Dynamic interaction for a viscous incompressible flow past a taut cable at moderate Reynolds numbers

Journal of Wind Engineering and Industrial Aerodynamics, 36 (1990) 361-370 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

361

DYNAMIC INTERACTION FOR A VISCOUS INCOMPRESSIBLE FLOW PAST A TAUT CABLE AT MODERATE REYNOLDS NUMBERS Peter Seellnger*

and

Jorg Hauer**

ABSTRACT The dynamics of a taut string in a steady viscous fluid flow is treated. The governing equations of motion are formulated in the sense of continuum mechanics. For simplification, an incompressible fluid and a plane vector field of motion are considered. In order to obtain an approximate solution, a decomposition of the state variables into a basic motion and superposed small disturbances is assumed. First, the mean flow past the cable is studied, where the small static deflection of the string is neglected. Then, the linear stability equations are solved. A critical flow velocity is calculated, at which the originally laminar fluid flow and the equilibrium position of the string become unstable. The influence of the flexibility of the string is discussed.

INTRODUCTION The vibration of overhead cables due to cross-wind, for example, represents an interaction problem of cylindrical structures in a fluid flow, which exhibits many interesting dynamic phenomena. The state of the art in this field has been described in the book by Naudascher (1974) and papers by Sarpkaya (1979) and Mahrenholtz (1986). The study of this and related literature shows that till now, due to the incomplete knowledge of the flow field about a stationary body and also to fundamental problems associated with the coupling of the structural oscillations and the flow field, a satisfying solution of the dynamic interaction for a viscous flow past a taut cable has not been obtained. The o b j e c t i v e of this contribution is a first step in this direction. The s t a r t i n g point is a formulation of the equations of motion in the sense of continuum mechanics. In particular, the

*Dr.-Ing., BASF AG, D-6700 hudwigshafen, **Professor Dr.-Ing., Institut f~r Technische Mechanlk, Unlversitat Karlsruhe, Kaiserstrasse 12, D-7500 Karlsruhe i, West Germany

016%6105/90/$03.50

© 199(N--EhevierScience Publishers B.V.

362

Navier-Stokes equations for the fluid and the one-dimensional wave equation for the string with corresponding boundary and transission conditions are the fundamental relations. In the present paper, an incompressible fluid and a plane vector field of motion at moderate Reynolds numbers are considered. Attention is focussed on the stability behaviour of the originally laminar fluid flow and the equilibrium position of the string. The resulting boundary value problem for superposed small disturbances determines a critical Reynolds number, at which the mentioned basic motion becomes unstable. A stability diagram for different stiffness parameters of the string is presented.

PHYSICAL MODELAND NON-LINEAR BOUNDARY VALUE PROBLEM The model of the dynamic system shown in Fig. 1 is a long f l e x i b l e string in a cross-flow. At both ends, the taut string (stretched by a constant tension force S) is fixed and the distance I between the supports is so large that no boundary influences disturb the flow f i e l d between them. The c i r c u l a r cross-section has the diameter 2a. The constant mass d i s t r i b u t i o n is u and damping influences of the cable are neglected. The f l u i d , with density p, is assumed to be viscous (viscosity v) and incompressible. The

-

~ ( r , ~ E)/

-

-

Y

L.~ ~

Fig. i.

I ~

-,-

:,

Dynamic System

cross-flow velocity and the corresponding pressure at a sufficiently far distance of the string are v~ and ~, respectively. As an appropriate coordinate system, a cylindrical r,~,z-reference frame with the unit base vectors e,e~,ez is assumed. For simplification, the static deformation of {he string in the direction of the steady flow will be neglected, as well the change of the position of the boundary (at H = a) due to the moving string. Additionally, a plane oscillation ~(~,[) of the cable perpendicular to v~ and a plane velocity field ~r(F,~,~,[), V~(~,~,~,~), which depend in the disturbed case on the z±coordinate, are assumed. Define the nondimensional quantities

363

r=--, a

v_.~,

t=

Vr =~-xV~

p

=

=

--

,

A

-

1

f

S

pv 2.

j

'r

pv ~.

