Drag reduction by dc corona discharge along
an electrically conductive flat plate for small
Reynolds number flow
Cite as: Physics of Fluids 9, 587 (1997); https://doi.org/10.1063/1.869219
Submitted: 20 February 1996 . Accepted: 21 November 1996 . Published Online: 04 June 1998
S. El-Khabiry, and G. M. Colver
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Physics of Fluids 9, 587 (1997); https://doi.org/10.1063/1.869219
© 1997 American Institute of Physics.
9, 587
Drag reduction by dc corona discharge along an electrically conductive flat
plate for small Reynolds number flow
S. El-Khabiry and G. M. Colver
Department of Mechanical Engineering, Iowa State University, Ames, Iowa 50011
~Received 20 February 1996; accepted 21 November 1996!
Corona-induced drag reduction was studied numerically over a finite region of a semi-infinite flat
plate having small Ohmic surface conductivity for low Reynolds number flow ~,100 000, based on
the farthest downstream electrode distance!. The model simulates a corona discharge along a surface
from two parallel wire electrodes of infinite length immersed flush on the surface and oriented
perpendicular to the flow. Charge deposition and removal with the conducting surface are included
as possible charge transfer mechanisms. The analysis is limited to ions of positive charge. Five
coupled partial differential equations govern the numerical model including continuity, momentum,
gas phase conservation of charge, Poisson’s equation of electrostatics, and conservation of charge at
the solid interface. The governing equations together with empirical breakdown and current–voltage
relationships ~F–I characteristic! were evaluated by finite differencing schemes. The calculated
results predict ‘‘corona thinning’’ of the boundary layer for a downstream ion flow and a
corresponding reduction in drag, in agreement with previous theoretical studies. Various parameters
of flow, electricity, and geometry, relating to corona-induced drag, are investigated. © 1997
American Institute of Physics. @S1070-6631~97!03003-1#
I. INTRODUCTION
A corona or partial gas discharge takes place as a localized electrical breakdown when the applied voltage exceeds
a critical value at a sharp electrode, resulting in localized
ionization.1 The field is highly distorted near the electrode
tip. In the farfield, the uniformity of the electric field depends
on the ratio of the linear dimensions of the electrode to the
gap length.2 The breakdown at the electrode is manifested by
light emission, audible noise, and fluctuations in the corona
current. The local charge carriers produced are of the same
polarity as the ion producing electrodes. For example, in a
point-to-plane arrangement the polarity of the charge carriers
produced is the same as the polarity of the point electrode.
With two parallel wire electrodes, ions of both sign are produced unless the potential difference is just above the corona
onset value, in which case only positive ions in atmospheric
air result.1 A positive ion corona ~unipolar ions! between
parallel wires is assumed in the present study with a positive
upstream electrode.
When directed along the surface of a body, a dc positive
corona generates an ionic wind force that can either increase
or decrease the thickness of the existing boundary layer depending on the polarity of the attached electrodes.3 The discharge momentum is coupled with the neutral gas molecules
through collisions and randomized ionic motion.1,4 By Coulomb’s law, the ions constituting the space charge transmit a
reaction force to the electrodes. The effect with mounted
electrodes is to produce a reaction force acting on the body
in a direction opposite to the ionic wind. ~For drag reduction
the electrodes must be attached to the body.! With unipolar
ions and a suitable choice of electrode polarity this reaction
force can be directed to oppose the existing viscous drag on
the body.
The difference between the drag force with and without
a corona is the drag reduction. With a sufficiently large curPhys. Fluids 9 (3), March 1997
rent, in principle, the discharge can become a thrust device
by completely overcoming the drag. If the electrodes are
detached from the body, an opposite effect would be observed, i.e., an increase in the body drag would result. In the
present study, the wire electrodes are taken to be attached to
the body of an electrically conductive plate.
Previous studies of corona discharge near surfaces have
identified several engineering applications relating to heat
and mass transfer, including drag alteration, surface cooling,
electrostatic precipitation, and flame kinetics.5–13 For example, it has been shown that an ionic wind can augment
heat transfer as much as 200%.5,7,10 Colver and Nakai14 observed that the electrical conductivity of a glass wall located
upstream from a flame had a significant influence on the
ionic wind in terms of visible distortions on the flame.
Bushnel6 and Malik et al.7 reported an ionic wind velocity of
several meters per second contributing to the momentum of
the boundary layer and a reduction in the drag. Malik et al.
studied Poiseuille flow at lowspeeds comparable to the ion
drift velocity and reported drag reductions of 20% for an
applied voltage of 15 kV. Their calculations of a laminar
boundary layer flow with a corona did not show a similar
reduction. They noted a dearth of knowledge on the physics
of a corona discharge near an insulating surface.
Soetomo15 experimentally investigated the drag effect
for both ac and dc corona discharges over an area of 28 mm
~width!320 mm ~electrode separation distance! on a small
flat glass plate ~2537531 mm3! of finite surface conductivity in a small wind tunnel at velocities up to 2 m/s. He
utilized razor blade electrodes mounted flat on the surface of
the glass. He observed drag reductions at low Reynolds number flow below 3600 ~based on distance along the plate! for
both ac and dc discharges. He noted difficulty in performing
dc experiments and found it necessary to subtract out electrostatic interactions of the drag apparatus with the surroundings.
1070-6631/97/9(3)/587/13/$10.00
© 1997 American Institute of Physics
587
II. MODELING AND NUMERICAL METHOD
A. Problem formulation
FIG. 1. Flow past flat plate with ‘‘corona thinning’’ ~drag reduction! of the
boundary layer d(x).
A dc theory developed by Colver et al.3 showed that
either ‘‘corona thinning’’ or ‘‘thickening’’ of a laminar
boundary layer is possible with an attendant drag reduction
~or increase!, depending on whether the polarity of the discharge aides or opposes the free-stream flow. Medvedev
et al.16 investigated the effect of a solid surface coating with
a magnetic fluid layer that is held by an outer magnetic field
on the free-stream flow structure. They concluded that their
technique could be used for flow separation control in lowvelocity flows. Nelson et al.17 studied positive and negative
corona to near spark-over and observed increases in the laminar friction factor to 250% for single and double wire corona
discharges running along the center of a 3.2 cm i.d. tube. For
Reynolds number greater than 5000 the corona effect was
slight.
