Journal of Wind Engineering and Industrial Aerodynamics, 33 (1990) 153-160 153 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

D ISCRETE VORTEX S IMULAT ION FOR FLOW AROUND A C IRCULAR

CYL INDER WITH A SPL ITTER PLATE

H. KAWAI

Department of Civil Engineering, Tokyo Denld University, Saitama 350-03 (Japan)

Summary

A discrete vortex model was developed to investigate unsteady flow in a wake of a circular cylinder with a splitter plate. In the model the splitter plate was expressed by a series of sources such as to cancel out the flow across it. Various flow features which had been observed experimentally by many research- ers were successfully reproduced in the simulation. The vortices suddenly rolled up between the cylin- der and the plate when the distance between them was beyond the critical value. The critical distance was hardly affected by the size of the plate when the plate was along the flow, on the other hand it increased proportionally to the size of the plate when the plate was normal to the flow. The critical distance was about a half of the experimental results at Reynolds number 104-2 104, because the length of the vortex formation region in the simulation was shorter than in the experiments. It was also clarified that the various physical quantities such as the base pressure coefficient, the fluctuating lift coefficient, the Strouhal number and the position of the separation point, etc. were systematically var- ied according to the change of the flow pattern.

Keywords Discrete vortex model, splitter plate, unsteady flow

1. INTRODUCTION

Various numerical methods for predicting the unsteady flow features in a wake of a bluff body have actively been developed in recent years. Among them, the discrete vortex model is relat ively simple and appropriate for investigation of vortex dynamics in a wake of a two dimensional body at high Reynolds number. Since its introduction by Rosenhead [1931], it has been applied to analyze various kinds of flows including a flow around a bluff body. Clements et al. [1975] wrote the excel lent review of the model and Sarpkaya et al. [1979] discussed precisely the principal diff icult ies of the model.

The paper firstly describes the discrete vortex model for investigation of unsteady flow in a wake of a circular cylinder with a spl i tter plate and secondary it describes a series of numerical experiments aimed at understanding ef fects of a spl i tter plate to the flow field and the relationship between the flow features and various physical quantit ies such as a base pressure coeff ic ient, the Strouhal number, etc. ,

0167-6105/90/$03.50 1990 Elsevier Science Publishers B.V.

154

2. DISCRETE VORTEX MODEL

In the discrete vortex model the flow is represented by the number of dis- crete line vort ices in a potential flow field . Each vortex is introduced near a separation point and is convected at the velocity induced by both the potential flow and all the other vortices. In case of a circular cylinder the velocity dis- tribution round the cylinder and the position of the separation points are unsteady. Therefore so the position and the strength of a nascent vortex vary with time. So, all the flow model should at tempt these two variations with time. In the present calculation, the separation points are determined by using the velocity drop of 4% below the maximum velocity, which was proposed by Stansby [19811, rather than solving the boundary layer equations such as the Schuhts unsteady momentum equation or the Pohlhausenls quasi-steady approximation. The rate d ]?/dt at which vorticity is shed into wake is approximated by 0.5U ~, where U is the velocity of the outer flow at the separation point, consequently the strength of a nascent vor- tex is given by 0.5U2At where At is the t ime increment. Nascent vort ices are introduced into the flow on a line normal to the surface through the separation point, as the non-slip condition is satisf ied at the separation point.

According to Taylorts idea [1944], the flow outside the wake of a porous sheet would be produced by a plane of sources of strength of 0.5U(k/(l0.25k) per unit area where k is a non-dimensional coeff ic ient related with the porosity and U the main stream velocity. Therefore the spl itter plate is expressed by a series of equi-spaced sources. The approximation is basically l imited to the very porous material such as k~4, but the velocity field upstream the sheet can reasonably be expressed in the case of k>4. The strength of each source is determined so as to cancel out the flow across the sources, or k~ , which is 2UmL/M where u m is the normal component of the velocity at the mth source position without the source, L the length of the plate and M the number of the sources.

The complex potential function W of the flow round a circular cylinder with a spl itter plate is given by

U a F n W=U Z+--~-~ +~ iX--fLg(Z-Z )-Lg(Z-Z*)+LgZ'~+~ qmfLg(Z-Z )+Lg(Z-Z* )-LgZ~ (1)

oo Z ~tL n n ) 2ztL. m m j

where U oo is the velocity at upstream infinity, a the radius of the cylinder, F the circulation of e lemental vort ices which are shed from the separation point ann Qm denotes the strength of the source. Z n and Z are the positions of the nth vortex and the ruth source; indicates the image of(nthe vortex and the source.

The drag force F D and the lift force F exerted on the cylinder are calcu- lated by the following equations on the Blasiu L theorem.

