Simulation of the turbulence influence on galloping vibrations

Journal of Wind Engineering and Industrial Aerodynamics, a l-44 (1992) 2023-2034

2023

Elsevier

Simulation o f the Turbulence Influence on Galloping Vibrations

H. Lindner lnstitut for Mechanik, Universit~t Hannover, AppelstraBe 11, W-3000 Hannover, Germany

Abstract

A method is presented to calculate the galloping behaviour in turbulent flow using the smooth flow lateral force characteristic and considering the turbulence mathematically. It is found that the agreement with experimental results depends strongly on the depth to height ratio b/h of a rectangular prism. At b/h=2.0 nearly exact agreement is found. The results for a square prism are as well similar to the measurements, but at considerable different turbulence intensities. The behaviour of a short rectangular prism b/h=0.5 can not be predicted by this method. It is shown, that the influence of the scale length of the turbulence is negligible. This suggests a direct calculation of the turbulent from the smooth flow characteristic. Indeed this procedure proved to be particularly meaningful.

1. Introduction Wind loads on buildings and structures are still one of the important problems in civil engineering. A small part of this field are the structures susceptible to self-excited galloping vibrations. This behaviour depends strongly on the shape of the structure and occurs e.g. on rectangular and D-shaped prismatic bodies and on ice covered transmission lines. In order to predict the galloping vibrations, Parkinson and Smith [1] developed a quasi-steady theory using the stationary measured lateral force characteristic CF. The agreement with experimental results is good, as far as the quasi-steady assumption holds. This is the ease if the flow velocity is high compared to that of the vibration [1]. Novak [2] and Laneville and Parkinson [3] have shown, that the quasi-steady theory also works in turbulent flow. But it is necessary to measure the lateral force characteristic in a flow with corresponding turbulence. In the present paper an attempt is made to calculate the behaviour in turbulent flow from the smooth flow characteristic. 0167-6105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

2024 2. Previous Investigations The mechanism of the galloping vibrations has been widely investigated for about 30 years. Thereby, a special interest has been taken on rectangular cylinders with different aspect ratios b/h (b section depth, h section depth). Parkinson [4] showed that galloping from rest occurs approximately in the range 0.75 < b/h < 4. This agrees with observations of Bearman and Trueman [5]. They pointed out that the shear layer trailing edge direct interaction has a significant effect on the vortex formation. This interaction commences for cylinders at b/h=0.62 and vanishes with stationary reattachment at b/h=4.

In reference [3] the validity of the quasi-steady theory in turbulent flow has been shown for rectangular prisms of b/h=0.5, 1.0 and 2.0. There is a distinct influence of turbulence intensity, but different for each shape (Figure 1). Laneville, Gartshore and Parkinson [6] explained this behaviour also by the shear layer trailing edge direct interaction. They have shown that turbulence increases the curvature of the shear layer leading to shem layer interactions tbr sections with b/h <0.62 (Figure 1). Accordingly turbulence causes e.arlier reattachment and rectangular sections with b/h<3.5 become stable.

//~,

c f,

.b_ _ 2

h

0

/

o/.

+m-I~

02

/

t

/

,:..

,,

,<- -+

b

\

:+

;;

;,

.

• +,,

,~\

,\

£>

,6,+

,,I);oi

.\

Q/o

h

t.+,! ,

,'

\

.... ,>,'\ \\

#/"/

o,,I) +,,':,.;<-Or

"

,\

~ \

")

" ':'I

~b- = 0,5 h ,I C, , 0,,'

!

01,

,i! q'l;l

Figure i. Sketches of expected shear layers, - ~ smooth, ........ medium turbulence, ........ high turbulence and corresponding measured [3] characteristics.

