Study of flow control using the MHD effect

TEcHNJScHE UNJVffiSREIT

vor

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16L015.788873.Fazi

O15-7a1 ABSTRACT

The

_study

on

the use of the MIlD

(Magneto-Hydrodynaniic) effect for controlling flows past a

wing

section

and a ship hull was made using CFD

(Computationá]. Fluid Dynamics) techniques.

The governing equations were derived for the

motion

of conducting viscous

flows

under the

influence

of the

electromagnetic forces.

They

are

the

incompressible Navier-Stokes equations

with

added MRD

terms.

Either of the two

tùr-bulence

models, i.e., the k-E two-equation model

or

the

Baldwin-Lomnax 0-equation model, was used

to include turbulence effects.

The

resulting

governing equations

were

discretized

using

the finite-difference method.

The

IAF scheme was used to solve the discretized

equations

by introducing pseudo-compressibility

into the continuity equation.

-The

flows

past a 2-D wing sectiorL. or à 3-D

ship

model

were

computed.

The results on the

wing

section

showed

that.,

by imposing the MRD

forces,

the lift increases and the drag-reduces,

resulting

in

the

improvement

of the lift/drag

ratio.

The results on the. ship hull showed that

the wake became thinner and the pressure recovery

In

the

aft

end

part

of

the

ship

body

was

Improved.

The efficiency of the MRD forces with

sea

water

used

as

media was found to be low,

mainly

due

to

the

low conductivity of the sea

water.

NOMENCLATURE

u : velocity vector

P :

fluid density

B : magnetic density vector

u

: magnetic permeability of the fluid

J

: electric current density

L : representative length

H: Hàrtnianp constant ([c/Pu1L'2BL)

p : pressure

kinematic viscosity

E : electric Intensity vector

electric conductivity of fluid

U

: representative speed

Re: Reynolds number (=UL/)

Rin: magnetic Reynolds number (=uciUL)

(x,y)

Cartesian coordinates

(,rm)

:

body-fitted coordinates

on Flow Control Using the MIlD Effect

Munehiko

Hinathu

Yoshia.ki Kodamna

Yoshitaka Ukon

Ship Research Institute, Japan

Ship Research Institute, 6-38-1, Shinkawa, Mitaka, Ibkyo, Japan.

i . IN11ODUC1'ION

Many

studies on flow control have been fflade

aiming

at

Increase

in hydrodynamic efficiency.

Some.

of

the.

súccessful

examples

are,

vortex

generators ,

bouñdary-layer

suction ,

multi-step

flaps,

and

leading-edge

slats

on

wings

in

aeronautics,

and

bulbous

bows,

bulbous tails,

ducted

propellers

In

nava].

- architecture.

However,

in ail of the above examples except for

the boundary-layer

suction, the flow field

coñ-trol is made using appendages.

- On

the other hand, Investigations wére made

by

PhIllips

and Yamaguchi 2to use

electromnag-netic forces for ships' propulsion.

K1tano3used

modél

ships

to conduct research on

electromnag-netic

propulsion.

They

reveàled

that the

propulsive

_efficiency

of

such a system Is very

16w, due to the high electrical resistance of sea

water, and that the magnetic field required needs

to

be

of very high intensity.

Both of the

dif-ficulties

prevented

the

realization

of

an

electromagnetic

propulsion

ship.

However,

progress

in

cryogenics and superconductivity is

quite rapid today, and the generation of magnetic

fields of high intensity and Its application for

engineering are becoming feasible.

In the

light

of

the

progress in today's

technology

above

mentioned,

the authors

inves-tigated

to

evaluate the

possibility

of using

electromagnetic

forces for flow control, because

the

authors

believe

that.

the local use. of the

forces

is

more hopeful, while the globàl use. of

electromagnetic forces directly to ships'

propul-sion can still remain infeasible.

A research

tool. used is cFD (Computational

fluid Dynamics), in which the governing equations

for the flow field are directly computed.

CFD is

In

rapid progress today, keeping pace with rapid

progress

in

computer hardwares.

Experiments on

MHO

(Magneto-hydrodynamics) are quite costly and

difficult, requiring special care to avoid

inter-ference

of

measuring probes

with the magnetic

field,

for

example.

CFD can become a powerful

substitute

for

such

difficult experiments,

be-cause,

in

CFD,

one

can

simulate a variety of

flows easily by changing parameters, and obtain a

lot of informations on the flow field.

In

Chapter

2,

the governing equations are

derived-

for laminar and turbulent f loWs with sea

water as the media, under the assumption that the

electrical

resistance

is very high.

IñChapter

3,

the

discretized

form of the governing

equa-tians

are shown.

In Chapter 4, computed results

are

shown

for flows past a two-dimensiònal wing

section

at

the Reynolds number of

(laminar)

and

1Ó6(turbulent),

and the MIlD effect for flow

control

is

evaluated.

Further,

computed flow

past

a Wigléy ship modél is shown and evaluated.

Finally, the òonclusions aré stated.

2 .GOVERNING EQUATIONS

2.1 MMD equatIons4'5

In

the present paper, only the sea water Is

considered as electrically conducting fluid.

The

motion

of

_{electrically conducting fluid In the}

electro-magnetic field Is governed by thé

follow-Ing equations.

(u.V)u=.Lvp+vAu+i(jxB)

_(2.1)

Vu = O

_(2.2)

where u is a velocity vector, P Is density of the

flúid,

p

Is pressure, j is the electric current

density, and B is the magnetic flux density.

j

and

B

satisfies

the

following Maxwell

equations.

2.2 EvaluatIon of Induced magnetic f leid

The equation of Induction (2.8) has the same

form

with

the

vorticity

equation.

From this

analogy,

the paramétér (uaY' IS called níagnetic

viscosity.

And the parameter Rn

E 1.JOUL

_(2.9)

is

éalled the magnetic Reynolds number.

In case

of sea wáter, Rth " O(lOs).