(2)

FL= p

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, * * * "

O@ Fn (Xn-X)Vn-(Yn-Y)Un (Xn-X)Vn-(Yn-Y)Un

* ,2 , * ,2 Xn-X~ +tyn-y~ (4)

In this form, the variations of the strength of the vortices and the sources are omitted because they are much smaller than the other terms. The drag and the lift forces can also be evaluated from the integration of the pressure on the sur- face of the cylinder. It was confirmed by the calculation that the two evaluations of the forces were little different.

It has been recognized that the circulation-reduction mechanism is inevitable to obtain reasonable results by the model. Sarpkaya et al. [1979] discussed the various mechanisms precisely and has proposed the heuristic model such as every vortex loses its strength proportionally to the current strength and position. After many preliminary calculations, the proportional constant k i~. the~ present calculation is determined only by the non-dimensional life time t=U:ot /a(t - is the life time) of the vortex as follows : X =0.002t for t40 when the flow is almost stable.

3. DISCUSSION OF RESULTS

3.1 Effect of a splitter plate attached to a cylinder

The result of the preliminary numerical experiment for a cylinder without a spl itter plate reasonably agrees with the experimental result: the drag coeff icient is 1.06; the base pressure coeff ic ient is -1.05; the rms lift coefficient[ root mean square value of the fluctuating lift) is 0.403; the Strouhal number is 0.194; the mean angle of the separation point is 78 from the stagnation point; the rms fluc- tuation of the angle of the separation point is 2.29 .

Fig. 1 shows the ef fect of the splitter plate attached to the cylinder to the base pressure coeff icient, the rms fluctuating lift coeff ic ient and the Strouhal number. The short splitter plate produces a significant change of the base pres- sure and the fluctuating lift: 20% increase of the base pressure and 80% reduction of the fluctuating lift for the plate of L/D=1/16 where L is the plate length and D the diameter of the cylinder. This change comes from the change of the flow patterns shown in Fig. 2 in which instantaneous pressure, drag and lift forces exerted on the cylinder are also shown.

As the plate prevents the surface flow along the cylinder, the position of the vortex formation moves down-stream and the flapping of the separated shear layer suppresses substantially

156

to reduce the f luc tuat ing lift. For the plate of L/D=I.0, the posit ion of the vor tex fo rmat ion is jus t downst ream the plate, so the posi t ive and negat ive vor- t ices in ter fe re each other to p revent the rol l ing up of the vor t i ces and the a lmost symmetr i c f low pat tern ap- pears round the cy l inder . For the longer p la te of L/D>I.5 , the vor t ices roll up a l te rnat ive ly behind the p late to increase the f luc- tuat ing l ift on the cyl inder. This change of the flow pat - tern is s imi lar to those ex- p la ined by Ape l t et al. [1973] based on the flow v isual izat ion.

However, there is the d i f fe rence in the base pres- sure and the Strouhal num- ber between the exper imen- tal resu l ts by Apel t et al. [1973] and Ger rard [1966] and the present numer ica l resul ts . One of the cause of the d i f fe rence is the e f - fec t of Reyno lds number . In the model, the separated shear layers is assumed ful ly turbu lent f rom the separat ion point which is cor respond to Re>5*10 ~. The Reynolds numbers of the exper iments by Apelt et al. [19731 an~l Gerrard[ l~66] are about 10 and 2"10 ~'. The length of vor tex fo rmat ion reg ion decreases as the increase of the Reynolds number . Ac - cord ing to the B loor 'exper iment [1964], the length is a lmost constant of 2.5D at Re=103 to 104and decrease to 1 .2D at Re=5*10 ' i wh ich is com- parable to the resul t of the present numer ica l exper iment . The other cause is the th ree d imen- s ional i ty of the flow. In the mode l , the f low is or iginal ly two-d imens iona l but it is expected that the vor- tex d iss ipat ion can express the th ree d imens iona l i ty . However, the character i s t i cs of the flow in the wake are not as s imple as to be

.22 ~

157

expressed by the dissipation mechanism adapted to the model.

3.2 Effect of the distance between a cylinder and a spl i tter plate

Fig. 3, 4 and 5 show the base pressure coeff ic ient, the rms lift coeff ic ient, the Strouhal number, the mean separation angle and the rms separation angle when the plate is distant from the cylinder. Fig. 6 shows the distribution of vort ices when the lift on the cylinder is maximum. Until the distance X between the cylinder and the plate is less than 1.5D, the vort ices roll up downstream the plate. For X/D,1.5, the vort ices begin to roll up suddenly upstream the plate. The base pressure, the f luctuating lift, the position of the separat ion point and the Strouhal number change abruptly with the change of the flow regime. When the vort ices roll up between the cylinder and the plate, the vortex formation reg ion is very c lose to the cylinder and the shear layers flap violently. Due to the violent flapping of the shear layer, the pressure on the cy l inder is g reat ly unba lanced between upper and lower sides. Consequently the f luctuating lift is considerably larger than that of a cylinder without a spl i tter plate. The larger the flapping of the shear is, the longer the spl i tter plate is. So is the f luctuating lift. The crit ical dis- tance at which the vort ices roll up between the cylinder and the plate is hardly af fected by the length of the plate and is a l i tt le larger than the length of the vortex formation region for the cylinder without the plate.