In the present study drag reduction is investigated numerically for a steady dc positive corona discharge over a
finite region of flat plate having finite ~Ohmic! surface conductivity ~Fig. 1!.18,19 The model simulates two parallel wire
electrodes of infinite length placed perpendicular to the flow
and immersed flush on the surface of the plate so as not to
have a disturbing effect on the flow. ~In our idealized model
it is assumed that a finite corona wire does not perturb the
flow, i.e. the wire is mounted flush with the surface and has
no other effect than to provide a corona source.! The analysis
is limited to ions of positive charge. For parallel wires, a
positive ion corona can be attained in practice if the dc potential is made somewhat greater than the positive corona
onset voltage but less than that for negative corona. @As suggested by the referees, an alternative and efficient method for
generating positive corona ~in practice! would be to increase
the diameter of the downstream electrode ~i.e., use unequal
electrode radii!. This would justify the production of unipolar ~positive! ions over the wide range of voltages used in the
present study.# To model drag reduction, the upstream electrode is placed at a positive polarity with respect to the second downstream electrode ~optionally held at ground potential! so that an ionic wind in the same direction as the freestream velocity is produced. Ion deposition and removal to/
from the conductive surface is included in the model.
The parameters investigated are the electrode potential
difference, ion deposition and removal, gap length between
the electrodes, diameter of the corona wire, the position of
the positive electrode as measured from the leading edge,
and the bias of the downstream electrode below ground potential. Ionic diffusion is neglected.
588
Phys. Fluids, Vol. 9, No. 3, March 1997
The continuum fluid model is governed by the boundary
layer equations for two-dimensional, incompressible steady
fluid flow over a flat plate at zero angle of attack ~gravitational forces are neglected!. A body force term couples the
equation of motion with the electrostatic equations to account for the collision interaction between positive ions and
neutral molecules. With the positive x axis taken along the
surface and the y axis perpendicular to the flat plate ~Fig. 1!,
the simplified thin boundary layer equations of continuity
and momentum with a body force term are given by20
]u ]v
1 50,
]x ]y
u
~1!
]u
] 2u
]u rc
1n
5
E x1 h 2 ,
]x
]y r
]y
~2!
where u and n are the flow velocity components in the x and
y directions, respectively, r is the fluid density ~1.20 kg/m3!,
h~5m/r; m5dynamic viscosity! is the kinematic viscosity of
the fluid ~1.4831025 m2/s!, E x is the electric field strength in
the x direction, and rc is the local ~positive! ion charge density. The first term on the right-hand side of Eq. ~2! is the
electric field body force per unit total mass due to the local
field. Equations ~1! and ~2! describe the fluid flow of the
boundary layer, except ~a! near the leading edge of the flat
plate where the Reynolds number based on downstream distance is small, and ~b! in the immediate vicinity of the electrodes where radial fields are large. Downstream from the
leading edge the momentum equation is valid at a Reynolds
number starting at about 1000.21,22 In the same way, for distances from the electrodes greater than about 5% of the electrode gap length, Eq. ~2! is valid ~this point will be discussed
further in Sec. III!.
An order of magnitude evaluation of the electromagnetic
forces shows that the electric field dominates the magnetic
field force so that the latter can be ignored.18 The electrostatic equations of Poisson and conservation of charge govern the electrostatics of the corona discharge and provide
local values of the electric field strength as well as the charge
density needed for the evaluation of the body force term in
Eq. ~2!. Using the current density including conduction and
convection terms J̄5 r c ~KE1n! and E52“f for the electric field strength together with Eq. ~1!, after eliminating the
variables J and E, the two-dimensional Poisson and conservation of charge ¹–J50 equations are given, respectively,
as18,23,24
] 2f ] 2f
rc
,
2 1
2 52
]x
]y
e0
u
S
~3!
D
r 2c ]r c ] f ]r c ] f
]r c
]r c
1n
1K
2
2
50,
]x
]y
e0 ]x ]x ]y ]y
~4!
where f is the electric potential, K is the positive ion mobility ~2.231024 m2/V s! and e0 is the permittivity of free
space.
S. El-Khabiry and G. M. Colver
The corona onset voltage between two isolated parallel
wires depends on the electrode diameter and the gap length
between the two electrodes as well as on the ambient conditions. A precise value of the field strength for the onset of the
corona is not critical to the development of our model; therefore, in the absence of a specific formula for this problem,
Peek’s semiempirical formula between parallel wires was
used as a crude means of estimating the onset field
strength,1,25,26
E 0 530m u ~ 110.301/ Aa u !
~ kV/cm! ,
~5!
where a is the radius of the corona wire in centimeters, m is
an irregularity factor having a value 0.72 for normal cable
wires ~unpolished!, and u is a factor relating to the gas. For
air,
u 53.92p/T,
V 0 5E 0 a ln~ S/a !
~7!
~ kV! ,
with V 0 the corona onset voltage and S the distance between
the electrode wires.
The relationship between the corona current and the potential difference between the electrodes ~the F–I characteristic! is determined largely by localized gas breakdown at the
surface of the electrodes. To handle this complex phenomena
for our problem would involve the development of a predictive breakdown model for two embedded electrodes on a
semi-insulating surface. Such F–I discharge characteristics
have been numerically calculated for less complex systems
such as point-to-plane discharges in the absence of
convection.27,28 In order to simplify the electrode problem
we utilized a semiempirical approach and least-squares fit of
Soetomo’s data15 for corona discharge over a glass surface
between sharp edged parallel electrodes together with the
development of Seaver29 ~details are given in Ref. 19!. @Soetomo’s dc data for ion current would be expected to generate both positive and negative ions for the voltage range
assumed in our model ~he used electrodes of equal radii!. His
results were used to fit Eq. ~8! since data for positive ions
alone ~unequal radii! were not available.# This gives the
semiempirical voltage-current characteristic as
kT
~ e x i q ~ D f 2V 0 ! /kT 21 !
x i qR
54.689 99310
u5 n 50,
u→U,
~6!
where p is the air pressure in centimeters of mercury and T
is the absolute temperature in Kelvin. For the present problem, T5298 K, p576 cm Hg, and u'1.0. The breakdown
voltage is then estimated by solving for the potential between
widely spaced parallel wires,1
I c5
The ~combined! constants kT/ x i qR and x i q/kT in Eq. ~8!
were evaluated by a regression fit of Soetomo’s data for 2
mm radius wires and a 25 mm separation distance at room
temperature. The resulting equation ~9! for the corona current was taken as applicable at a potential difference exceeding V 050.680 kV ~E 05672 kV/cm! and standard atmospheric conditions for Soetomo’s particular electrode
geometry. Values of V 0 for other electrode spacing S and
wire electrode radii a were then calculated from Eq. ~7!,
approximating R as a constant in Eq. ~8!.