2025 Considering these investigations the shape of a lateral force characteristic becomes more clear. The inclination between the flow and the prism (Figure 2) leads to a stronger shear layer trailing edge interaction on the lower side. This results in less pressure on the lower side resulting in a lift force [7]. Increasing the inclination causes reattachment. In the range of the miminum reattachment angle the CF-characteristic reaches its maximum [4]. At higher angles with stationary reattachment the lift mechanism breaks down (Figure 1). Thus the CF-characteristic is determined by the shear layer and its interaction with the afterbody. This, and the apparent effect of smoothing the characteristics due to turbulence, leads to the idea that the galloping behaviour may be calculated using the smooth flow characteristic, considering the turbulence mathematically. Novak [2] mentioned this also but because of the difficult analysis of the equation it was not examined further. 3. Simulation of Turbulence

m

,

17x

Figure 2. Aerodynamic forces on a vibrating square prism.

Figure 2 shows a rectangular prism and the wind loads in a cross flow. This galloping oscillator is described by the equation my + d y + c y = F where m is the mass, d the damping coefficient and c the spring constant.

(1)

2026 The aerodynamic force F is given by p2 F = ~u. beC F(q))

(2)

with the air density p, the prism length e, the flow velocity u.. and the lateral force characteristic CF OP). Now a two dimensional velocity field I =

[o:Ul

(3)

is assumed, where u. is the mean flow velocity and u, v are the turbulent distortions. The turbulence intensity is defined by J

Tu = -U- , U.

(4)

where u' is the rms (root mean square) value of the longitudinal turbulent fluctuation.

ure~....---~ ~ LIm

LI

v

Figure 3. Influence of turbulence on the resulting flow.

Figure 3 shows, that the instantaneous velocity fluctuations alter the magnitude and the angle of attack q)" of the resulting velocity ~ , . Application to equation (2) yields

F = §be(u..+u) 2c F(q~') ,

(5)

with tan¢' - :Y + v Urn+ U

.

(6)

The fluctuations u, v have been simulated at first as uncorrelated bandlimited white noise [9]. The wind ,mnel turbulence can be assumed as homogeneous and isotropic, that means that in the experiments performed the fluctuations are in fact uncorrelated [10]. The

2027 break-frequency of the white noise is calculated with the d~pth b of the prism and the flow velocity u. as

u..

(7)

where only distortions are simulated with wavelengths longer than b. Numerical simulations have been conducted for a square [9] and a rectangular prism with b/h-2.0. The results are shown in the following.

4. Results of the Galloping Simulation For the simulation of the behaviour of the square prism the parameters and the smooth flow lift characteristic are taken from the experiments published in [11]. The data for the rectangular cylinder are taken from reference [3]. All simulations in the present paper have been conducted using either the measured values or interpolated intermediate values [9].

O =0,0057

D = O,0009

I n o.

0.8 L

'

/.q.

* O

/~/,//'+~

0,6

II

I:1

,,

n

&

A

/+

'

/'aZ~

0,4

,

!

÷o//

&

0,2

I

/

+ /+

,~' .......

0

i

i

°//

II/,

OLL

Simulotion: - - - - - T u =0 % .... Tu=3% ..... Tu=6%

/

/

/

Experiment. + Tu=0%

.

a

.

./

Tu = 6 % Tu = 1 0 %

& J ....

....

t ...........

2

l

.......

t ..........

4

l ...............

1_ . . . . . . . . .

~

........ -J.

6

..........

8

1. . . . . . .

J. . . . . . .

10

Vr= uoo/((B0b) Figure 4. Displacement vs wind speed for square prism. Comparison between simulation and experiment [11].

2028

Figure 4 shows the computed relation of the dimensionless amplitude ¥,=y(rms)/b to the reduced flow velocity Vr=u=/(¢oo b) for two damping ratios D=d/(2 m ¢Oo). It can be seen that the onset of galloping agrees quite well with the experimental data. The course of the amplitude shows the same inaccuracy as the quasi-steady theory (Tu =0%). Particularly for the higher damping ratio D=0,0057 the simulation agrees well with the measured values. And in the simulation as well as in the experiment the curve for Tu=0% intersects those for Tu,0%. For higher flow velocities the influence of the turbulence intensity on the hysteresis effect is simulated, see Figure 7. With increasing turbulence the hysteresis disappears. Experimental results of Novak [2] showed the same behaviour. Unfortunately, these results are calculated with a significantly lower turbulence intensity compared to the experiment [9], [Ill. The simulation of the rectangular section with b/h=2.0 (D=0,00237) shows a slightly different behaviour. In Figure 5 the simulated behaviour agrees with that calculated by the quasi-st,~dy theory [3]. Both methods show differences from the measured values. The overestimation of the response results from the quasi-steady approximation which is also the base of the; simulation method. Therefore the simulation results have to be compared with the calculation in reference [3] based on the turbulent lateral force characteristic. At the high turbulence intensity the simulation as well as the measurements show a slightly earlier onset of galloping, the quasi-steady theory is not abl¢ to reproduce this behavior. The different influence of the turbulence intensity on these bodies has been investigated further. But first the effect of the turbulence structure will be discussed.