Therefore, Eq. (2.8)

can be appfoximated as

tBs0

-

- (2.10)

Furthermore, J normally satisfies the

Qin's

lai.

ja(E+uxB)

(2.7)

where

u

is

the magnetic permeability, E Is the

electric

field,

and

a

is

the

electric

condúctivlty.

In general casés, the motion of electrically

conducting. fluid

is

determined by solving the

equations

(2.1)

and

(2.2)

together

with

the

equation of induction, which is derived from eqs.

(23),(2.4), and (2.7), i.e.

. a

That is, the magnetIcÑi'tion

of

sea water can be neg1ectdipared With the

applied magnetic field and ¿!héimosed m4etic

field

can

be

obtáiñed

Laplace equation.

On

the

other hand, when the electric field

is

applied, a magnetic field is generated around

the

electric

current.

The

magnitude

of

the

generated magnetic

field

B Is evaluated by

in-tegrating

Eq.

(2.4)

in

a

region

where

the

elèctric field is applied

B 'u _(2.11)

where I: total electric current

S: Area of applied electric field

u

is of 0(10e) for sea water, and assthning that

I is of lO3 Amperes at ost, the induced magnetic

field

can again be neglected, In comparison with

the

applied, magnet-ic field.

Due to the reasons

stated

abOve,

the.

induced magnetic

field

is

neglected

in all subsequent discussions, and the

magnetic field is treated as given.

2.3 GovernIng equations

The

governing equations

used

here are as

follows.

=. - Vp + vAu + (E + ÜxB)xB

(212)

Vu

O _(2.13)

where

u

and

p are unknowns.

The equations are

then nondimensionalized.

*

*

2

u=uIJ,p=ppu, r=rL,

B=BB0, t=tL/U,

*

_E=EUBo

*

(2.14)

+ +

(E + w'B)xB

where

Re

Is

the

Reynolds number

Hartmann númber defined as follows.

(2.15)

(2.16)

and Ha Is the

The

Hartmann

number

is the square root of the

ratio

of the electromagnetic force to thé viscous

force.

In

2-D

case,

assuming

that

the body

surface is inconductive, the electric field E can

be

_explicitly

given, once the region where E is

applied Is determined.

2.4 Modlf led k-E equation6

When

a fluid moves at high Reynolds numbers

in

an

applied

electro-magnetic field, the flow

becomes

turbulent

in general, and therefore the

turbulence analysis is necessary.

(2.3)

V X B = (2.4)

VB = O

(2.5)

Vj = O

(2.6)

Re = tJL/v

(2.17)

+ (uV)B

(BV)ú +

(2.8)

_{Ha u}

_v'7

_LB0 (2.18)

In the 0-equation model, eddy viscojity is.a

function

of

mean

_{velocity components throgh}

which

the MIlD effect on turbulence is refIétéd.

In

the 2-equation

turbulence

model,

the

MRD

effect occurs both in. the length scale and in the

velocity scale of turbulence, and therefore it is

possible to take into account the MRD effect more

closely than the 0-equation fflodèl.

Now,

the velocity

components and pressure

are

decomposed

into

mean

and

fluctuating

components.

Assuming that there is no

fluctuat-ing component in E and B,

u =

+ u', p =

+ p', ii' = 0,

' =0,

_B=,

B =

(2.19)

The

above

equation

is substituted

into

Eq.

(2.15),

and

time-averaged

to obtain, In tensor

form

au.

au.

a2i.

_:1.

3(u!u!)

_1]

ax.z -

ax.

at

ax

3Xj

J J

(2.20)

+

ijkjk +

ijkEjlmUlBmBk

The

equation

for

the

turbulence

kinetic

ener' k

u2/2 is derived similarly with ord1

nary non-MRD, flows.

+

--

.i+ s

at

U1

3x.

T

ax.

ax.

ax.

E+a[{V+IakI

_ffaBk (2.21)

BB]

The rightmost term in tq.(22ì)'i5 a sink, WhIch

shows

that the

turbûlence

kinetic

ener' is

absorbed

by

the action of magnetic field.

This

is

because,

in case u and B are orthogonal, the

electromagnetic

_force

( uxB )xB acts òonstañtly

in opposite. direction with u, and the fluctuating

velocity

component

orthogonal with B is damped.

The

process

Is nonisotropic

and

therefore an

experimental

constant

Ca

is

multiplied.

According

to Kitamura

eta1.7, Ca=3/8, which is

adopted here.

The equation for E is in an exact form,

Dt

ax

ax

3x

p axt,. ax

3x.

2

3u

a2

2va

auj a(u'lBmBk)

y u1

_ax

axax1

+ p

ijkCjIm ax

ax

-(.224)

This

form can not be directly solved.

Recently,

using the

TSDIA

(Two-Scále

Direct-InteractXon

Approximation)

method,

Yoshizawa8

has derived

theoretically the following form for E.

= C.y.

C1= constant

(2.25)

The

k-equation

is

substituted

Into the above

equation.

Then

Dc a

vTac

DE

v-

-i kax1 Tax1

'v

ax1

2k3pbj

C C

2 A

(2.26)

An

empirical

coefficient cb in Eq. (2.26) is set

to.3/4.

nondimens ionalizat ion

The

Reynolds

stress term, the 3rd term in

RHS

of

Eq.

(2.20),

is

modelled

using the

gradient-diffusion hypothesis.

=

V[

}

-

(2.27)

(2.22)

'Using

the

values shown in Eq. (2.]4),

nondimen-sionalization

is

made on the momentum equations

ánd the k-E equations.

(2.23)

Ç Ç

1

a21

a:

.

Ç

;+Ui_ax+.+

-.

4

Cijkjk +

(2.28)

---v

-at

i 3x

T ax

aç.

,Ii

V,

' 2

_C_4CaBj2k

2B1B J 3

ai'

(2.29)

.

a. au! au'

3u

au!

i e +

2v2

ax1 ax

2'v

ax

a* a1

aic

where

au! au!

₁

3.

3x.

ax.

