Layer ModelAn Integral Boundary Layer Modelfor Corona Drag Reduction/Increase on for Corona Drag Reduction/Increase on

a Flat Platea Flat Plate

5th Electrohydrodynamics International Workshop5th Electrohydrodynamics International Workshop

August 30-31, 2004, Poitiers, France

Department ofMechanical Engineering

IOWA STATE UNIVERSITYOF SCIENCE AND TECHNOLOGY

Gerald M. Colver Professor of Mechanical EngineeringProfessor of Mechanical Engineering

Frans Soetomo Graduate StudentGraduate Student

(MS Thesis)(MS Thesis)

Department of Mechanical EngineeringIowa State University,

Ames, IA. 50011

.

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This part work was supported in part by a grant from IFPRI (International Fine

Particle Research Instutute)

AcknowledgmentsAcknowledgments:

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(1) Use Karman-Pohlhausen boundary layer (b.l.) integral method

(2). Formulate a simplified model for drag increase (reduction) due to dc corona discharge along a flat plate

(3). Seek closed form solution for velocity profile, boundary layer (b.l.) thickness, plate drag, etc

(4) Plot boundary layer profiles in dimensionless form

(5) Compare b.l. growth to experimental results

The Problem The Problem :

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Model Assumptions :Model Assumptions :

• Steady, thin boundary layer approximation, constant fluid properties (density and viscosity)

Electrostatics DC corona discharge :

• Parallel electrodes mounted flush plate (one at leading edge; one downstream at infinity) – perpendicular to flow

• Single polarity ions (ionic wind can oppose/aid free stream flow) and constant ion mobility

• Ion current flow is confined to the momentum boundary layer (vertical current is negligible)

• Convective (bulk flow) ion current is ignored – small free stream velocity

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E

U

I J x )x

y

U p s t r e a mE l e c t r o d e

20 kV discharge glass slide; 25x75x1 mm3 (Soetomo - 1992)

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FormulationFormulation::

Karman-Pohlhausen - Integral Momentum EquationIntegral Momentum Equation

U – free stream velocity u(x,y) – b.l. velocity (profile)(x) – b.l. thicknessf – electrostatic body forces inside b.l.f – electrostatic body forces freestream (=0)

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More … More …

Continuity

Free stream momentum

Current density

Mobility

Body force/volume

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Gives body force…Gives body force…

Integrate (above) Integrate (above) body forcebody force across b.l. across b.l.

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Assumed velocity profile (Assumed velocity profile (need 4 constantsneed 4 constants))

Evaluate Evaluate 4 constants4 constants from boundary conditions from boundary conditions

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Gives…Gives…

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Substitute above velocity profile into integral Substitute above velocity profile into integral momentum gives d.e. for b.l. thickness momentum gives d.e. for b.l. thickness (x) (x)

For + j (Downstream directed ionic wind) -> boundarly layer “thins”

Ionic wind force:

For - j (Upstream directed ionic wind) -> boundary layer “thickens” (grows)

where

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Special Case I: r.h.s. =0Special Case I: r.h.s. =0

Boundary layer thickness remains Boundary layer thickness remains constant along plateconstant along plate

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Special Case II: j=0 (I =0)Special Case II: j=0 (I =0) (Ionic current zero)(Ionic current zero)

Field-free boundary layer growthField-free boundary layer growth (solution checks)(solution checks)

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The general solution(s) for a The general solution(s) for a finite ionic current (j≠0) taking finite ionic current (j≠0) taking =1=1

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Boundary layerBoundary layer “thinning/thickening” “thinning/thickening” from ionic windfrom ionic wind

Plot of dimensionless boundary thickness: Plot of dimensionless boundary thickness:

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(Dimensionless) (Dimensionless) velocity profiles along velocity profiles along flat plate for flat plate for ± ionic ± ionic wind directionswind directions

U

u*

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The total ionic wind force The total ionic wind force FFionion acting on the plateacting on the plate

(Integrate f=nqE along the plate 0->x)(Integrate f=nqE along the plate 0->x)

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Numerical ExampleNumerical Example

• nominal value of current I = 10‑4 A• x = 2.54 10‑2 m• K = 2.110‑4 Cm/Ns• U=0.67 m/s = 1.9810‑5 kg/ms = 1.1774 kg/m3, and =1 Paper: Using Eqs. (17), (23), and (26)

Gives j = 1.4106 m‑1, max = 1.4310‑6m, and, Fion = 1.2 x 10‑2 N.

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• Karman-Pohlhausen integral method simplifies to the know field-free solution

• Boundary layer thinning and thickening are predicted depending on the direction of (net) ionic current flow

• The calculated ionic force Fion is too large (~10‑2 N) compared to experimental values (~10‑4 N)

- Summary -- Summary -

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Questions ?Questions ?

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CFD Solutions/ El-Khabiry – Colver, CFD Solutions/ El-Khabiry – Colver, 19991999