The condition of no slip between the fluid and the wall
and the recovery of the outer-flow velocity far from the wall
gives the fluid boundary conditions
26
~e
0.365~ D f 2V 0 !
~8!
21 ! ~ A/m! ,
~9!
where I c is the source corona current per length of electrode
wire, Df is the potential difference between the two electrodes in kiloVolts, V 0~50.680 kV for Soetomo’s data! is the
breakdown voltage given by Eq. ~7!, q is the ion charge, k is
the Boltzmann constant, T is the absolute temperature, x i is
a ratio known as the ion-to-neutral excess momentum concentration factor, and R is a constant representing the gasphase resistance outside the corona wire with units (M V).
Phys. Fluids, Vol. 9, No. 3, March 1997
~10!
y50,
~11!
y→`.
The surface and bulk electrical conductivities of typical
insulators with adsorbed ionic molecules ~glass, fly ash, dielectrics, ceramics, etc.! are usually reported as being
Ohmic.30–32 However, very high fields ~.107 V/m! in bulk
solids such as glass and gas discharge over surfaces can lead
to non-Ohmic behavior.33–36 For example, a voltage–current
relation of the form E.AI 2n is postulated for a surface discharge ~e.g., n50.62 for tin oxide deposited on glass!.34 For
the electrostatic boundary conditions, the plate surface is
taken to be Ohmic and semi-insulating, that is, an insulator
with a small but finite surface conductivity of value
ss 510214 S sq. This magnitude of electrical conductivity
falls within the range of experimental conductivities for the
adsorption of water molecules on a glass surface.
The wire electrodes are assumed to be immersed flush
with the flat plate so as not to disturb the flow. The finite
wire curvature discussed previously is utilized only for considerations of estimating the breakdown and current magnitude. The upstream positive electrode acts as a source of
positive ions being driven by the electric field toward the
downstream ~negative! electrode. Between the electrodes,
steady ion deposition and removal is permitted with the surface thereby supplying ~or removing! additional charge carriers to the adsorbed ionic molecules already on the surface.
The conservation of charge is derived for an elemental control volume on the interface length Dx along the surface with
width w, including the normal component of gas phase current density J c ~per area! and surface conduction current density J s ~per width!,
~ J s u x1Dx 2J s u x ! w5Jc –nw Dx,
~12!
y50
or
]Js
5Jc –n,
]x
~13!
y50,
where n is the unit normal vector to the plate surface and
where
J s 5 s s E s ~ A/m!
and
Jc 5 s c Ec
~ A/m2 ! ,
~14!
for the surface and gas phases, respectively, with s c 5 r c K.
In the limit of a vanishing control volume, the gas phase
S. El-Khabiry and G. M. Colver
589
the condition that the electric field tangent to the surface
remains constant across the solid–gas interface,
E c,t 5E s,t ,
~16!
y50.
For the normal components of the electric field we can, if
needed, evaluate Gauss’ continuity condition ex post facto
across the solid interface to relate surface charge density and
electric field strength. @The role of Gauss’ law applied to
volume/surface current conduction is given by Haus and
Melcher37 ~Chap. 7!.# This was not utilized in the present
study.
The electric potential on the plate surface upstream from
the positive electrode as well as downstream from the negative electrode are each constant from the condition J s 50
giving the boundary conditions
f5fu ,
f 50,
FIG. 2. Vector field plot of electric field at various distances above the
surface for symmetric potential at the upstream electrode ~f1512 kV! and
downstream electrode ~f2522 kV! placed 100 and 125 mm from the leading edge showing ion deposition/removal at the surface ~between electrodes!; U50 m/s.
conduction parallel to the surface interface does not contribute to Eq. ~13!. Combining Eqs. ~13! and ~14! with E52“f
and taking ss as constant gives
] 2f
]f
,
ss
52 s c
]x2
]y
x l <x<x 2 ,
y50,
~15!
as a boundary condition accounting for ion deposition/
removal along the surface between the two electrodes with
x 1 and x 2 defined as the respective positions of the positive
~upstream! and negative ~downstream! electrode wires from
the leading edge, Fig. 1. A constant value of surface conductivity is admittedly unrealistic in view of our previous discussion; however, this assumption expedites an approximate
solution of charge deposition/removal. With charge deposition, Eq. ~15!, nonlinear surface conduction is possible
through the first of Eqs. ~14! since E s can vary along the
surface. With the electrode potential difference Df specified,
Eqs. ~13! and ~14! lead to the conclusion that E s ~and J c ! will
take on both decreasing and increasing values between the
electrodes for the nonlinear case, i.e., current will both leave
and enter the surface to conserve charge ~see Fig. 2!.
In the case of gas phase current, nonlinear conduction is
possible in the second of Eqs. ~14! through the variation of
rc . In the absence of ion deposition and removal we have
J c 50 at y50 and so recover the two surface boundary conditions between the electrodes as ]2f/] x 250 and ]f/] y50,
where the first of these equations leads to a linear Ohmic
variation in potential between the electrodes f5c 1 x1c 2 .
The second equation infers a zero ~gas phase! current normal
to the surface, i.e. without deposition or removal of charge.
Equations ~14! and ~15! are taken as a boundary condition for the potential along the surface and applied to Eqs.
~3! and ~4! in the gas phase. Consequently we have satisfied
590
Phys. Fluids, Vol. 9, No. 3, March 1997
0<x<x 1 ,
x>x 2 ,
~17!
y50,
~18!
y50.