J J VT

= C

C

= constant

a +

ax

3c

i

uji=_

i

The underlined term in Eq. (2.28) was omitted in actual computations, because, with the term present, the pressure contouis showed skewness near the body surface. _{The values of the coeff}

i-dents

C1, and C2used are

C=O.O9, a,=1..0, o=i.3, C1=1.44, Cfi.92 (2.31)

They are

standard

values, in non-Mil]) condition.

Though it Is better to use values which take Into account the MHD effect, they are not used here because they were not available.

BoUndary. conditions

The

boundary

conditions

for laminar flows

past a

wing section are as follows.

_{At upstream}

boundary,

a

uniform

flow

was

Imposed,

and

at

downstream

boundary,

zero

extrapolation

was

imposed. On the wing surface, the ist and 2nd derivatives of pressure with respect to n (normal

to

the wall)

were

set

zero, and non-slip

and linear extrapolation conditions werè used for velocities.

In the computation of high Reynolds number turbulent flows using the k-z lliodèl, it is common

to use the local equilibrium hypothesis for k and

z and the logarithmic law for veloc1ty

In that

cáse,

the flow around a body is treated as fully

turbulent

in

the

entire

region,

and,

near the

point

of

stagnation,

the

velocity

component

tangential with the. wall becomes siiíall, änd

the

friction velocity becomes small. Thereforé, the logarithmic velocity distributiOn and

local

equilibrium do not hold there.

In order to circumvent this problem, the flow region near the wall .is decomposed as a function of y= ReUY which is a nondimensional distance

from

the

wall)0

_{uT iS the friCtIon}

velocity.

+ lo 30 6°

velocity.

Spal Ing

formula

(composite region)

logarithmic law

In the cmpösite region, the functions in inner and oUter layers are smoothly connected using a..

cosine fUnctiOn.

3. FINITE DIFFERENCE DISCEETIZATION

(2.30)

3.1 Governing equations in ladmar flow

The IAF (Implicit. Approximate Factorization) Which s popular iñ the

aeronautics

field was used.

Thi

scheme

neces-sitates

the

presence

of the time derivative of

each dependent variáble. Therefore, the time derivative of pressure was artificially added to the continuity equation (2.16). The resulting equation was computed together With Eq. (2.15).

1

2

=

-Vp u

+

-(E

+

uXB)xB

+ B(Vu)

O

When

a solution

reaches steady state, the

aplat

term

vanishes

and

the original continuity

equa-tion

is

exactly

satisfied.

The Eqs. (3.1)

and (3.2) can be written in vector form..

q I+

=

H(%+ q)

+ C... (3.3)

Here,

q=[u,v,p]i'

where

u

and,

y are velocity

components

in

x

and

y directions, respectively.

The coefficient matrices are

Next, Eq. (3.3) Is transformed into body-. fitted coordinates

(.

n) defined as

=

(x,y),

r = _T1(X,y) . (3,5). resulting in

q+ ai= H(aq+

¼ q1.11

dq+ eq)

-

wflfl1)+

+

C

(3.6) where

a=J

b=-Jy, c=_JXi

d=Jx, J=l/(xY_xY)

A=aIri.i,B=bL+dM

(3.7)

a, b, c. d are metrics, a, b, etc. can be defined similarly.

At high Reynolds numbers, the boundary-layer

developing on the body surface is very thin and

the

mesh

size

needs to be very small there, in

order to get enough resolution. Also, as the

u O i

y

O O 1/Re O O L=

O u O

B O O , O

y

1 O O , O

1/ReO

O O O -B 2

BB

O

-EB

Ha2

S=-

Ha2

EB

(3.4) ' Re O O O O behavior at inner layer

k, z

(composite region) Loca]. equilibrium condition

diffúsion tenu In the governing equations becomes very small lt Is difficult to

get

enough

rniinèri-cal

stability. Therefore, the 4th' deriütivé

térms

are added In E4. (3.6) and treated time-Implicitly. _WZ

and

arepositive cOnstants.

Thè

time derIvatives are approximated by the Euler Implicit formula written in the

form

of Padè time differenclng.

=

fl,

gt=fl+l_ gtfl 1qt=fl

Here the truncation error is 0(At). Substituting Eq; (3.8) Into Eq (3.6)

q" t{(Aq)'1+ 4(Bq)1-

_{)"_}

c(Eq). d(Hq)'1_ e(Bq)'_

(Sq)nl+

+ =

t{Aq'

-1i(aq

+bg

+ cq

+ dq

+

q) - Sq

+ C} (3.9) (3.8)

Approximate factOrizatiOi is nadé to the LHS Of Eq. (3.9), separating Z

and ri

derivatives. The final

form

is

{'i+4t(s i-A +A

-Hag _HE2«Z)}

{i +At(

a a a2

3kflfl

A A A A A =

-Sq

A

n-1

+At(q)

(3.12)

where the mixed derlvative term Is shifted by one time stép. The aboVe equation can be solved in and n directions separately, similar to the API

(Alternating Direction Implicit) method

The spatial derivatives in the above equa-tion are approximated by 5-point central differencthgs

a. 1

-2

-1

12

EZ - 8EZ +

8Er- E)

-4Er +6E

-4E

+

Substituting, the above equatiOns into Eq. (3.13) results ithe following systems of equations for the Z-sweèp.

Z-sweep

*

*

*

*

J&j*2+K4q1

_+I4q

_{+MAq41 +Naq2=f}

where.

J=

(A

+H(a-d) +12wI], K=-[A +H(2a-.d)

I

I

+1-S +A +Ba

_+6wZI]At

(3.16)

M*

-H(2ad) -6wI], N= 4[A H(ad)_l2wZI]

f=

-Sq

+ttHb(

The above equation forms a block pentadlagonal system. Solving them with suitable boundary conditions produces qTl, and qr Is ObtaIned by Eq. (3.9). The procedure is repeated until

convergence is obtained.

3.2 Governing equations in turbUlent 'flows

In solying for

the. turbulent

flow using Eqs.