,

pl - pa 3

S W

,

F

V~ = V-~ V~

e

z |

z

a

~

o a a v 2. '

(1)

and the Reynolds number

Re

2

=

v~ a -v

(2)

Then the Navier-Stokes equations are

av~ 8t

~ +

Vr

Uav~ +

8r

r

-

a~

~

ap

+

r

2 Fay/~_~ lav_u = r-e L ar 2

av~ 8t

~-~ +

Vr

8r

+ r -

r

=

Re

L

ar ~

1 a'v v,v~

+

8~

2 Fa_i ~

r-

Or

.

v=

2 %1,

r2

r2

J

1 Op r a~

i 8~v~ U +

O~

• + ....

r

I~ +

=

8r + .r 2 . 6~ z.

!m av~ +

-8r-

-r -z

O~ 2

-

r~

2 av_u] r-{

+

a~ ]

(3)

and the continuity equation is 8(rVr) + ~ = O. 8r 8~

(4)

Assuming the above mentioned simplifications, the vibrations of the string can be described by the linear wave equation 2~

2~

a2W aZWT - I p(r= l , ~ , z , t ) s i n ~ d ~ + ; T ( r = l , ~ , z , t ) c o s ~ d ~ = 0 (5) A a--~TB a-~0

0

in which the last terms represent the interaction with the surrounding fluid due to the pressure p and the shear stress r at the surface of the string r = i. The pressure p(r=l,~,z,t) is obtained by an integration in the form

p(r=l,~,z,t)

= p® +

~rr l~:°dr + ~ i ap Iap 0

and, for the shear stress,

lr:id~

(6)

364

r=~e e

r ~ rr

(7)

+ -r a~

is valid. The corresponding boundary and transition conditions read v~(®,~,z,t) = sin~,

v r (~,~,z,t) = - cos~, w(O,t) = O,

w(1,t) = 0

(8)

and Vr(l,~,z,t) = wt(z,t)sin ~,

v~(l,~,z,t) = wt(z,t)cos~,

(9)

respectively. Define the stream function • by 1 8~ vr

r a~

8~ ,

v~ -

(10)

ar

Then the continuity equation (4) is satisfied. Substituting the relations (i0) into the Navier-Stokes equations and eliminating the pressure p leads to

AA~ + 2r

ar

ar

a~

r

at J

where A denotes the Laplacian operator. Using the relations between ap/Sr, ap/a~, r and ~, the equation of motion (5) for the string and the boundary and transition conditions (8), (9) can also be expressed by stream function quantities. Together with the field equation (ii), they describe the considered dynamic system as shown in detail by Seellnger (1987).

ANALYSIS Small disturbances &~(r,~,z,t), aw(z,t) about the stationary state ~o(r,~), wo(z) ~ 0 will be considered. For this purpose, a solution of the boundary value problem (5), (8) and (ii), is assumed in the form 9(r,~,z,t)

= 9o(r,~) + a~(r,m,z,t),

w ( z , t ) = 0 + AW(Z,t), where the disturbances w i l l be s p e c i f i e d as

(12)

365

AF(r,~,z,t)

- g(r,~)sin n z e ~t.

hw(z,t) = a o s i n N z

e xt .

(13)

Relations (13) are now substituted into the mentioned value problem. Expanding To(r,~), g(r,~) into a series

boundary

• o(r,~) = fi(r)sin ~ + f2(r)sin 2~ + ..... g(r,~)

- go (r) + gi( r)cos 9 + .... + hi(r)sin ~ + ....

yields, in connection with the classical Galerkin method neglecting the h-terms, a boundary value problem

fl + -r f* - r-~fi + r-3 fl - --r 4 fi + ~r r 2 f2 + r-{ f2

r 2 fi

fz

+

+

r

fi

r'

+ f~

r2

f~ + r f~

r2

r f~ - --r 2 f2 + --r, fz + ~r

fz

- 0

=o ,

1 fl r

0,

- i,

fl (®) " i.