A corona-free region exists upstream of the plate leading
edge with the electric potential described by Laplace’s equation. Conformal mapping for a thin flat plate was applied in
the negative x direction from the leading edge ~y50! to obtain the boundary potential over a characteristic distance H
as Eq. ~19!. ~The characteristic length H is adjustable and
taken equal to the final height of the computational domain
in the vertical y direction.! Accordingly we approximate the
complete solution of Laplace’s equation along the negative x
boundary as
F S DG
f 5 f u 12
f 50,
2x
H
1/2
,
2H<x<0,
2`<x<2H,
y50.
y50,
~19!
~20!
For the remaining gas phase boundaries ~far from the electrodes! we have
f 50,
x→6`,
x5x,
y5y,
.
y→1`
~21!
Both the voltage drop through the positive corona sheath
and its thickness can be neglected in approximating the radius of the sheath by the wire.27 @It was pointed out by one of
the referees that for the wire radius assumed ;2 mm the
corona sheath thickness is determined by gas kinetics and the
mean-free path ~among other things! to be of the order of the
radius of the wire; whereas, for larger macroscopic wires
~e.g., .100 mm! the sheath may be neglected in comparison
to the wire radius.# To further simplify the electrode problem, the corona sheath was replaced by a line current density
of magnitude I given by Eq. ~9!. Conservation of current is
satisfied at the positive electrode according to
I5
1
W
E
J–dA1J s ~ A/m! ,
~22!
where the gas phase current density in Eq. ~22! is given by
J5 r c ~ KE1 n!
~ A/m2 ! .
~23!
@Equation ~8! does not include surface conduction in its derivation. However, Eq. ~9! was fit to Soetomo’s data and
would necessarily include the effect of surface conduction.
The inclusion of J s in Eq. ~22! is justified for consistency
S. El-Khabiry and G. M. Colver
TABLE I. Effect of ion deposition/removal on net drag force between electrodes. Electrode gap 25 mm positive electrode distance from leading edge
25 mm: electrode length w51 m; wire diameter 2 mm, f154 kV, f250 kV
~electrode source current I is constant!; U53 m/s, ReL 55068; shear drag
x
calculated from D(x 1 ,x 2 ) shear 5 w * x 2 m @ ] u(x,y)/ ] y # u y50 dx ~N!.
1
Without
corona
Corona with
ion
deposition/
removal
Corona
without ion
deposition/
removal
Electrode
current density
I c ~A/m!
Surface
current density
J s ~A/m!
Shear
drag
~N!
~Net! drag
Eq. ~24!
~N!
0
0
0.0010
0.0010
1.131025
~variable!
0.001
0.0000
1.631029
0.0011
1.131025
~Fig. 1!. This definition emphasizes the corona region contribution to the drag while excluding the corona-free regions.
Other useful definitions of C D could equally be justified, for
example, to include the noncorona region of the plate just
downstream from the leading edge. Similarly, we define a
‘‘gap length’’ Reynolds number ReL based on the separation
distance L5x 2 2x 1 between the electrodes,
ReL 5
UL
h
~26!
.
The percentage drag reduction with and without a corona
over L is then given by
~2!0.0003
Drag reduction ~ % !
5
D ~ x 2 ! net, without corona2D ~ x 2 ! net, with corona
3100. ~27!
@ D net~ x 2 ! 2D net~ x 1 !# without corona
B. Numerical method
when setting Eq. ~22! equal to Eq. ~9!. This detail is of little
practical consequence since the surface current is entirely
negligible in comparison to the gas phase current in Eq. ~22!
~see Table I!#. The ion mobility K in Eqs. ~4! and ~23! depends on the electric field strength11 but becomes approximately constant below a critical value that depends on the
particular ion.1 The positive ion mobility was taken as
2.231024 m2/V s.13
With the potential difference Df specified in Eq. ~9! and
taking I5I c , Eqs. ~1!–~4! constitute a set of four simultaneous equations to be solved for four unknowns u, v , rc , and
f, together with the boundary conditions Eqs. ~10!, ~11!,
~15!, ~17–21!, and the current source term Eq. ~22!.
Predicting drag reduction was a primary objective of this
study. The net drag force on the plate results from the combined but opposing effects of viscous drag acting along the
surface of the plate and the electrostatic reaction force acting
on the electrodes. As a result the ionic wind reduces the
momentum deficit of the flow and consequently reduces the
~net! drag. A momentum balance from the leading edge to
any position x along the plate gives the ~net! drag as20
D ~ x ! net5 r w
E
h
0
u ~ x,y !@ U2u ~ x,y !# dy ~ N ! ,
~24!
where h is the height of the computational domain at location x ~h@boundary layer thickness d50.99U!, U is the
free-stream velocity, w is the flat plate width ~electrode
length51 m for calculations!, and u(x,y) is the local velocity across the boundary layer. Equation ~24! includes any
force effect of the corona.
For purposes of the present study the drag coefficient is
defined in terms of the drag force reduction acting between
the two electrodes, i.e., by evaluating Eq. ~24! for D(x) net at
each electrode and subtracting the result to give
C D5
D ~ x 2 ! net2D ~ x 1 ! net
1
2
r AU 2
,
~25!
where A is the wetted area of the plate and x 1 and x 2 are the
upstream and downstream electrode positions, respectively
Phys. Fluids, Vol. 9, No. 3, March 1997
Various numerical approaches have been used to investigate simple flat plate boundary layers ~Refs. 38–40!. The
present formulation involves an equilibrium problem, governed by the elliptic equation ~3!, and two marching problems, one from Eqs. ~1! and ~2! ~parabolic when combined!,
and one from Eq. ~4! ~parabolic or hyperbolic!. ~This equilibrium problem is the Dirichlet problem.! These four equations constitute a coupled system with the four unknowns: u,
v , rc , and f.