(2.28) (2.30), it freqUently occurs that either.,

k or E becoùies negative and the computation blows

up.

In order to circivent this

dlffIcult, one

ofiilthbrs

(Kodauia13)

expressed k and a

in

exponential forms

shown 'below.

k ='exp(Ìn)

,

c =

_expçn)

(3.17.)

where n and

n are computed instead of k and . Using the above expréssions, it Is obvious that k or E never becomes negative. Using the above equation, the governing equation corresponding to Eq. (3.6) becOmes, in. vector form

A A A

+T

i-C (3.18)

Time

derivative in Eq. (3.18) is approximated by Padé time differencing. The approximately fac-tored equation for Eq. (3.18) is

(ï +at(-S -T1-i-A +(A

-Ha2+w

j)}

{

+tt(

}.qfl

=

-at(-.-a-H(

-Sq

-c -T

+atI(ag)

A

n-1

(3.19)

here the following relations aré used.

5

The noñllnear terms in the abôve equation are locally linearized as follows.

Mkk)%

where A

OA.q+

auZ CUZ O

A.cQ'= q'' AAq

(3.10)

A= avZ

CVZ O (3.11.) 0

00

The term with respect to B is treated similarly.

a2

Ing pressure and shear stresses along the wing surface. Also, the lift and drag forces gener-ated as reaction to the MILD force were computed by spatially integrating the MRD force. First. the relation between lift and the applied electric field is discussed. Fig. 4.8 shows the rélatlon between the electric field intensity and the lift or the lift components. AS seen in the figure, the lift comes mostly from the surface pressure integration. The lift component which arises as reaction to the MIlD force Is propor-tionál to the electric field intensity. This is due to the fact that the (ExB) term Is much larger than the (uxB)xB term, 1n the momentum equation. The lift increases linearly with lEI in the

range 0IEll00.

At higher IEI, the relation becomes nonlinear. Its reason Is as follows.. In the momentum equation, the MRD force term affects mainly the pressure gradient term, and as the fOrce still increases, it affécts the nonlinear convection term.

Next, the relation between drag and the applied electric field is considered. Fig. 4.9 shows their relation, together with the drag components. The magnitude of the drag components arising from the frictional stress and the sur-face pressure is of the same order, but, as lEI Increases, the friction component becomes predominant, which can be understandable from the fact that flow separation has disappeared. On the other hand, the leading-edge suction In-creases as lEI Increases, resulting in the decrease in the pressure component of the drag. These two cancel out. The drag as reaction to the MIlD force is negative, which means it acts as thrust, which is proportional to lEt. The mag-nitude of the thrust Is greater than the frictional component at E=-200, and it plays an important role In decreasing the total drag, as a result.

4.2 Evaluation of the turbulent flow coiputatioms wIth MRD forces

In order to check the validity of the present formulation and computations for the turbülent flow with MRD forces, a channel flow was computed under the same condition with Broüillette's experiment.15 In the experiment,

conducting fluid flows through a channel with rèctangular cross section surrounded by an insu-lated material. The magnetic field is imposed in normal direction to the flow, and the electric field which is normal to the magnetic field satisfies the Rartsiann condition. The Reynolds number based on the mean flow speed and the hydraulic diameter was l.8xlO hen the size of the rectangular cross section is 2ax2b, the hydraulic diameter is 4ab/(a+b).

In the computation, the magnitude of the electric field was updated at each time step, so that total electric current across the channel becomes zero. The pressure gradient was deter-mined so that the mean flow speed becomes unity.

Fig. 4.10 shows the comparison of the com-puted velocity distribution with the Brouillette's experiments. The horizontal axis is plotted logarithmically. The agreement is very good. The computed results simulate well the flow behavior, in which the velocity near the

= T1q +T2q +T3q

_(3.20)

o _(3.21)

Since Eq.(3.21) corresponds to Eq.(3.12), Eq.(3.2l) can be solved numericälly similarly to section 3.1.

4. COMPUTED RULTS

4.1 Flow control of laiinar flows past a wing sect1on14

C-grid was use4 for flows past a wing sec-tion, whose topology and numbering is shown in Fig. 4.1. The wing section Is NACAOO12 With an attack angle x=5 deg. Fig. 4.2 shows the grid used. The parameters used In the computation are, Re=10

_w=w,=5.O,

minimum spacing=O.002, t=O.2. Convergence was obtained after

500

iterations. The convergence criterion was

U..=1o05.

The CPU time was 30 sec/iteration using the Fujitsu FACOM M18OIIAD.

Fig. 4.3 shows the flow with no MIlD forces. Large flow separation exists on the back side. At the stagnation point, C=l.0. and C becomes negative on both sides of the wing surface.

The magnetic field was imposed on the back side of the wing as shown in Fig. 4.4, so that the flow is accelerated there by the Lorentz force, in order to avoid the flow separation on the back side. The distribution of the magnetic field was obtained by placing magnets In the aft part of the wIng. (70 % of the ôhord length), with N poles on the back side and S poles on the facé side, and

by computing

the field using the poteñtial theory. Further, the electric field is Imposed such that the electric current occurs 1h normal-to-paper direction inward. Fig. 45 shows the generated MIlD forces.

FIg. 4.6 shows the flow under the effect of MRD forces. The intensity of the imposéd mag-netic field corresponds to H=l2.64, nond.imenslonalized by the maximum intensity of the magnetic field. The intensity of the electric field was E=-200. In dimensional units, they correspond to 2 tesla (1 tesla=104 gauss) for the maximum magnetic field intensity, and 40 V/m for the electric field intensity, when the chord length is 10 cm and the uniform flow is 10 cm/sec. As seen in the figure, the flow separa-tion vanishes. The pressure on the back side is made smaller, and the pressure oñ the facéside

is made larger. The difference is moré clearly seen in Fig. 4.7, where the pressure distribution along the wing surface is shown. Thè MRD force has made the negative peak on the back side steeper. Further, the pressure at the trailing edge changes from negative to positive, and the pressure on the face side is nade larger. The kink in the distribution on the sûction side shows that MRD force Increases in a step-wisé fashion there.

integrat-wall increases as the magnitude of the-applied magnetic field increases, and

the. f1-SV

retarded away from the wall due to theUàFtmànn effect. Thus the validity of the present com-putation has been demonstrated.

