Numerical simulation of flow past a circular cylinder under a magnetic field

Numerical simulation of flow past a circular cylinder under a magnetic field

Computers Fluids Vol. 21, No. 2, pp. 177-184, 1992 Printed in Great Britain. All rights reserved 0045-7930/92 $5.00+0.00 Copyright © 1992 Pergamon Pr...

340KB Sizes 1 Downloads 136 Views

Computers Fluids Vol. 21, No. 2, pp. 177-184, 1992 Printed in Great Britain. All rights reserved

0045-7930/92 $5.00+0.00 Copyright © 1992 Pergamon Press pie

NUMERICAL SIMULATION CIRCULAR CYLINDER UNDER

OF FLOW PAST A A MAGNETIC FIELDt

YOSHIHIRO MOCHIMARU Department of Mechanical Engineering, Tokyo Institute of Technology, Tokyo 152, Japan

(Received 24 April 1991; received for publication 8 November 1991) Abstract--Steady-state laminar flow fields around a circular cylinder placed normally t o a uniform flow under a uniform magnetic field are analyzed. The fluid is assumed to be incompressible and to have moderate magnetic diffusivity. A hydrodynamic assumption regarding the magnetic field is introduced. A Fourier spectral method is used to decompose the vorticity transport equation. Using a semi-implicit method, the resulting differential equations can be integrated easily by a time marching method to obtain a steady-state solution. It is found that the flow characteristics vary remarkably according to the Hartmann number, the Reynolds number and the direction of an applied magnetic flux density vector, although the dependence on the magnetic Prandtl number is small as long as it is < 1, as in liquid metals. NOMENCLATURE a= B= CD = CL= E= g= Ha = J = p = r= Re = Re D = R H= R, = t= U~ = V=

Radius of the cylinder Magnetic flux density vector Drag coefficient = drag per unit depth/(pU~a) Lift coefficient = lift per unit depth/(p U2~a) Electric field vector Gravitational acceleration vector Hartmann number based on a diameter = 2 ~ Conduction current density vector Pressure Cylindrical polar coordinate Reynolds number = paU~/~ Reynolds number based on a diameter = 2 Re Magnetic pressure number = IB~ [2/(jtepU~) Magnetic Reynolds number = U~ aalac Time Free stream velocity Velocity vector

(x,y,z) = Cartesian coordinate system, the x-axis being downward = Dimensionless vorticity ~ = Vorticity at r = I O=Amplitude in a cylindrical polar coordinate system /~ = Viscosity o f fluid ~, = Magnetic permeability ~=lnr p = Density of fluid a = Electric conductivity of fluid ~b = Direction o f B~ measured counterclockwise from the positive direction of the x-axis ~b = Dimensionless stream function A = Laplacian operator

Subscript oo = At the free stream

1. I N T R O D U C T I O N

A practical situation concerning magnetohydrodynamic behavior in liquids was encountered, e.g. in liquid-metal MHD [1], when liquid metals were being investigated as disordered condensed matter [2]. In this paper, a two-dimensional steady flow field around a circular cylinder placed normally to a uniform liquid metal flow is analyzed numerically. The external magnetic field applied far from the cylinder is assumed to be uniform and perpendicular to the axis of the cylinder, and it is also assumed that the externally applied electric field is such that no pressure gradient (except a contribution from the gravitational field) exists far away from the cylinder. 2.

ANALYSIS

2.1. Basic equations Under the usual magnetohydrodynamic approximations to the constitutive equations, the equation of motion can be expressed as D p ~ V = - V p +pg +taAV + J x B,

(1)

tPresented at the 2nd Japan-Soviet Union Joint Symp. on Computational Fluid Dynamics, Tsukuba, Japan, 27-31 August 1990. 177

178

YOSHIHIRO MOCHIMARU

where the fluid is assumed to be incompressible and to have no electric charge density, and the fluid viscosity is assumed to be constant. In two-dimensional flow, taking the curl of equation (1) gives D p ~ (V x V) = ~a(V x V) + ( a . v ) a - (J.V)B,

(2)

where the Maxwell equations V.B=O

(3)

V. J = 0

(4)

J = a(E + V x B).

(5)

and

are used. Ohm's law is

From the assumption regarding the far field, J~ = 0,

(6)

E~ = - V : ¢ x Boo.