Two computational grids were defined for the solution
domain, one for the electrostatics problem and one for the
fluids problem. The two grids were congruent on their outer
boundaries. Finite differencing was applied to approximate
the governing equations @~1!–~4! and ~15!#. First-order accurate forward differencing was used for Eq. ~1!. A fully implicit scheme was applied to Eq. ~2!, which was linearized by
lagging coefficients in the convective terms. The five-point
scheme used with Eq. ~3! is second-order accurate, explicit,
and conditionally stable. A first-order accurate upwind difference scheme was used for the convective terms in Eq. ~4!.
Backward differencing for the first derivative and central differencing for the second derivative were used in Eq. ~15!.
Gauss quadrature was employed to integrate the electrode
current, Eq. ~22!.
The computational domain was a two-dimensional rectangle with the flat plate and its extension in the negative
~upstream! x direction comprising the bottom side of the
domain. The velocity and potential on the boundaries are
given by Eqs. ~10!, ~11!, ~15!, and ~17!–~21!. The value of
the charge density rc at the positive electrode was numerically adjusted in Eq. ~23! to satisfy Eq. ~22! @5Eq. ~9!#. The
charge density was assigned zero values on all gas phase
boundaries of the computational domain ~i.e., excluding
gas–solid boundaries, where it was computed!.
The computational domain was discretized with a uniform grid in the x direction and two ~separate! clustered
grids in the y direction. A coordinate transformation was
employed for grid clustering with a stretching parameter ~initiated with trial values! for each grid. With the two outer
congruent grids defined over the domain, the electrostatic
S. El-Khabiry and G. M. Colver
591
grid satisfies the stability conditions of the five-point scheme
while the fluids grid satisfies the higher resolution requirements of the velocity gradient inside the boundary layer. All
derivatives were approximated directly on the clustered
grids. A cubic spline function interpolation scheme was used
to transfer the solution velocity components from the fluids
grid to the electrostatic grid and transfer back the electrostatic body force solution found from the charge density and
potential on the electrostatic grid.
In the numerical simulation the two electrodes were considered as line sources having prescribed potentials. This
point-electrode model avoided the complex analysis of the
electrode discharge discussed previously and should be considered more accurate at a grid distance far removed from the
ion sheath. The charge density at the positive electrode was
evaluated so as to satisfy Eq. ~22!, the total electrode current,
as well as the local conservation of charge, Eq. ~4!.
The numerical solution was initiated by estimating a
charge value at the positive electrode. An iterative procedure
was carried out on this initial value until the charge density
satisfied the predicted corona current, Eq. ~22!. The potential, charge density, and velocity at all other grid points were
obtained numerically. The elliptic equation was solved by
iteration over the closed domain. A marching process was
used to solve the parabolic equations. The conservation of
charge equation ~4! was marched from the positive to the
negative electrode. The boundary layer equations were
marched from the leading edge of the plate in the positive x
direction.
When a solution on the electrostatic grid was converged,
the body force term was interpolated and applied back to the
fluids grid. A marching process on the fluids grid followed,
and the velocity components were interpolated back to the
electrostatic grid. The process was repeated, alternating solutions back and forth between the two grids until simultaneous convergence was obtained satisfying all defining equations and the boundary conditions. If a converged solution
did not satisfy the assigned electrostatic boundary conditions
at the three gas phase boundaries of the computational domain, the domain was enlarged in the appropriate direction
and the procedure repeated. The solution was considered
converged when the three variables, velocity, charge, and
potential, fell within prescribed tolerances at all grid points
in successive iterations.
With a final solution obtained, the drag force was found
by Eq. ~24!, either with or without corona discharge. Ion
deposition and removal to/from the Ohmic surface was an
option that could be specified in the program. All computations were performed on a VAX 4000. A typical computation of the FORTRAN code involved approximately 105 nodes
and required about 10 min CPU time. This performance attests to the high efficiency of the numerical method.
III. RESULTS AND DISCUSSION
A. Corona characteristics
With the exception of Figs. 2 and 14, the downstream
electrode was held at ground potential, and located 50 mm
from the leading edge, while the upstream electrode was held
592
Phys. Fluids, Vol. 9, No. 3, March 1997
FIG. 3. Positive corona onset voltage with wire size at different gap lengths
between the electrodes; U50 m/s.
at 14 kV and was located 25 mm from the leading edge.
With the ground potential configuration, increased charge is
lost to the surroundings from the positive electrode compared to the symmetrical electrode potentials of Figs. 2 and
14.18
Figure 2 is a vector field plot of the electric field strength
showing the field direction ~without regard to magnitude! at
zero free-stream velocity with ion deposition/removal. In
Fig. 2 the upstream potential was positive 2 kV ~rather than
14 kV!, with the downstream potential held at negative 2 kV
~rather than 0 V! to emphasize both ion removal and deposition phenomena along the surface. Upstream and downstream from the electrode region, the field lines are normal to
the constant potential surfaces at zero current since the surface is conductive; however, between the electrodes, the field
lines run very nearly parallel to the surface reflecting a finite
surface current. Positive ion removal from the surface is observed just to the right of the positive upstream electrode,
whereas ion deposition occurs downstream just to the left of
the negative electrode.
While the surface mechanisms of deposition/removal of
charge may or may not be phenomenologically correct, Fig.
2 gives the field solution should they occur ~the authors are
not aware of any studies addressing parallel gas flow with
surface ion deposition or removal!. In the absence of ion
deposition/removal, the electric field lines in the near boundary layer itself are found to be deflected vertically away from
the surface, whereas, in the far boundary layer the field lines
coincide with the ion deposition/removal model. As noted
previously from the boundary conditions at the surface, in
the absence of ion deposition/removal, the normal potential
gradient to the surface becomes zero, reflecting no current to
the surface, and the tangential gradient in potential reverts to
a simple linear potential drop along the surface.
Figure 3 gives the calculated corona onset voltage for
different wire and gap sizes. For a decreasing wire size to
S. El-Khabiry and G. M. Colver
FIG. 4. Positive corona current for 2 mm wire with the potential difference
at various gap distances between the electrodes; U50 m/s.
