Department of Mechanical Engineering I OWA S TATE U NIVERSITY OF SCIENCE AND TECHNOLOGY An Integral Boundary Layer Model for Corona Drag Reduction/Increase on a Flat Plate 5th Electrohydrodynamics International Workshop August 30-31, 2004, Poitiers, France

Department of Mechanical Engineering I OWA S TATE U NIVERSITY OF SCIENCE AND TECHNOLOGY Professor of Mechanical Engineering Graduate Student (MS Thesis) Gerald M. Colver Professor of Mechanical Engineering Frans Soetomo Graduate Student (MS Thesis) Department of Mechanical Engineering Iowa State University, Ames, IA

3 This part work was supported in part by a grant from IFPRI (International Fine Particle Research Instutute) Acknowledgments Acknowledgments:

4 (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 :

5 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

6 20 kV discharge glass slide; 25x75x1 mm3 (Soetomo )

7 Formulation: Integral Momentum Equation Formulation: Karman-Pohlhausen - Integral Momentum Equation U – free stream velocity u(x,y) – b.l. velocity (profile)  (x) – b.l. thickness f – electrostatic body forces inside b.l. f – electrostatic body forces freestream (=0)

8 More … Continuity Free stream momentum Current density Mobility Body force/volume

9 Gives body force… Integrate (above) body force across b.l.

10 Assumed velocity profile (need 4 constants) Evaluate 4 constants from boundary conditions

11 Gives…

12 Substitute above velocity profile into integral momentum gives d.e. for b.l. thickness  (x) For + j (Downstream directed ionic wind) -> boundarly layer “thins” Ionic wind force: For - j (Upstream directed ionic wind) -> boundary layer “thickens” (grows) where

13 Special Case I: r.h.s. =0 Boundary layer thickness remains constant along plate

14 Special Case II: j=0 (I =0) (Ionic current zero) (Ionic current zero) Field-free boundary layer growth (solution checks) (solution checks)

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

16 Boundary layer “thinning/thickening” from ionic wind Plot of dimensionless boundary thickness:

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

18 The total ionic wind force F ion acting on the plate (Integrate f=nqE along the plate 0->x)

19 Numerical Example nominal value of current I = 10 ‑ 4 A x = 2.54  10 ‑ 2 m K = 2.1  10 ‑ 4 C  m/N  s U=0.67 m/s  = 1.98  10 ‑ 5 kg/m  s  = kg/m 3, and  =1 Paper: Using Eqs. (17), (23), and (26) Gives j = 1.4  106 m ‑ 1,  max = 1.43  10 ‑ 6 m, and, F ion = 1.2 x 10 ‑ 2 N.

20 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 F ion is too large (~10 ‑ 2 N) compared to experimental values (~10 ‑ 4 N) - Summary -

21 Questions ?

22

23 CFD Solutions/ El-Khabiry – Colver, 1999