(7)

which leads to Thus, the hydrodynamic approximations B ~ B~

(8)

J ~, a (V - V~ ) x B~

(9)

and

are introduced in the equation of vorticity transport, so that equation (2) can be replaced by D

P b-~ (v x v ) = ~ A(V × V) + ~ (B~. V) [(V - V~ ) x a ~ ].

In the Uo~, a z-axis of the

(l 0)

following, velocity, length and time are assumed to be made dimensionless with respect to and a/U~, respectively. Let (r, 0, z) be a cylindrical polar coordinate system such that the is coincident with the axis of the cylinder and that the direction 0 = 0 corresponds to that uniform flow. Then the nonzero component of equation (10) can be expressed as

8~ 8~b8( O g ~ 1 ( 0 2 ~2) r2~-t 4 000~ 0~00=R--~ ~-~5+~-~ ( + R n a o x -rZ( sin2(0 - ~ b ) + c o s 2 ( 0 - ~ ) ~ - ~

+ sin2(0 -~b)

where q~ is the direction of the applied magnetic flux density vector field, ~0 is the two-dimensional stream function and ~ = Inr. The nonzero component of vorticity, (, is related to ¢ as {02 02 ) r2( = -\~-~-5 + ~--0-7,,1~k. (12)

2.2. Fluidflow behaviorfar from the cylinder As the distance r increases, the fluid flow field and the electromagnetic field tend to a uniform flow and an applied uniform electromagnetic vector field, respectively, Thus, at a location far from the cylinder, the equation of motion, the equation of continuity, the Maxwell equations and Ohm's law can be linearized with respect to the quantities deviated from those in the uniform field. After manipulation, we get 0 2 A O A{RH( cOs~b--~-061+ sin d ~ y ) - ( ~ x R-e)(~x ~-o)} ( 0 - y ) : 0 " (13)

Flow past a circular cylinderunder a magnetic field

179

From the wake character, if R." sin ~b ~ 0, then A[A - l(Re + R~) ~x - ~ ( c o s x A-½(Re+R~)

4~d-~ + sin ~b~ y ) ] +

~

cos~b~x + sin ~ ~y)](~O-y) ~0;

(14)

and if sin qb = 0, then t~ - ~I[(R e - R~)2+ 4 ReR~R.],/2 ~x } A{ A-½ (Re+ R~)~x x { a - ½ ( R e + R o ) o x~--0+~,[(Re_R~)2+4ReR,Rn],/2~x}(~b - Y) = 0. (15) The asymptotic solution of equation (14) is of the form d/(r~,O)-y = ,.= ~x/r~ e x p[_/ - - -4 r~ (0 -/3~)2 -C°lnr°~+Cl

(16)

1

and ~(roo,0)=g-~. 5 a,.Km 1 2rg m = l

r~(O-flm) 2 exp -

r~(O-fl,.) 2

(17)

where K. exp(ifl~)~ = Re + R~ + x/ReR~Rn(cOs ~b + i sin ~b), Ks exp(ifl2)J 2 The asymptotic solution of equation (15) is of the form ~O(r~, O)--Y= ,.=1 ~ Sm{ e r f ~~- ~ ~ ( 0 - - f l m ) l

(K,, g2 > 0).

0 -~fl,.}

(18)

(19)

and

(20)

~(r~, 0) = 2x//_~ .,=, where

KI

} Re + R~

/(2 exp(ifl2) =

2

+ [~(Re - R~)5 + ReR~RH]I/2,

(K2 > 0, fit = 0).

(21)

2.3. Relation between flow characteristics It is assumed that the cylinder possesses high magnetic permeability, so that the contributions from the Maxwell stresses to the total drag, total lift and total angular momentum on the cylinder surface become zero. Thus, considering the balance of the linear and angular momentum, we obtain for RH" sin q~ ~ 0: 2

CD= --2x/~RHRam~=l= ~mm am [sin/3m -- cos fl~ sin(2fl., -- 2~b)] - ~ RMR~ sin2 ~b

(22)

and 2

CL = --2nCo + 2x/~RnR ~ ,.=~ ~ am [cos fl~ + sin fl~ sin(Eflm - 2~b)] + ~RHR~ sin ~b cos ~b, (23) with 2 am

m~l ~/K,.

sin2(t~ - tim) = 0;

(24)

180

YOSHIHIRO MOCHIMARU

and for sin ~b = O, CD

=

S~ - 2 RHR ~

- -

m=l

-l

(25a)

gm

and (25b)

C L ~0.