According to the experi- mental result by Roshko [1954] for the spl i tter plate of the length of 1.14D at Re=14500, the crit ical distance is 2.75D. The disagreement of the result between the experiment and the present calculat ion seems to be come from the di f ference of the length of the vortex formation region induced by the deference of the Reynolds number, which is ment ioned in the prev ious section. According to the ex- per iment carr ied out by Bloor [1964], the length of the vortex

-I .6

- I .4

-I .2

-I .0 Cpb -0.8

-0 .6

-0 .4

-0.2

O. 0

L

i

I

1 2 3

X/L

Fig, 3 Base pressure coefficient on a cylin- der with a detached plate: L/D:0.25, o L/D=I, L/D=4, m.~ experimental result, L/D=I.14, Roshoko[1955].

.20

.18 1

.14 '~._ ',

.I0

0.6 / ' -~k

CLrms O. 3

o . , \/ o.li I ",>.~__ (~+

~ .....i X

O0 l 2 " 3 4 X/L

Fig. 4 Strouhal number and" rms fluctuating lift coefficient: for symbols of legend see Fig. 3

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fo rmat ion reg ion a t Re=14500 is about 2.2D which is about twice as long as that of the present calculation.

The longer the plate is, the smaller the Strouhal number at the crit ical dis- tance is. Therefore, the strength of the circulation in the cluster of the vort ices increases, and so does the lift on the cylinder. The product of the S t rouha l number and the rms fluc- tuat ing l i f t coe f f i c ient is almost invariant: 0.088 for L/D=0.25; 0.089 for L/D=I.0; 0.087 for L/D=4.0.

3.3 Effect of the splitter plate normal to the flow

When the plate is set normal to the flow, the roll- ing up of the vort ices is also prevented upstream the p la te unt i l the c r i t i ca l distance, and then the vor- t ices roll up suddenly be- tween the cylinder and the plate. When the rolling up does not occur, the two shear layers are separated to the each end of the plate and form a very wide wake region upst ream the p late shown in Fig. 7, which is also observed in Okanan et al . [1986] in the f low visualization. The pressure on the downward face of the cylinder is a f fected by the p la te and recovers greatly. On the other hand, the pressure near the f ront s tagnat ion point is hardly af fected by the plate, which agrees well with the exper imenta l resu l t by Lesage et al. [1987].

Fig. 8 and 9 show the ef fect of the distance to the base pressure, the fluc- tuating lift and the Strouhal number. The broken lines show the critical position of the vortex rolling up. The c r i t i ca l pos i t ion moves downstream for the longer p late, which is d i f fe rent from the plate along the

800

78

Om 761. ~ ~-,,~-"' ~..-.- ...~

741

6 . . . . . .

5

4 0 rms 3

O~ 1 2 3

X/L

Fig. 5 Mean and rms fluctuating angle of a separation point from a front stagnation point: for symbols of legend see Fig. 3.

o .~, .o.~ L/D=1 ,X/D=1

~oR, g~" "~" o"::

, . . :

"" ~o L/D=I,X/D=I.5 o,o o oo

.~ o @ o oo o

?C:~.;o

%oo C~% ~ ~o- . So.. " ' "

. .o

: ; .,~8,'.'.

: e..

:LID=4,X/D=I .5 oo oo %% Oo

o~o o ~oo ~. .%0

C~. o.-.. o o oOZe ~-

Fig. 6 Vortex arrangements and pressure on a cylinder with a detached plate.

159

flow. The short plate has -1.2 the same ef fect as the plate -1 .0 along the flow. It amplifies the flapping of the shear -0.8 layers to induce the large fluctuation of the lift. On -0.6 the other hand, the long Cpb -0 .4 - - plate stagnates the flow, so the strength of each vortices is weakened and the f requency of the vor tex shedding decreases.