FIG. 6. Velocity profiles, ahead of the positive electrode and 25 mm downstream from the positive corona wire ~negative electrode! with/without a
corona and with/without ion deposition/removal; f154 kV, f250 kV,
U53 m/s, ReL 55068.
gap ratio the onset voltage becomes a weak function of the
gap distance, approaching a value of about 600 V for a line
source. Figure 4 shows the calculated exponential increase in
corona current per unit length of wire by Eq. ~9! versus the
increase in potential difference between the electrodes for 2
mm wire electrodes. By Eqs. ~5!, ~7!, and ~8!, the corona
current will increase with either an increase in the potential
difference between the electrodes or a decrease in the wire
diameter. For our semiempirical model the corona current is
essentially independent of the gap length for voltages to 10
kV and small 2 mm diam wire electrodes since all curves
overlap ~Fig. 4!; however, the current becomes dependent on
the gap length when the wire diameter is increased to about
500 mm ~Fig. 5!.
FIG. 5. Positive corona current for 500 mm wire with the potential difference at various gap distances between the electrodes; U50 m/s.
Phys. Fluids, Vol. 9, No. 3, March 1997
B. Drag reduction
The parameters studied for corona drag reduction include the potential difference, separation distance between
the electrodes, distance of the positive electrode from the
leading edge, electrode wire diameter, ion deposition, freestream velocity, and negative electrode bias. Of additional
interest in this study are the effects of corona discharge on
velocity profile, local shear stress, and drag force magnitude.
The velocity profiles at the downstream ~negative! electrode with/without corona and ion deposition/removal in Fig.
6 show that the local surface shear stress will increase as the
positive corona drives up the velocity in the boundary layer
and as charge deposition is reduced. The latter effect is probably associated with a decrease in ion concentration in the
boundary layer. Figure 6 also shows the upstream coronafree velocity profile just ahead of the positive electrode. The
increase in velocity gradient at the wall with the corona is
more clearly displayed in Fig. 7, where the local shear with
ion deposition/removal is calculated with/without corona and
ion deposition/removal. The shear stress was calculated by
the finite difference from two adjacent grid levels, starting at
the wall using Newton’s law.
The increase in plate shear drag is more than compensated for by the momentum addition to the boundary layer
from the field forces resulting in a net reduction in the plate
drag shown in Fig. 8. @Figures 8–13 are plotted against the
dimensionless Reynolds number ReL , which should be interpreted here as a variation in the free-stream velocity since U
was the independent variable used in the calculations ~i.e.,
variables h and L held constant!.# Figure 8 reveals the magS. El-Khabiry and G. M. Colver
593
FIG. 7. Local shear versus downstream position with/without a corona and
with/without ion deposition/removal; f154 kV, f250 kV, U53 m/s,
ReL 55068.
FIG. 9. Drag coefficient versus Reynolds number ~ReL ! with ion deposition/
removal at various electrode potentials ~f154 kV, f250 kV, electrode gap
25 mm, positive electrode distance from leading edge 25 mm, wire diameter
2 mm, U53 m/s!.
nitude of the drag force and the drag force reduction with
corona using Eq. ~24! against the Reynolds number equation
~26! over a 25 mm electrode gap and 1 m electrode length,
with the positive electrode located 25 mm from the leading
edge for 2 mm wires at 4 kV. The drag force without a
corona can also be calculated directly from the shear ~Fig. 7!,
with the results giving close agreement with Eq. ~24!. The
negative values for the dashed line in Fig. 8 represent positive thrust on the plate at low Reynolds number.
A comparison of the calculated drag force for positive
ion deposition/removal was made with the experimental co-
rona drag results of Soetomo15 in Table II, for which we
have computed one set of drag forces at 2 kV for free-stream
velocities up to 2.1 m/s. The order of magnitude net drag is
seen to be comparable for the calculated and experimental
results, but with the calculated corona drag being lower ~i.e.,
more effective in reducing drag! than the experimental results at all velocities. This comparison may be explained
since Soetomo used electrodes of equal radii thereby generating ions of both sign, whereas the calculated values are for
positive ions. The presence of ions of opposite charge would
FIG. 8. Drag force per unit width of flat plate versus Reynolds ~ReL ! number with/without a corona and with ion deposition/removal; f154 kV,
f250 kV, electrode gap 25, positive electrode distance from leading edge
25 mm, wire diameter 2 mm, U53 m/s.
FIG. 10. Percentage drag reduction versus Reynolds number ~ReL varied by
the free-stream velocity! with ion deposition/removal and electrode potential
as a parameter ~f154 kV, f250 kV, electrode gap 25 mm, positive electrode distance from leading edge 25 mm, wire diameter 2 mm!.
594
Phys. Fluids, Vol. 9, No. 3, March 1997
S. El-Khabiry and G. M. Colver
FIG. 11. Percentage drag reduction versus Reynolds number ~ReL !, with ion
deposition/removal and electrode gap distance as a parameter as ~f154 kV,
f250 kV, positive electrode distance from leading edge 25 mm, wire diameter 2 mm!.
FIG. 13. Percentage drag reduction versus Reynolds number ~ReL !, with ion
deposition/removal and wire diameter as a parameter ~f154 kV, f250 kV,
electrode gap 25 mm, positive electrode distance from leading edge 25 mm!.
result in opposing currents that would reduce the effectiveness of corona on drag reduction. It is noteworthy that Soetomo’s experiments were difficult to carry out, particularly
for dc corona, because of the sensitivity of the drag force
measurement and the electrostatic attraction of the drag apparatus to the surroundings that had to be subtracted out in
Table II.
Figure 9 shows that the variation of the drag coefficient,
Eq. ~25!, is diminished with increasing Reynolds number.
The electrode-wire configuration is the same as that used in
Fig. 8. The effect of increased voltage is to increase the drag
reduction to negative values. In this case the momentum
added by the ionic wind exceeds that of the viscous drag
alone. Negative drag implies a positive thrust device ~see
Fig. 8 at a low Reynolds number!. The trend of decreasing
~net! drag force with an increase in corona voltage is consistent with the experiments of Soetomo.15 At a Reynolds number ReL ,20 000 the drag coefficient is seen to be markedly
influenced by the application of corona voltages up to 7 kV.