Yr

Experiment ,, Tu = 6.7%

1,/,

1.2

•

....

Tu=9

Tu= 7.%

%

v

~

quasi =steady theory Simulation

/Tu=6.7% ~

S

/

,,~,/

1.0

-//

10.5%

Tu=9

,8 o v

11//,

.

.6

Y

v

.4 .2

//

t

1

2

3

s

6

•

8

g

1'o

Vr

Figure 5. Reduced amplitade vs reduced velocity for prism with b / h - 2 . 0 . Comparison between simulation, quasi-steady theory and experimental results [3].

2029 5. Turbulence Structure The turbulence intensity Tu describes the entire energy of the fluctuations u. The distribution in the frequency domain is conveniently described by the scale length Lx of the longitudinal distortion u. L,, can be interpreted as the length of a mediumsized eddy [10]. The influence of the scale length L,, has been examined at particular velocities for the square prism with D=0.0057 and Tu=6%. The fluctuations u, v were both simulated as white noise but with different length scales. In a homogeneous, isotropic turbulence the relation is

[IO1 t,, =

1

t.,

.

(8)

The range of L,, is varied from L,,/b=0.2 up to L,,/b=600. The latter value is much higher than usually produced in a wind tunnel (maximum L,,/b =20, [8]). The question of the length scale influence on bluff body flow and galloping vibrations is examined by many authors. Partly the previous research showed an influence on the vortex formation and the drag (e.g. Lee [12]), but most results show little effect on the integral values like lift and drag ([3], [8] and Petty [13]). Indeed, the influence on the rms value of the galloping response over a long simulation time is found to be negligible. However, there are small differences in file time histories. The time history calculated with L,,/b-0.2 in Figure 6a is smoother than that simulated with the large value L,,/b-600 (Figure 6b). The reasons are the higher values of single disturbances, because the turbulent energy is contained in a much smaller frequency range.

O)

oo ....

-O05-

0 ""

b) -Ou3I- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

o

1 . . . . . . . . . . . . . . . . . . . . . . . . . . .

~o

Time [ sec]

I- .....

loo

Figure 6. Calculated influence of scale length Lx on time histories of a vibrating square prism (520 cycles, D=0.0057, Tu=6%, V,= 10): a) Lffb=0.2, b) L,,/b=600.

2030 It can be assumed that the predominanteffect on the presented turbulence simulation is the random disturbance of the quasi-steady process. The structure of turbulence has at most a secondary influence. If this assumption holds it must be possible to calculate the turbulent C&-characteristicfrom the smooth one, using the relations (5) and (6). 6. Direct CaIcuIationof the Lateti Force Characteristic From the smooth flow characteristic, a developing angle a0 is considered. With an arbitrary flow velocity u, a galloping velocity .. YO

=

(9)

u, tana0

is calculated; Now, a certain turbulenceintensity Tu is simulatedby a probability generator which creates a normaldistributionof the values u,v. The standard deviation u is calculated with the chosen intensity and the velocity u, (a=Tu~). With these random values of u and v the equations (5) and (6) can be written as F’

= ;

be

(u,

+u)*C,(O

9

with

The calculation of F’ is continued until the mcc;1n value of F’ is n man value F’ the lateral force coefficient

is calculated. Be ting this for a series of angles a0 1 to a complete C,-characteristic. Pi response of the squa prism with the damping ratio B lence simulationsand ther from simulations using the direct calculated force characteristic. The a reement is nearly complete. Only in the ranges where the galloping is very sensitive, which are the onset of galloping and the jumps to the upper limit cycle, differences occur. But this is not surprising, because in the case of turbulent simulationhigh values of the disturbancescan influence the we& equilibrium. An other source of error may be the problem to achieve stationary results of the rms values, because it takes long simulation times particular for small vibrations and high turbulence intensities, ence th of calculation time is one of the advantages of the direct calculation of the characteristic. The other advantage, a direct evaluation of the galloping sensitivity in turbulent flow, will be used to examine the turbulence influence on different rectangular prisms.