3. N U M E R I C A L S O L U T I O N P R O C E D U R E 3.1. Primary variables

The stream function 0 and vorticity ( can be expanded into the following Fourier series of 0: 0 = ~ ~bs.(r, t) sin nO + ~ ~O~.(r,t) cos nO n=l

(26)

n~0

and oc = ~ (~(r, t) sinnO + L ~ ( r , t)cosnO. n= I

(27)

n=0

Then, using the addition formulas of the trigonometric functions, equations (11) and (12) can be separated into each Fourier component of 0, which constitutes a system of differential equations with respect to 0~ s, 0~ s, (,~ s and (~ s in r (or ~) and t in the interior region (except on the boundary r = 1 and the virtual boundary r = r~) (a kind of Fourier spectral method). As an example, the zeroth cosine components of equations (11) and (12) are shown below: r2d

1~

~

n=l

__½ R H R , [ _ r2(e0

. I'r 2 d2 d + sm 2 4 ~ - (s2 + ~-~ 0s2 + ~-~ Os2 - 20s2)

+ cos 2q~ ~- (¢2 + 0-~ 0~2 + ~-~ ~b~2- 20¢2

= ~

~ 5 (~o

(28)

and r 2(~0 = -- (d 2/d~ 2)0~0,

(29)

respectively. 3.2. Boundary conditions at r = 1

At the surface of the cylinder, no-slip flow is specified: 0 ( ¢ = 0) = 0,

o0

~

(¢ = 0) = 0.

(30)

The second condition can be replaced by 2 ~ 0 ( ~ = h) + ~(~ = h) = 0 ,

(31)

where h is the coordinate at the grid adjacent to the cylinder surface. These conditions can be separated to give 0.,(¢ = 0) = 0,

2 ~ ~k.,(~ = h) + (.,(~ = h) = 0,

(32)

where .n stands for cn (n/> 0) or sn (n t> 1). In particular, from the continuity of pressure, it follows that 8 d--~(~o(~ = 0) = 0.

(33)

Flow past a circular cylinder under a magnetic field

181

-2

Fig. 1. Streamlinesat ReD = 2, Ha = 0.89, 4~= 0. ----, indicates the direction of the uniform flow.

Fig. 2. Streamlines at Reo = 2, Ha ---2, q~= 0.

3.3. Virtual boundary conditions at r = r ~ ( ~ 1)

The coefficients Co and

C L are

given by

cD - - R e ~ - ~ s l - - ~ s l ) ~=0 - ~

(34)

sin2~RoR.

and

L = Ree ~ -- ~-~ (cl "q- ~cl,/~=0 + n sin q~ cos ~bR, RH,

(35)

which specify the factors ares or S~ (S: = 0 is assumed). Thus, equations (16) and (17), or equations (19) and (20), can be expanded into Fourier series, at least numerically. 3.4. Numerical integration scheme

By truncating the series (26) and (27) up to a certain order, and discretizing the resulting system of equations, e.g. equation (28), in space and time by a finite difference method incorporated with boundary conditions, the system of equations can be integrated with respect to time by a semi-implicit method to give a steady-state solution, where a potential flow field with no magnetic field as initial values is assumed, being superimposed with an approximate developing boundarylayer flow field. 4. N U M E R I C A L

RESULTS

4.1. Flow patterns

Streamlines for low Reynolds numbers ReD = 2 are shown in Figs 1 and 2, which show that the extension of the wake width behind the cylinder is smaller in the latter case of the higher Ha. Streamlines at Reo = 20 and q~ = n/4 are shown in Fig. 3; in this case the flow patterns are

o 1 ~

-0.5

-1

_.-----3 Fig. 3. Streamlines at ReD = 20, Ha = 8.9, ~ = n/4.

Fig. 4. Streamlines at

R e D = 100,

Ha = 6.32, ~b= =/20.

182

Y O S H I H I R O MOCHIMARU

-3 2 1 0.5

/

0

.----------1 ---2 - - - 3 Fig. 5. Streamlines at Re D = 100, Ha = 14.1, q~ = n/20.

Fig. 6. Streamlines at ReD = 200, Ha = 14.1, ~b = 0.