4.3 Turbulent flow past a wing section using the

k-E

turbulence iodel

In order to see how accurately the turbulent flow past a Wing section can be computed using

the k-s turbulence model, the flow past a NACAOO12 wing section with an attack angle x=5 degrees

at Rel06 was

computed with no MID forces. The parameters used are; t=O.002,

w=w2O.

and The convergence

criterion was The CPU time

was 2 minutes per timestep using a Sun3-llOC workstation (with FPA). Convergence was obtained after 1300 tlmesteps. Fig. 4.11 shows the com-puted pressure distribution along the wing surface. In the same figure, the computed result using the Baldwin-Lomax zero-equation turbulence modeI16 Though both results show similar ten-dency, the negative peak at the leading edge of the result using the k-E model is lower than the other, and the negative pressUre zone on the face side is greater. A reason for the difference may be that, in the 0-equation model the transition between laminar and turbulent state is taken into account, and the flow near the leading edge is judged as lamiñar, While, in the k-s model, the flow Is cOmputed aS turbulent In the entire domain. Another reason may be the magnitude öf and w1., which are equal to 5 in the 0-equatloñ computation. However, from the fact that, In a channel flow computation, the friction coeffi-cient varied by only 1.5 % by changing w from 5 to 20, the effect of the difference in transition Is considered to be much greater than that of w.

The lift and drag coefficients were computed by integrating the computed pressure and friction force. The lift coefficient CL was 0.416 using the k-s model, while 0.500 using the 0-equation model, and 0.57 from the experiment by Abbott7 The k-s model produces only 73 % of c1 by experiment. This Is due to the tendency that the negative peak is insufficiently computed and the negative zoñe on the face side Is over-estimated. The drag coefficient CD was 0.039 using the k-s model,

0.015

using the 0-equation model, and 0.008 by experiment. In the computed result sing _the

_k-s

_{model, ,the ,mgnitude of. the;} fric-tional and pressure èomponents in the drag coefficient is of the same order at an attack angle x=5 degrees. It Is considered that the friction is over-estimated in the present k-s computation, compared to the 0-equation computa-tion, because the whole flow field is computed as turbulent, and there Is difference in the pres-sure distrjbutiôn. How to include transition from laminar to turbulent into the k- model is an important future task, and it Ïs inevitable to obtain drag with high accuracy. Further, in the present computation, the effect of the pressure gradient is not taken into account in the

bound-ary condition för velocIty. It is not yet clear how 1Icäntly that affects the computation near the leading edge, where the pressure gradient is very large. Therefore, the boundary condition on the wing surface is another future

taSk)8

It has been shown that the present computa-tion method has a certain problem in estimating lift and drag forces. However, the authors believe that the present computation method Is still useful In estimating general tendency of the effect of the MED forces on the flow control, and the turbulent flow field with MilD forces was computed.

4.4 Flow control using MilD forces on the turbulent flow past a wing sectIon10

Computations were made for flow control using the MRD forces for turbulent flows past a wing section. The applIed magnetic field Is the saine as in Fig. 4.4. The zone where the electric field was imposed was confined, In the same way as in laminar flow, to the aft half of the back side df the wing.

The nondimensionalized magnetic field inten slty was kept constant at Ha=126.4. This corresponds to the maximum magnetic fIeld Inten-sity of 2 tesla, when the unIfOrm flow is i n/s and the chord length Is 1 iu Nondlmensionalized electric field intensity was varied up to -800. Then the . order of magnitude of the electromag-netic term (E+uxB)xBxHa2/Re becomes 0(10) In maximum. Also, the nondlmensional electric field magnItufie:of -800 corresponds to -1600 v/n under

tffàt3&d1tion

The results are shown in the foilo*iñg

FIg. 4.12 shows the velocity and pressure distributions wIth no MHD forces.

Fig 4.13

shows the similar distributions with Efl-800. It is seen clearly that the jet-like flow occurs at the trailing edge. under the action of large MED forces. In the pressure distribution, negative pressure increases in magnitude near the leading edge. This is more clearly seen in the surface pressure distribution shown in Fig. 4.14. The negative pressure zone in the back side becomes wider, because the MilD force drives the flow in the trailing edge. direction. Where the MED force acts, the discontinuity in pressure is observed. The positive pressure on the face side increases slightly. These tendencies are similar to those in the laminar flow.

The computed result was integrated to get lift and drag. - Fig., 4.15- shows the relation between the -lift 'coefficient añd the imposed electric field intensity. The friction force component In the lift Is very small. The lift increase as the Intensity of the applied electric field increases. The increase in lift becomes. smaller when the intensity of the electric field exceeds -600, whIch is considered to be due to the fact that the MED forces acts mainly to the nonlinear convection term in the governing equations. However, the gain in lift in tur-bulent flows is smaller than that in laminar flows, which implied the difficulty in control-ling turbulent flows using the MHD effect. At Re=l04 In laminar flow, flow separation occurs

under

in

non-MilD

condition, and the separation

disappears with the action of MilD forces.

But.

in

turbulent flows, no flow separation occurs in

non-MEW

condition, therefore the MilD force could

not have such a great effect as in laminar flow.

Fig.

4.16

shows

the

relation between the

drag coefficient and the intensity of the imposed

electric field. As the intensity of the eléctric

field

increases, the frictional component of the

drag increases

due

to the suôtion force toward

the

trailing edge.

On the other hand, the

pres-sure

component

of

the

drag decreases

as the

intensity

of

the

electric field increases, and

therefore,

the

sum of the pressure and friction

components

of the drag is nearly constant,

inde-pendent

of

the

change

in

the

electric field

intensity.

The

same

tendency was observed In

laminar flow case.