1 r

,

r~®

fz r ~ i

(15a)

'

(15b)

f2 (®) " 0.

t

fi(1) - 0,

f, + -r fl

fl f~ + r- fl - r-f fl +

+--f,-~f, f ~ + r f ~ - - -r 2f ~ r s I

i

fi(1) - 0

and first

fl f~ + r f2 -

. ]. 2,fz . [f[ , ., ]. fl[ .,fz + - fl

_

(14)

f2(1) = 0 '

i

f2(1) - 0,

= d dr

'

(16)

determining the functions fl (r), f2 (r). A more detai led formulation of these governing equations describing the steady basic flow can be found in a paper by Underwood (1969). Next,

subtracting

the mean-flow quantities,

.... 2 ,,, 1 . 1 , Re [ go + r g o - r ~ go + r' go + ~r

[-~i

1

2r

,,

1

,

,,

1

,

f~ g* + 2 f* g~ + 2 7

II1 =

,,,

3

,,

3

,

., 1 ,, , fl g i - ~ fl g i + 1

.,

1

f~ g* + ~ f~ g~ + ~

2r 2 fl gl - ~ [+ r g o + go .... 2

one obtains

3

,,

f~ g~ + (17a)

0, Re r

1

,,,

g' + ~ g~- V g' + V g~ - nr g~+ 7r [-~ f' g~-

366 . . . . . .

fi go

-

f2 gt

-

,,

1 -

1 ~ - i,

fa gi

~r

,

- - r- -

go

1 ~

+

,

,,

f2 gi

1

,

,

~r f2 gl

+

2 . . . . . . . 1 ,, 1 ,, 2 , + --r ~ fa gl + f l g o + fa gl + r- fl go + ~ f2 gl + r-f fa gl fa

+ X

gl + r

= O,

(17b)

.... 2 ,,, 3 ,, 3 , 3 Re r 1 . . . . . . hi + r- hi - --r2h i + ~-~h± - --r 4 hi + ~r [ 2 f2 hi + f~ hi + 1 ,, i f,2 hl,, + i , , _ -2 f'~ hi _ f2 hi,,,_ _i fz hl,, + 2r f2 hi - 2 ~r f2 hi r2 r r a fa hi + r-f fa hi + X

rh i - h i + - h i r

,

]

712A + X2B a° - ~ee g i ( 1 ) - g (1) + g , ( 1 ) .

.

2 go ( I )

+ g~ ( i )

.i

+ gl ( i ) -

2

X

-

[_g o ( l )

go ( ' ) = O,

+

gi(1)

-

- 2 go (1) + ]

2 go ( i )

2

=0,

-

2 g'i(1)

,,

i(1)

f1(1)

-

~

+ 2 gi(1)

,, ]}

fz(1)

gi(l

-

= O,

(18a)

gl ( ' ) = O,

r~

g'o(1) : O,

"

+

i

+

r1 gl

'

(17C)

= O,

'

g i ( 1 ) - xa o = 0 ,

g'i(1) - xa o = 0

(18b)

p

rl- hl r ~ - = 0 ' Using the eliminated, p

go ( ' ) = O,

h'i ( - )

= O,

hi(1) = O,

(18c)

hi(l) =.0

relations (18b) in (18a), the variable a o can be and instead of (18a,b) the modified boundary conditions

I_ t gl

gl (') = O,

= O, r ~ -

p

go(l) = O, - + X

XB g i ( 1 )

-

~ee g 1 ( 1 )

P

-

2 X gi(1)

+

~ee

.

+ 2 gi(1)

-

2

go

(i)

+

gl (1)

-

2 go(l)

-

.

f~(1)

-

gi(O) f 2 ( 1 )

= O,

i

g i ( 1 ) - g i ( 1 ) = O,

supplement the equations that the two eigenvalue are decoupled.

(19)

of motion (17). R e m a r k a b l e problems (17a,b), (19) and

is the fact (17c), (18c)

367

To approximately calculate the basic flow, procedure (see Fletcher, 1984)

a generalized

Galerkin

N Ua(r)

:

Uo(r)

+ Zai4) i(r), i=$

¢o

1

is applied with the governing differential operator L of the differential equations (15), the shape functions @i, i = 1,2 .... N and an additional weight function f(r), here selected as 1 f(r) = -

(21)

r

to stronger weigh the neighbourhood of the cylindrical 1-term Ritz series

string. Now,

f~(r) = r - - + -- + ai~ 1 - - + r r2 r f2(r)

= a2~

1 -

r

(22)

+ --

r 2

satisfying all boundary conditions (16) results in an inhomogeneous non-linear equations Re

are introduced. This system of algebraic

Re

K3 a11 + - ~ - K 2 a21 + ~ - K 4 a , , a21 = KI, Ka a21 +

z = - Re Re K~ a ~ + Re K7 all

Ks

(23)

for ai~, ai2, where the constants K i till K 7 will not be specified here. A numerical solution of (23) leads to the governing ai~ and a2~ values so that the corresponding stream functions Fo(r,?) can be determined. Fig. 2 shows the results. Due to the low number of shape functions, the solution is inaccurate but qualitatively utilizable. Obviously, the results for low Reynolds numbers are ambiguous (see Fig. 2a) and the streamlines (see Fig. 2b) are only plotted for a11-, ai2-values corresponding with the solid line solution branch. To solve the stability equations (17a,b), (19) and (17c), the Galerkin procedure (20) is applied again. Appropriate comparison functions are now

(18c), 1-term

368

a12

all

\ ss.o

~ - , . ~

\,.