2031

/J

2.5 CF- Calculation Simulation

Yr

Tu=3% . ~ , ~ - / "

2

1.5

f

/

/"

.f

,,:,~-~ "Tu=6% /- f

//

/

0.5 ~ "

I ~ "

7

i

8

l

I

9

i

i

10

I

i

11

i

I

12

I

I

13

I

I

I

1L, Vr

Figure 7. Comparison of the galloping response between the simulation and the direct calculation of the CF-characteristic.

7. Results of the Direct Calculation of the CF-C'haracteristic The results of Laneville and Parkinson [3] for three rectangular cylinders (h/b=0.5, 1.0, 2.0) were used to examine the different turbulence effect and its meaning for the presented method. The smooth flow characteristics were taken from [3] and for different turbulence values the CF-characteristics have been calculated. Figure 8 shows a) the calculated lateral force characteristic of a section b/h--2.0 and b) the measured one. It can be seen that the figures are nearly identical, exept that turbulence intensities are slightly higher for the calculated curves. The agreement of the results for the square prism in Figure 9 is not so good as for the long rectangular. The decrease of the maximum value and the gradient angle in the origin are similar in both cases (Figure 9 a/b). But the shapes of the curves are different: The calculated curve remains similar to the smooth flow characteristic whereas the measured curve shows a remarkable change. It seems that the conditions, which are responsible for the turning point (which produces the hysteresis) are disappearing in turbulent flow (Figure 9b). Whereas the calculation only smoothes this range. Considering the good results for a square prism in Figure 4 these differences are surprising. This means that the direct calculation is more sensitive than the galloping simulation because the latter is mainly affected by an entire energy equilibrium.

2032

a) Calculation

OF ----!

/"~ !/

0.6

0.4

---ru

\

,~

= o%

..... Tu: 70/0 -----Tu =10.5%

//•b}

CF' 0.6

Experiment: Tu=0% --- Tu=6,7% -

-

--'-

U ""

0.20"4

0.2

2

4

6 "-. \

-0.2

I

\ , \ 12

'\.

.

"~\

|

14 qS[o]

4

6

"

.

~,~12

,,\\.

14--

~o[o]

-0.4

\ \',\~.

=0,6

2

-0.2

\'"'~.~~

=0.4

0

-0.6

Figure 8. Comparison between calculation and experiment [3] for a rectangular prism with b/h=2.0.

I'

c~ I~1

a) Calculot)on Tu = 0% ___o Tu =2,5%

_ V~I

-

..............]

A

=To°

s O/o

....... To--6%

"'

0.2

..i.l

._~___=j~._=j,.~

s

~o

20

~o[~]

b} Experiment: ~Tu= 0% . . . . Tu =6,7% ~Tu= 9% / \ .......Tu = 12%

CF 0,4

~" ..,/'v'~, ,.,,'/ /..~ \ , ~,' ~#I~I

0,2

, y

0

s

,,',\ ,o

'.,I 2o~r°I

=0,2

J

":~ \

Figure 9. Comparison between calculation and experimental results [3] for a square prism.

2033 For the short rectangular prism b/h : 0 . 5 it is not possible to achieve similar results, see Figure 10. Even if the calculation is conducted with Tuffi12% there is a negative gradient in the origin and the maximum of the curve is also much too small. The reason for the failure of the method in this case is that at low angles there is no shear layer trailing edge direct interaction and therefore no galloping excitation. But these qualitatively different conditions can not be reproduced by the presented method, This indicates a lower boundary of the applicability of the method at about b/hffi0.75 where galloping from rest occurs.

a) Calculation: ----Tu : 0 %

OF

I-'-I

A

-.-..--I I / \ .... Tu= 6.7% L.J / \

0.2 0.1,

- . - Zu = 9% ....... Zu= 12%

/ .~_'_-'\.