The MilD force acts as thrust,

and

the total drag of the wing section decreases

as the electric field Intensity increases, and at

E=-800 the total drag becomes approximately zero.

4.5

Si.ulat Ion

of Flow Control about Turbulent

flow Past a Wigley Huil

So

far,

2-D flows have been considered for

the flow control using the MHD effect.

But flows

of

engineering

Importance are mostly 3-D flows,

which are investigated in the present section.

A

turbulent flow past a Wigley hull was computed at

the Reynolds number Re=lO

Computation

of

high

Reynolds number flows

past a Wigley hull was already made by one of the

authors

(Kodaina19),

where

the Baldwin-Loniax

O-equation

turbulence

model

was

used.

1h

thé

present computation, the same 0-equation model is

used,

for

simplicity, the effect of

electromag-netic forces on turbulence is neglected, and the

electromagnetic

forces are given as body forces.

Then,

It

becomes

possible to use the cóniputer

code

developed

by

Kodaina

with

minor

modifications.

Fig.

4.17

shows

the

grid: used.

The

Cartesian

coordinates

are (x,y,z), which are in

lengthwise,

width,

and

depth

directions,

respectively.

The

body-fitted coordinates are

(E,r1,). which are in lengthwise, glrthwise, and

normal-to-wall directions, respectively.

The

Imposed

magnetic

field

was

given as

follows.

On

the

hull

surface,

a rectângular

circuit

of

electric current was placed, and the

magnetic

field

Indúced by the circuit was

com-puted

using

the

Biot-Savart's

law.

The

contributions

from mirror images were álso

com-puted

to

Impose

symmetry.(see Fig. 4.18)

Fig.

4.19

shows

the

actual location of the electric

circuit.

The

electric field was computed by assuming

that the electric current goes from

=2 to

=l8

along

r'-axis

In a hexa-hedron bordered by

, r1,

or

=constant plane, as shown in Fig. 4.20.

This

electric

current

approximately

corresponds

to

that

obtained when two electrodes aré placed in

hatched

areas

shown in Fig. 4.20.

The electric

circuit

was

placed In that position In order to

avoid velocity defect near the water súrface.

The

_{magnitude of the applied magnetic field}

was Ha=126.4, and the nondimensionalized electric

fléld Intensity

was 800, using the maximûm

mag-netic

field intensity adopted as representative.

This

corresponds

to

the

magnetic field of 0.5

tesla,

and

the

electric field of 100 v/m, when

the ship length Is 4 n and the speed Is 0.25 m/s.

First,

computed

results with no MilD forces

are

shown.

Fig.

4.21 shows the comparison

be-tween

experiments

and

computations on the wake

distribution at

A.P..

The

agreement is good.

Fig.

4.22(a)

and 4.23(a) show comparison of the

computed distribution without or with MRD forces.

It

Is

seen clearly that the flow Is accelerated

due

to

the

MRD

forces

and the boundary layer

becomes

thin.

The wake shown in Fig. 4.22(a) is

somewhat

thicker

than

that shown in Fig. 4.21.

Both

results

were

obtained using the same

com-puter

code

and

grid.

The

difference

comes

possibly

from

the

difference

in the number of

tlmesteps

computed.

Fig.

4.22(b)

and 4.23(b)

show the computed hull-surface pressure

distribu-tion,

where dashed lines show negative pressure.

The

pressure recovery near A.P. Is significantly

improved by the MRD force, because It accelerates

the flow there.

Finally,

the

Input

power correspondIng to

the

present

computation condition is estimated.

Assuming the

ship

length of 4 w, the volume of

the

imposed

electric

field is

approximately

2.76x103

ni3

per

one

side of

the hull.

The

electric field

density

is

400 A per unit area

when E=800.

Then the.input power Is estimated to

be

110.5

W, or 0;147 PS, totaling 0.294 PS with

both

sides

of the hulL

On the other hand, the

effective horse power of the 4 m-long Wigley hull

Is estimated to be 0.124 PS at Re=lO

Therefore,

the

power

required for the present flow control

amounts to 2.37 tImes that of the effective horse

power.

It may be said that the efficiency of the

present flow control Is very low.

5. CONCLUSIONS

In

the

present

study,

the possibility of

using

electromagnetic

forces

for

controlling

flows

and

the

relation

between

applied

electromagnetic

forces

and

ôhanges in the flow

were

_{investIgated by}

simulating the flows with

CFD

techniques.

Followings are the conclusions

obtained.

GovernIng

equations

for

fluid motions

in

electromagnetic

field was derived for fluid with

low electrical coductivity such as sea water.

As

the

induced magnetic

field

can

be

neglected

compared

to the applied magnetic field in such a

fluid,

the

magnetic

field

can

be

treated as

static

for steady-state flows, thus the problem

can be significantly simplified.

Further, in 2-D

flows,

the

electric field can be given aprIori,

and

It

finally results that only the velocity

components

and

pressure are dependent variables

tobe computed as solutions.

The flow past a 2-D wing section was simulated

at

the Reynolds number Re=l04, and the effect of

electromangetic

forces

on

the

flow

field was

investigated.

The

result was that, when a wing

section moves through the electromagnetic field,

the

lift

Increases

as the Intensity of the

ap-plied

electric

field

Increases.

The

rate of

Increase

was

linear at

first, and became

non-linear

as

the intensity of the applied electric

field became large, which is due to the fact that

the effect of the diffusion (viscous) term became

smaller

The

total

drag decreased becaethe

reaction

of

the electromagnetic force w'rked as

thrust.

Consequently,

it

seems

possible

to

increase

lift

using electromagnetic forces even

when the speed of advance cannot be made larger.

In

order

to

simulate flow controls of

tur-bulent flows using electromagnetic forces, the

k-L

_{equations for turbulence with the MilD effects}

included were

derived.

A

simple channel flow

simulation

was

compared with

experiments, and

good agreement was obtained.

Next, as in laminar

flows,

turbulent

flows

past a 2-D wing section

were

simulated

with or without the MMD effects.