3 ~

2

2

o

O

t

-

-

2

.3 Fig. 7. Streamlines at Re o = 200, Ha = 14.1, ~b = n/4.

Fig. 8. Streamlines at Re D = 200, Ha = 14.1, ~b = n/2.

asymmetrical, which is generally true if RH" sin 2~ # 0. Streamlines at small angles of q~ are shown in Figs 4 and 5. Two standing circulating zones with a vortex are found only in the case of the lower Ha flow (Fig. 4). Streamlines at ReD = 200 and Ha = 14.1 are shown in Figs 6-8. A pair of standing vortices are found in the case of ~b = 0. 4.2. Drag coefficient T h e r e l a t i o n b e t w e e n the d r a g coefficient a n d H a for q5 = 0, ~b = rt/4 a n d ~b = n / 2 is s h o w n in F i g s 9 - 1 1 , r e s p e c t i v e l y . I n this i n v e s t i g a t i o n t h e d r a g coefficients o b t a i n e d are g r e a t e r t h a n t h o s e c o r r e s p o n d i n g to H a = 0 ( p u r e l y v i s c o u s flow), a l t h o u g h n o t s h o w n . I n a d d i t i o n , the d r a g coefficients a r e a l m o s t i n d e p e n d e n t o f the m a g n e t i c P r a n d t l n u m b e r ( R ~ / R e ) , as l o n g as it is <1.

5

A

4--

4

&

Q

Q

3!

3

2--

2 -0

O

O

O •

i

i 0

o

(~ I0

9 Ha

I

1 20

Fig. 9. C D VS Ha at $ = 0. &, Re D = 20; O, ReD = 100; • , R e D ----- 200.

0

10

Ha

20

Fig. 10. CD vs Ha at ~ = n/4. A, ReD = 20; O , Re D = 100;

• , ReD = 200.

Flow past a circular cylinder under a magnetic field

183

IC

4

O 0

3 •

2-0

0

0





0

I 0

o.

i0



[

10

20

Ha

Fig. 11. CD vs Ha at ~ = 1t/2. A , ReD = 20; O, ReD = 100;

~, ReD----200.

20

Ha

Fig. 12. CL VSHa at t~ = 1t/4. A, ReD= 20; O, ReD = 100; @, Rev = 200.

4.3. Lift coe~cient If sin 2~b = 0, the streamlines are symmetric and thus no lift is produced, i.¢. CL = 0. The relation between the lift coefficient and H a for ~b = 7r/4 and ~b = lt/20 is shown in Figs 12 and 13, respectively.

4.4. Vorticity distribution Vorticity distributions along the surface o f the cylinder at ReD = 200 and H a = 14.1 are shown in Fig. 14. 5. C O N C L U S I O N F l o w fields a r o u n d a circular cylinder under a magnetic field can be analyzed numerically, using a Fourier spectral method. It is f o u n d that the wake region just behind the cylinder is suppressed

30

20

V--%.

10

O 0

-i0

-

O

~. / /

0.1 - -

-20

\

--A

I 10 Fig. 13.

C L v$

20

Ha at ~ = ~t/20. &, Re D = 20; O, ReD ----100.

0

0

Fig. 14. Vorticity distribution along the surface of the cylinder at Revffi200, Ha ffi 14.1. - - . - - , ~ •0; ~b = ~/4; - - - ,

CAF 2112--D

/

I

-30

Ha

//

¢, = ~/2.

7r

184

YOSHIHIRO MOCHIMARU

by a m a g n e t o h y d r o d y n a m i c effect, a n d t h a t the d r a g is increased c o m p a r e d with that u n d e r no m a g n e t i c field. REFERENCES 1. G. F. Berry, E. S. Pierson, M. Petrick and S. Sukoriansky, The application of liquid-metal MHD to renewable energy sources. In Proc. 25th Syrup. on Engineering Aspects of Magnetohydrodynamics (Edited by R. Kessler), ~thesda, MD, Paper 6.2.1 (1987). 2. G. Fritsch and E. Lfischer, Macroscopic and microscopic properties of liquid metals and alloys: the experimental situation. In Proc. NATO Advanced Study Institute on Liquid and Amorphous Metals (Edited by E. Lfischer and H. Coufal), pp. 3-61. Sijthoff & Noordhoff, Zwiesel, Germany (1980).