\ \

m

,

0

....

- .

• ~---~"~-

,s

2o.

-0.1

".,\

:~o,p~o]

" ". "2~i ~~~ el e

b) Experiment: - - - - T u = 0%

C~.

....

0.2"

--.--

Tu = 6 . 7 % Tu: 9 %

,

,~.."/~'."j¢",

........ ~u: ,~O,o_,,,*

0.1"

'"

...1'..' '

A .,, I \

/ _./

/,',,

/

\

\

-'., \ \

. '::x",\

o_.___.

.

ea e

•

\

Figure 10. Comparison between calculation and experiment for a rectangular prism with b/hffi0.5.

2034 8. Conclusions It has been shown, that the method presented in this paper is capable to describe the turbulence effects for the square and the long rectangular cylinder (b/h=2.0). In the latter case even a nearly exact agreement with measurements has been found. Further, in the simulations no significant influence of the scale length was found, suggesting the direct calculation of the turbulent characteristics. For the galloping response y (rms) this method is shown to be ne~ly identical with the simulation, but it saves computation time. Further, the more sensitive direct calculation gives evidence about the main effects of the turbulence. An agreement between the calculated and the measured turbulent flow C~-characteristic indicates that the main effect of the turbulence is a distortion of the galloping mechanism. On the other hand a discrepancy indicates a qualitative change of the shear layer trailing edge direct interaction due to turbulence. By that a separation of these main effects is possible. It can be seen that the galloping behaviour is substantially affected by the turbulent distortions. To get a validation of the presented method further investigations have to be made for a wider range of rectangular and other sections like D-prisms. References

10 11 12 13

Parkinson, G. V.; Smith, J. D.: The square prism as an non-linear oscillator. Quart. Journal of Appl. Math. 17 (1964), pp 225-259 Novak, M.: The effect of turbulence on the aeroelastic instability of ~luare prism. BLWT-7-68. University of Western Ontario, London; Canada 1968 Laneville, A.; Parkinson, G. V.: Effects of turbulence on galloping of bluff cylinders. Proc. 3rd Int. Conf. on Wind Effects, Tokyo 1971, pp 787-798 Parkinson, G. V.: Aeroelastic galloping in one degree of freedom. Symp. on Wind Effe~:ts, Teddington 1963, pp 582-609 Bearman, P. W.; Trueman, D. M.: An investigation of the flow around rectangular cylinders. Aero. Quart. 23 (1972), pp 229-237 Laneville, A.; Gartshore, I. S.; Parkinson, G. V.: An explanation of some effects of turbulence. Proe. 4th Int. Conf. on Wind Effects. Heathrow 1975, pp 333-341 Nakamura, Y.; Tomonari, Y.: The aerodynamic characteristi~.s of D-section prisms in a smooth and in a turbulent flow. Aero. Quart. 32 (1981), pp 153-168 Laneville, A.; Williams, C. D.: The effect of intensity and large scale turbulence of the mean pressure and drag coefficients of 2-D rectangular cylinders. Proc. of the 5th Int. Conf. on Wind Engineering, Fort Collins 1979, pp 397404 Lindner, H.: Turbulenzeinflu8 auf Galloping Schwingungen des quadratischen Prismas. Z. Angew. Math. und Mechanik ZAMM 70 (1990) No.4, pp T74-T77 Rotta, J. C.: Turbulente Str6mungen. Stuttgart: B. G. Teubner, 1972 Bearman, P. W.; Gartshore, I. S.; Maull, D. J.; Parkinson, G. V: Experiments on flow-induced vibration of a square-section cylinder. J. of Fluids and Structures 1 (1987), pp 19-34 Lee, B. E.: Some effects of turbulence scale on the mean forces on a bluff body. J. of Ind. Aerodynamics 1 (1976), pp 361-370 Petty, D. G.: The effect of turbulence intensity and scale on the flow past square prisms. Proc. of the 3rd Coll. on Ind. Aerodyn., Aachen 1978, Part 1, pp 281-291