The non-MUD results showed that the flow obtained

using the k-s

equations was somewhat different

from

the

flow

obtained using

the

0-equation

turbulence

model.

The differences were that the

negative

pressure

peak

at the leading edge was

lower,

and

the

drag was greater than the

0-equation

result,

due

to

the

fact

that

the

transition

is

not taken into account tri the k-s

equation result.

Inclusion of transition effects

into the k-s equation Is a future task.

Further,

in

the

boundary

condition

for velocities, the

logarithmic

distribution

without

pressure

gradient

effect

Is

used.

Modifications in the

boundary

conditions are another important future

tasks.

Electromagnetic

forces were imposed on the

turbulent

flow

past

a wing section In order to

control

the

flow.

The simulated results showed

that

the

pressure

on the back side of the wing

section

became

lower

by

the action of imposed

electromagnetic

forces,

and the friction stress

Increased

in

the zone where the electromagnetic

forces

were

imposed.

As

the result, the lift

Increased,

and

the

total

drag decreased.

HOwever,

the

gain

in the lift was smaller than

that

In

laminar

flows,

th1ch implies the

dif-ficulty of controlling turbulent flows using the

MHD effects compared to laminar flows.

The numerical, simulation of flow control of

3-D

flows past a WIgley hull was made.

The result

showed

that

the flow is accelerated In the wake

region by the MIlD forces, and a more uniform wake

distribution

was

obtained.

However, the Input

p7wer

_{was very largé compared with the effective}

power

required

for propulsion, which means that

the

_efficiency

of

the

present flow control is

very low.

AcKNOWLEDGNENTS

The present study was conducted as a part of

the work made at the.Surveyànd Research Group on

Ship

Flow

Control,

organized by

the Maritime

Technology

and

Safety

Bureau,

the Ministry of

Transport,

Japan.

The

authors

would

like to

thank

Prof.

H.

Maruo

of

Yokohama

National

University,

who

was

_{the chairman of the group,}

and

all

the

other

members

for valuable

dis-cussions.

The

authors would also like to thank

the

CFD

members

at the Ship Research Institute

for

many

assistances given In the course of the

study.

REFflENCES

-z 4

PhIllips,

0.M.,"The

Prospects

for

MagìietohydrodynamIc

ShIp PropulsIon", Journal of

Ship Research, Vol.

, 'pp.43-51, March, 1962.

' Yamaguchi,

H.

and

Kato,

H.,"Survey

and

Consideration on ElectromagnetIc Propulsion", 9th

:jÇ' March, l986.(in Japanese)

Kitano, M., Iwata, A., and Sail, Y.,"The Basic

Study

of

ElectromagnetIc Thrust

Device

Using

Superconducting Magnet 1,11", Rev. of Kobe UnIv.

of

Mercantile Marine, Vol. 26, pp.219-262, 1978.

(in Japanese)

ImaI,I. and Sakural,A. ,"Magnetohydrodynamlcs",

Iwanaml Shoten, 1959.(in Japançse)

Hughes,

W.F.,

and

Young,

F.J.,"The

Electromagnetodynamlcs of Fluids", John Wiley and

Sons, 1966.

. ,,

Hinatsu,

M.,"k-s

EquatIon

in

MIlD Flow and

Channel

Flow

Computation",.

2nd

NST Symposium,

Institute

for Industrial Science, University of

,

Tokyo, January, 1987'. (in Japanese)

Kitamura,

K.

and Mirata,. M.,"Turbulent Heat

and MOmentum Transfer for Electrically Conducting

Fluid

Flowing

in

Two-DimensIonal Channel Under

Transverse MagnetIc Field", Proc. 6th IHTC Vol.3,

M-18, 1978.

. , ,

Yoshizawa,

A. :"Statlstical

Modeling

of

a

Transport

Equation

for

the Kinetic

Energy

Dissipation Rate",

Physics

of FluIds, Vol.30,

No.3, 1987.

. for example,

RodI, W.,"Turbulence Models and

Their

Applicàtion

in Hydraulics -A State of the

Art Revie*-", June, 1980.

Hinatsú,

M. ,"Numerical

Simulation

of

Tubu1ent..- Flôw around

Wing Section

COntrolled

by-E1eçtrornagnetic

Forces",

Journal

of

the

Syôf NavaL Architects of Jap,Vol.l63

JUne, 1988.

Beam,

R.M.,

and Warming, R.F. "An lùiplicit

Factored

Scheme

for

the

Compressible

Navier-Stokes

Equations", AIAA Journal, Vol. 16, No. 4,

April, 1978.

Kodama,

Y.,"Computatioñ

of

the

Two-dimens tonal

Incompressible

Navier-Stokes

Equations for Flow Past a Circular Cylinder Using

an

Implicit

Factored Method",

Papers

of Ship

Research Institute, Vol. 22, No. 4, Jly, 1985.

Kodaijia,

Y.,"A Method 'to Assure Positiveness

of

k

and

s

in

the

Computation

of

the

k-s

Turbulence

Model",

Journal

of

the

Society of

Naval

Architects

of Japan,

Vol.160, pp.21-27,

1987.

ilinatsu, M. "Study on COñtrol of Flow past a

Wing

SectIon

Using Electromagnetic Forces",

Papers

of Ship ResearCh Institute, Vol.24, No.5,

Séptember,. 1987. (in Jaanese)

''

--'Brouillétte,

E.0

and

Lykoudis,

P.S,

"Magneto-Fluid-Mechanic

Channel

Flow.

I

Experiment",

PhysIcs

of

FluIds,

Vol.10, No.5,

pp.995-1001, May, -1967:

Kodama,

Y.,"Computation of the NS Equations

for

High

Reynolds

Number Flows Past a 2-D Wing

Section

UsIne

the Eddy VIscosity", 46th General

Meetings

of

'Ship

Research Institute, November.,

1985. (in Japanese)

Abbott,

1.11.

and Doenhoff, A.E.,"Theory of

Wing Sections", Dover, 1959.