The Blasius solution @corrected for our Reynolds number,
Eq. ~26!# is also shown in Fig. 9 for comparison with coronafree numerical calculations. Very good agreement is found
for the two solutions. @The Blasius drag force was calculated
between the electrodes from the wall shear t w
5 0.332U 3/2Ar m /x by integrating from the leading edge to
each electrode (x 1 ,x 2 ) to give D 1 5 0.664bU 3/2Ar m x, and
subtracting the two results.#
An alternative drag expression is the percentage drag
reduction in Fig. 10 using Eq. ~27! for the same geometry
and flow conditions as in Figs. 8 and 9. The drop-off in drag
reduction with increasing Reynolds number reflects the decreasing influence of the applied field on the boundary layer
momentum at higher free-stream velocities. The effect of
increased voltage is to increase the percent drag reduction.
At 7 kV, cut-off Reynolds numbers of 20 000 and 12 000 are
seen to correspond to 30% and 100% reductions in the drag,
respectively, with the latter value identifying the onset of a
positive thrust mode. A small percentage reduction in the
drag of about 8% is seen possible up to a Reynolds number
ReL of 50 000. This value of ReL corresponds to a conventional flat plate Reynolds number Rex 5100 000, based on
the distance from the leading edge as the upper limit for
effective corona drag reduction for the particular dataset of
voltages, electrode spacing, etc. being evaluated.
Figure 11 shows that the percentage change in drag is
FIG. 12. Percentage drag reduction versus Reynolds number ~ReL !, with ion
deposition/removal and positive electrode distance x 1 from the leading edge
as a parameter ~f154 kV, f250 kV, electrode gap 25 mm, wire diameter
2 mm!.
Phys. Fluids, Vol. 9, No. 3, March 1997
S. El-Khabiry and G. M. Colver
595
FIG. 14. Ratio of electrostatic y/x body forces versus downstream distance
with ion deposition/removal and the y position as a parameter ~f151.5 kV,
f2521.5 kV, U53 m/s, ReL 55068, positive electrode distance from leading edge 25 mm, wire diameter 2 mm!.
increased with the gap distance between the two electrodes at
a given Reynolds number at a constant potential of 4 kV.
Figure 12 shows that the percentage drag reduction increases
with the distance downstream of the electrode pair from the
plate leading edge for conditions of constant gap separation,
Reynolds number, and potential ~4 kV!. Both Figs. 11 and 12
imply that the efficiency with which corona reduces the drag
is improved when the discharge is located downstream from
the leading edge of the plate, where the shear forces to be
overcome are reduced.
In Fig. 13, the percentage drag reduction is inversely
related to the diameter of the corona wire at constant voltage
and Reynolds number. This follows since both the corona
current and positive ion concentration in the main flow decrease with the diameter of the corona wire for a given electrode potential difference by Eqs. ~5!–~8!.
To evaluate the distance over which the thin boundary
layer equations approximation Eqs. ~1! and ~2! remain valid
near the electrodes because of the strong y component of
force, the program was run for one case in which the two
electrodes were maintained at the same magnitude but with
opposite polarities of the electric potential ~61.5 kV!. @A
somewhat related problem to the electrode problem occurs at
the leading edge in boundary layer theory, as noted previously below Eqs. ~1! and ~2!.# Figure 14 shows the ratio of
the vertical to the horizontal component of the electrostatic
body force in the region between the two electrodes. The
results show that the force ratio increases with height above
the plate surface. At the surface ~y50! the vertical force
component of the electric field falls to near zero values for
distances from the electrode exceeding 5% of the gap length.
For the thin boundary layer approximation in which the y
component of momentum is neglected ~gradients are as596
Phys. Fluids, Vol. 9, No. 3, March 1997
FIG. 15. Boundary layer growth ‘‘corona thinning’’ with ion deposition/
removal ~circles are electrodes; f154 kV, f250 kV, U53 m/s,
ReL 55068, positive electrode distance from leading edge 25 mm, wire diameter 2 mm!.
sumed to be negligible in the x direction in comparison to the
y direction!, this 5% distance can be taken as the nearest
distance to either electrode for which our analysis is valid.
The full Navier–Stokes equations would be needed to correct the near-electrode region ~,5%!, as well as for the leading edge approximation.
Figure 15 shows the calculated ‘‘corona thinning’’ effect
of the boundary layer for a downstream directed corona compared to a corona-free flow ~see Fig. 1!. As noted previously,
an upstream directed corona leads to ‘‘corona thickening’’ of
the boundary layer and a corresponding increase in the plate
drag.
An additional drag effect is that of the negative bias of
the downstream electrode below ground potential ~see Fig.
1!. Our calculations show that for the same electrode potential difference, electrode geometry, and free-stream condition
that corona drag reduction is diminished as the negative potential is driven below the ground potential. This corresponds
to increasing the current flow to the surroundings.
C. Ion deposition/removal
The influence of ion deposition/removal at the plate surface is indicated by the net drag force calculations in Table I
for Df54 kV ~constant source current density! for which an
effective increase in net plate drag from 20.0003 to
;0.0000 N is observed by Eq. ~24! with ion deposition/
removal ~negative drag implies a positive thrust on the plate
as in Fig. 8 at small Reynolds number!. An evaluation of the
spatial charge distribution above the plate shows that simultaneously a significant reduction of charge concentration occurs by a factor of the order 103 when charge deposition/
removal is included in the model. It is this reduction of ion
concentration that apparently accounts for the reduction in
drag effectiveness with ion deposition/removal. An obvious
S. El-Khabiry and G. M. Colver
TABLE II. Comparison of calculated ~net! drag ~N! on plate with positive
ion deposition/removal compared with the experiment drag of Soetomo.15
Soetomo’s data based on electrode separation 20 mm; distance downstream
of positive electrode 2.5 mm from the leading edge; electrode width 28 mm
~corrected to 1 m above!, downstream electrode grounded.
Free-stream
velocity m/s→
2 kV
calculated drag ~N)
2 kV
experimental drag ~N)
0
0.66
1.14
1.62
2.09
0.0000
0.0001
0.0004
0.0008
0.0013
0.0000
0.0007
0.0030
0.0062
0.0114
explanation for the decrease in corona effectiveness on drag
is the ‘‘short circuit’’ of gas ions through the conducting
surface ~see Fig. 2!. However, a closer examination of this
mechanism below suggests that it is probably not the correct
explanation.