The re lat ionsh ip be- tween the critical distance and the p late length is summarized in Fig. 10. The cr i t i ca l d is tance increases proportionally to the plate length. The data are plotted against of the crit i - cal distance divided by the plate length shown in the solid c i rc les in Fig. 10. .18 The value of X/L near ly constant at L /D>3. .16 Therefore, the main scale S for the flow seems to be .14 represented not by the the .12 diameter of the cylinder but by the plate length. This conclusion agrees with the 0.6 exper imenta l resu l t s by 0.5 Okanan et al. [1986] and Lesage et al. [1987] for 0.4 L/D>3. However, the C critical distance, X/L oh- Lrms0.3 rained by the experimental 0.2 resu l t of Lesage et al. [1987] for L/D=0.33 and 0.] L/D=0.17 are about 2.2 and i .7 respect ive ly and are 0 again twice as long as those of the present calculation.

The distribution of the mean pressure on the plate is also dramatically changed with the abrupt change of the flow regime, as shown in Fig. II. When the vor- t ices do not roll up and the wake is spread to the both ends of the plate, the mean pressure near the end increases. On the other hand, the pressure distribu- tion is the same as the or- dinary flat plate after the vortices role up. The dramatical change was also observed in the Leasage 's experiment [1987].

-0.2

0

0.2

0.4 0

/ .a t

I..D.I_x|~ ..--.--. _.---r ,,"

l 2 3 4 X/L

Fig. 7 Base pressure coefficient on a cylinder with a detached plate normal to flow:

L/D=0.25, O L/D=I, L/D:2, ~ L/D=3, A L/D=4.

.20

Y

i J

~J f

f,,,, I !

I!, x~ ,'

I

T I

1

f

s /s st i

2 3 4 X/L

Fig. 8 Strouhal number and rms fluctuating lift coefficient on a cylinder with a detached plate normal to flow: for symbols of legend see Fig.7

s

X/D .p~ "f 2

o,, . - f ~ .~.

l ' - " ~ i 0 0 1 2 4 5

L/D

Fig. I0 Critical distance of vortex formation between a cylinder and a plate: O X/D, X/L.

160

1.O

0.8

0.6 ~-D ~- X4

0"4 1 0.2 - -

C9 2 0

-0.2 --- /1.5 -~z~

-0.4 5 '~'~. '':

-0.8 ~'~m~ --~0. ~'n~ '~ 5 -I .0

-0.5 -O.l O.l O. y/L

Fig. 11 Pressure distri- bution on a plate: L/D:I.

4. CONCLUSION

According to the dis- crete vortex simulation, it is clarified that the model in which the plate is expressed by a series of sources can reproduce various important

L/D=4,X/D=3"5 ~ .- .... . .~ -

:r.-.~'."-: . ~;" "~o

~"~,lb~.~.},....~.-.'- , . e

.o . e t t . .~,

,.~o I

I

. . . ;

6# ~l~m ~

, ,

LID=4,XlD=5.5 i . . . . . o . .

o..:.""; 5" "'.; ": " ", ~: ." . .::... -...

, o : . , : : ;g : : y/ / ? ~ . . ' , . . : .~

, o

g o

~

Fig. 9 Vortex arrangements and pressure on a cylinder with a detached plate normal to flow

o

O O

. o

features in the unsteady wake of a cylinder with a splitter plate. However, the critical distance from the cylinder to the plate where the vortices begin to roll up upstream the plate is about a half of that of the various experiments. The dis- agreement seems to come from the difference of the Reynolds number. As the Reynolds number of most experiments is around 10 4 to 2"10 ;4 , the shear layer is laminar in some distance after its separation and the large vortex formation region with the length of about 2.2D is made. On the other hand, the discrete vortex model simulates the fully turbulent shear layer to produce the smaller vor- tex formation region: the corresponding Reynolds number is more than 5x10" and the length of the vortex formation region is 1.2D. The difference of the vortex formation region seems to lead to the discrepancy about the critical distance.

REFERENCES

Bloor, M.S. 1964, J. Fluid Mech.~ Vol. 19, pp.290-304. Clement, R.R. and Maull, D.J. 1875, Prog. Aerospace Sci.~ Vol.16, No.2,pp. 129-146. Gerrard, J.H. 1966, J.Fluid Mech., Vol. 25, pp.401-413. Leasage, F. and Gartshore, I.S. 1987, J. Wind Engineering and Industrial

Aerodynamics~ Vol.25, pp.229-245. Okanan, H., Shiraishi N. and Matsumoto M. 1986, P roc . 9th National Symposium

on Wind Engineering, pp. l15-120. Roshko, A. 1954, NACA Tech. Note 3169. Sarpkaya, T. and Shoaff, R.L. 1979, AIAA Journal~ Vol. 17 pp.109-128. Stansby, P.K. 1981, Aeron. Quart.~ Vol.32, pp.48-71. Tamura, Y. 1985, Journal of Wind Engineering, No26, pp.49-59, in Japanese. Taylor, G.I. 1944, Repor.t_s and Memoranda of A.R.C.~ No.2236, pp.383-386.