Chen,

H.C.,

Patel,

V.C.,"Calcuiation

of

Trailing-Edge, Stern

and

Wake Flows by a

Time-Marching

Solution

of

the

Partially-Parabolic,

Equations",IIHR Report. No.285. April, 1985. 19. Kodaina, Y. "Computation of 3-D Incompressible Navier-Stokes Equations for Flow Around a Ship Hull Using an Implicit Factored Method", Proceedings of Osaka Colloquium on Ship Viscous Flow. 1985.

----==-

-(a) Velocity distribution

Fig. 4.3 Computed result of flow around wing section in non-MHD condition (Re=iO4, angle of attack = 5 deg.) Fig. 4.1 Coordinate system for C-grid topology

-

LO

Fig. 4.2 Mesh division for computation of flow around wing section

(angle of attack S deg.)

Fig. 4.3 (b) Vorticity distribution

Fig. 4.3 (c) Pressure distribution

I.e

Fig. 4.4 Distribution of magnetic field around wing section

Fig. 4.5 Lorentz force distribution around wing section

0.2 0.4 0.6 0.0 1.0 Velocity distribution

u'

'T

-2

(e) Pressure distribution

Fig. 4.6 Computed result of flow around wing section (Ha=12.64, E=-200)

(Re=l04, angle of attack = 5 deg.)

!00012 01.12.64 0.2 0.10

_{r.s)

-I

J .4 0.6 0.0 1 0

"e

(a) case In E=O _{(b) case in E=-200} '.0

TO

Vorticity distribution

Fig. 4.7 Comparison of computed pressure distribution

(Re=104, angle of attack = 5 deg.)

Lift CL 0.4 0.2

00.

0

O -50 -100 -150 -200 E

Fig. 4.8 Relation between intensity of applied electric field and lift coefficient of wing

(Re=l04, angle of attack = 5 deg.)

D Ç1/2)pUL o

o

I I I CDP + CDf 0.04 CDf(frictiofl)

Ö

V

CD (pressure) I I -50 i-100 -150 -20O E t (e. ZOce) -0.04

Fig. 4.9 Relation between intensity of applied electric field and drag coefficient of wing

(Re=104, angle of attack 5 deg.) C _{+ CLf}

V C(pressure)

O CLZ(frictioro)

O Cmf

(electro-onagnetic force)

o

D

D

11 0.02 -0.02

-2 0

cp

o

.

.0

0.5 e...

e

1.0.

Fig. 4.10 Comparison of computed result of velocity profile of Hartmann flow with experimental data

NACAOO12 o

Re].06

O

Rodaina's cal.

p

_{(Baldvin-Loinax's}

-1.0 -

0-equation model)

0.5

-'Q

'Q

.

'e

. o

x,c

--

__-__

_____s__

(a) Velocity distributioñ

i s '..

\

/ ---_-;.;--

'-S

\

'S. 'S

'

s -s I o

'

_.:(::

.'..\\

',

\

I; (b) Pressure distribution / = 10 deg.) :

present cal.

(k-c model)

Fig 4.12

wing section (k-E model, non-MILDComputed result of flow around condition, Re=106, angle of attack

0 -- - - 1-. 0

Fig. 4.11 Pressure distribution along wing surface (k-e model, non-MHD condi-tion, Re106, angle of attack = 5 deg.)

- 1.o,

--- ---

--

--.-

____________-_

-t

cp -4.0--1.0

(a) Velocity distribution

s,-I,,

\

0.2 _,'

-- --.. -0.6 s

i'

*'.'-"-.

-s'

\

-z-x/c OOACA0OI2 Re - 106 k-c moe1 ncn-0HD condition n 10 -2.0

(a) case in non-MHD condition Fig. 4.14 Comparison of computed pressure

distribution

(Re=106, angle of attack = 10 deg.) 1.0 o -4.0

t

_C C *10 1.0 0 0.5 - 1.0 lACRO 012

M 106

k- odel o- 10' Ra- 126.4 E --800 'J

Fig. 4.14 (b) case in Ha=126.4, E=-800

Lift

CL (1/2)pU2L

-200 -400 -600 -800

Fig. 4.15 Relation between Intensity of applied electric field and lift coefficient of wing

(Re=106, angle of attack = 10 dèg.)

13 (b) Pressure distribution

Fig. 4.13 Computed result of flow around wing section (k-E model, Ha=126.4 E=-800, Re=106, angle of attack

= 10 deg.)

\.0.2

I

-3.0

Drag CD

_(1I2)ptJL

0.10

0.0

8h-C + C 0.06

0.04

-0 .-02-0

0.02

0.04

0.06

--0.08

CD (Pressure)

D

C(°

Upstream boundary -200 -400 -600 -800 Water plane E

II

Fig. 4.16 Relation between intensity of applied electric field and drag coefficient of wing

(Re=106, angle of attack = 10 deg.)

A.P

F.P. (x-z) symmetry plane

Fig. 4.17 Grid division around Wigley ship form

AP.

Fig. 4.18 Arrangement of rectangular circuit for computation of magnetic field

A.?.

Fig. 4.20 Illustration of domain of applied electric field

rA

water surface pww

A P.

LA

bottom

Fig. 4.19 Actual location of rectangular circuit

--0.8

0.9

0.8

2y/B

-- - - Measured

Computed

0.7

r-0.0

- 1.0

z/D

Fig. 4.21 Comparison between experiments and computations on the wake at A.P. (Wigley model, Re=106)

Fig. 4.22 Computed result of flow in non-MHD condition (Re=106, Wigley model) (a) Computed wake distribution at A.P.

Fig. 4.23 Computed result of flow (Ha=126.4, E=800, Re=106, Wigley model) Computed wake distribution at A.P.

15 water surface FLOW

A.P. bottom 7-P.

Fig. 4.22 (b) Computed hull surface pressure distribution

wMer sIIrfeA FLOW

V(

r'

A.P. bottom F.P.

Computed hull surface pressure distribution