-..... e.o

..... i ...... '............ - ....... -4.o

-llll.O

a.o

ss.o

14.o

~.o

~.o

a.o

le.o

=4.o

z.o

Re

Fig. 2a= Steady Flow Past the Cable, a,~, azl versus Re

3.3-

2.2-

1.t-

J

0.0 -4.00

-2.40

-.80

.BO

2.40

4.00

Re=l. 6

3.3.

2.2"

i.i-

0.0 -,4.00

-2.40

-. 80

.80

2.40

d.

Re=6.4

O0

369

3.3-

2.2.

1.1¸

0.0 -4.00

-2.40

-.80

.80

2.40

4.oo Re=lO.O

Fig. 2b: Steady Flow Past the Cable, Streamlines #o = const, for selected Re

go (r)

=

goo

i

-

r

+

'

gi(r) . g,o[l _ 3_~] + [a g,o + b goo] [_ir 3r2 2 ] ' (24)

hi(r) " hi°[1 - r-2+ r~ ] and the linear, homogeneous equations for goo, g,o, hlo read 2 Cz)goo + [Cs + C4 al,]glo = O, [Ci x + ~ee + C 6 ai,

goo +

7 X + Re

io~ + ~ee Cil + Ciz a,2 hlo = O.

2

o

=0,

{25a) {25b)

where the properties a(A,B,~,Re,all,ai2), b(A,B,x,Re,ai,,ai2) and the constants C i till Ci2 will not given here as well. Non-trivial solutions goo, ~ , ) hlo can be obtained if the determinants of the equations vanishes. The numerical solution of this so-called characteristic equations yields an approximation for the lowest eigenvalue )~minas a function of A,B and Re. The evaluation shows, that for I/A ~ 0 (rigid cylinder) the elgenvalue corresponding with the steady state solution of Fig. 2b

370

is negative, i.e., the basic solution 9oi, Wo~ 0 (0 0 (flexible cable), the amount of the governing negative eigenvalue branch decreases, but for a realistic flecibility the real part remains negative.

CONCLUSIONS The stability behaviour of a taut cable in a steady viscous cross flow at moderate Reynolds numbers has been studied. Approximate methods based on direct variational approaches have been utilized to solve the governing boundary value problem originally formulated in the sense of continuum mechanics. A critical flow velocity has been calculated. The flexibility of the cable qualitatively influences the eigenvalues of the stability equations and increases the sensitivity to disturbances. However, a change of the critical velocity could not be found. Discussing the problem if the mentioned neglects ate included, modified results can be expected. While a nonzero value w o insignificantly changes the calculation expense, the change of the position of the boundary (at ~=a) due to the moving string significantly complicates the calculation procedure, because much more complicated stability equations appear. Results will be presented in a future paper.

REFERENCES Fletcher, C.A.J., "Computational Galerkin Methods", Springer, New York, 1984. Homann, F., "Einflu8 groSer Zahigkeit bei Stromung um Zyllnder", Forschung im Ingenieurwesen, 7, 1-10, 1936. Mahrenholtz, O., "Fluidelastische Schwingungen", Zeitschrift f~r Angewandte Mathematik und Mechanik, 66, 1-22, 1986. Naudascher, E. (ed.), "Flow-Induced Structural Vibrations", Springer, Berlin, 1974. Sarpkaya, T., "Vortex-Induced Vibrations", ASME Journal of Applied Mechanics, 46, 241-258, 1979. Seelinger, P., "Dynamische Fluid-FestkOrper-Wechselwirkung", Dr.-Ing. Thesis, University of Karlsruhe, 1987. Underwood, R.L., "Calculation of Incompressible Flow Past a Circular Cylinder at Moderate Reynolds Numbers", Journal of Fluid Mechanics, 37, 95-i14, 1969.