We can examine the role of ion deposition/removal on
net plate drag over separation distance L of the electrodes
with the approximate drag equation of Colver et al.,3 which
neglects charge motion by fluid convection,
D ~ x 2 ! net2D ~ x 1 ! net'2
I c wL
K
1w
E
x2
x1
m
] u ~ x,y !
]y
U
dx ~ N ! ,
y50
~28!
where the first term on the rhs is the drag reduction due to
positive gas phase ions and the second term is the surface
shear. @Equation ~24! when evaluated for D(x 2 ) net2D(x 1 ) net
equates to Eq. ~28!.#
For the first term on the rhs in Eq. ~28!, we note that the
surface conductivity is small but finite for our problem
~ss 510214 S sq!. As a consequence, the quantity of ions removed or deposited at the surface is expected to be limited to
values not significantly different from the surface current
density J s ~A/m! in the absence of ion deposition/removal, as
shown in Table II ~J s ;1.631029 A/m!. To account for the
decrease in the net drag in Table II by a reduction in gas
phase ion current alone using our values of L and K in Eq.
~28! would require a substantial increase in the surface current to J s →331026 A/m in the region between the electrodes. This increase in the surface current density J s would
imply a corresponding increase in the electric field strength
E s by a factor of 331026/1.631029 –103 from the first of
Eqs. ~14!. However, no such increase in field strength can be
observed, as indicated by the surface potential in Fig. 16
with ion deposition/removal; rather, only a modest nonlinearity in the potential is observed compared to the linear
case. We conclude that the calculated drag reduction due to
space charge reduction is not simply explained as a ‘‘short
circuit’’ or bypassing of ion current. However, the bypassing
effect of current should become increasingly important on
corona drag as the surface conductivity increases.
To consider the second term on the rhs in Eq. ~28! as an
explanation for the net drag increase with ion deposition requires that the integrated surface shear between the elecPhys. Fluids, Vol. 9, No. 3, March 1997
FIG. 16. Electric potential along the surface and in the gas phase showing
nonlinear conduction on an Ohmic surface ~y50! due to ion deposition/
removal ~f154 kV, f250 kV, U50 m/s, positive electrode distance from
leading edge 25 mm, wire diameter 2 mm!.
trodes with and without ion deposition/removal account for
any differences in drag. In the numerical calculations of
Table II and Fig. 7, no such change in the calculated shear
force was found to four significant digits with and without
ion deposition/removal ~viz., 0.0011 N!. It follows that an
alteration in the surface shear does not account for the drag
effect with ion deposition.
A remaining possibility is that spatial charge reduction
and the attendant drag increase are a consequence of an alteration in the boundary conditions along the Ohmic surface
that takes place with ion deposition/removal ~see Fig. 16!.
This decrease in surface potential also conforms to a decrease in current flow to the surroundings, with the latter
effect having been shown to alter the corona drag effectiveness. Regarding the accuracy of the calculations, it is noteworthy that Table II values of the drag force between the
electrodes without corona ~0.0010 N! are in close agreement
as calculated by the momentum integral equation, Eq. ~24!,
or by integration of the surface shear. Table II also shows the
expected result that the shear drag is increased by corona
0.0010→0.0011 N, although there is a reduction in the net
drag.
IV. CONCLUSIONS
The present numerical study confirms ‘‘corona thinning’’ of the boundary layer previously identified theoretically by Colver et al.3 with a corresponding reduction in the
drag of a flat plate for a downstream directed dc positive
S. El-Khabiry and G. M. Colver
597
corona. The electrodes were treated semiempirically as line
sources. The reverse effect of ‘‘corona thickening’’ and drag
increase occurring for an upstream directed current was not
investigated in the present study. The corona drag reduction
is found to be effective ~.8% improvement! for Reynolds
number Rex ,100 000 ~ReL ,15 000! based on the distance
of the farthest downstream electrode from the leading edge
for a particular set of voltages ~,7000 V!, electrode positions, wire diameters, and corona discharge characteristics.
Ion deposition and removal along a semi-insulating
Ohmic surface were included in the model as possible charge
transfer mechanisms affecting the drag. A decrease in the
effectiveness of corona drag with ion deposition/removal is
thought to be due to a reduction in positive ion concentration
associated with a corresponding alteration in the surface potential. However, the relationship between drag and corona
current has not been resolved. The possibility of a ‘‘short
circuit’’ of the ion current to the surface does not appear to
explain the drag reduction. With the negative electrode held
at the ground potential ~rather than negatively biased and
opposite in magnitude to the positive electrode! additional
charge is conducted to the surroundings.
Our simulation indicates that dc positive corona drag
reduction ~corona effectiveness! increases with; an increase
in potential difference ~corona current!, a decrease in gap
length between the wire electrodes, a decrease in free-stream
velocity, an increase in the distance of the corona electrodes
from the leading edge of the plate, a reduction in ion
deposition/removal at the surface of the plate, a decrease in
the diameter of the wire electrodes, and a decrease in the
negative bias of the downstream electrode below the ground
potential.
The drag calculations in the present study with a corona
are thought to be numerically meaningful, showing general
agreement with trends found experimentally. However, experimental drag reduction with dc corona is found to be less
than that predicted by our model. The present modeling can
be further advanced as follows: including ion diffusion and
ion mobility dependence on electric field strength; incorporating limiting mechanisms for charge deposition and removal; simulating a non-Ohmic surface conduction model;
incorporating a theoretical development for F–I discharge
characteristics ~presently treated semiempirically!; and by
the application of the full Navier–Stokes equations to remove the boundary layer approximations at the leading edge
and the electrodes while incorporating a finer electrostatic
grid and fluid grid to improve numerical accuracy.
ACKNOWLEDGMENTS
The authors would like to thank Professor Joseph M.
Prusa at Iowa State University for helpful discussions on the
numerical aspects of this study and Professor James F.
Hoburg at Carnegie Mellon University for his comments on
electrostatics. The authors are further indebted to the reviewers for their constructive comments and careful attention to
detail.
598
Phys. Fluids, Vol. 9, No. 3, March 1997
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