Prediction by discrete vortex method of aerodynamic forces on smokestacks of various cross sections

Journal of Wind and Industrial Aerodynamics

ELSEVIER

Engineering 65 (1996)

309-319

Prediction by discrete vortex method of aerodynamic forces on smokestacks of various cross sections K. Kushioka”.“,

T. Itob, A. Hondaa,

K. Hirao’

“Fluid

L$nomics & Heat Transfer Laboratov, Nagasaki Research & Dewlopment Cerzter, .Mit.whishi Heavy Industries. Ltd., I-I Akunoura-machi, Nagasaki X50-91, Japan hilrchitectural & Structural Technical Department, Tolq~o Electric Power Services Co., Ltd. I-I, Uchisaiwai-cho 2.chome, Chiyoda-ku. Tobo 100, Japan ‘Hiroshima Machineqv Works, Mitsubishi Heavy Industries. Ltd., 5-1 Ebaoki-machi, .Nakaku. Hiroshrma 730. Japan

Abstract Recently, the number of prismatic smokestacks of various cross sections are increasing in Japan. When evaluating the aerodynamic stability of those smokestacks, the conventional standards so far used for the smokestacks of circular cross section can no longer be applied. The aerodynamic characteristics of one of the prismatic-bodied smokestacks by wind-tunnel test and 2-D discrete vortex method of numerical analysis are studied.

1. Introduction In Japan, the number of smokestacks not circular in cross section is increasing for their aesthetic value. The conventional standards so far used in evaluating the smokestack aerodynamic stability, being formulated presupposing circular cross section, can no longer have application to the prismatic smokestacks of various noncircular cross sections. Evaluating the aerodynamic stability of the latter types of smokestack, therefore, involves trial-and-error wind-tunnel experiments or elaborate theoretical analysis. The authors analysed typical prismatic-bodied smokestacks for aerodynamic stability

by 2-D discrete vortex method [1,2] and compared the calculated results with reality by tests on a scale model of the smokestack in two different types of wind tunnels, for assessment of the validity of the numerical analysis through comparison of the calculated and measured time-average drags and Strouhal number.

*Corresponding

author.

0167.6105/96/$15.00 I$‘# 1996 Published PII SO167-6105(97)00049-4

by Elsevier

Science

B.V. All rights

reserved

310

K. Kushioka

2. Numerical

et a1.i.I

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calculation procedure for the discrete vortex method

2. I. Formulation To obtain an unsteady aerodynamic force acting on the smokestack of noncircular cross section, unsteady air flow around the cross section was calculated using the 2-D discrete vortex method. Fig. 1 shows a conceptual illustration of the separation, or bubble, of air flow around a bluff body. In the discrete vortex method developed based on the potential theory, the fluid flow is represented by the distribution of velocity potential 4. The velocity potential is obtained by superposing the velocity potential of approaching flow $“, that of a boundary layer along the body surface &, and that of a separated shear layer &. Each of these velocity potentials is expressed as follows: $Lp = U, (x cos CI+ y sin a),

(1)

il!k ddx,Y)= YBYB(L v)tan-,Y x Pn

da

5s

ds,

(2)

(4)

Y) = dhJ(X~Y) + &3(x, Y) + $ss(x,Y).

The explanation of the symbols employed is as follows: U, is the velocity of approaching flow, a the angle of attack, B, Sm the body surface and mth separated shear layer, yB,yS,,,the circulation per unit length of boundary layer on body surface and separated shear layer, Ns the number of separated shear layers, and (5,~) the

Point Fig.

1. Flow

with

separated

shear

vortex

layer

around

a bluff

body.

K. Kushioku

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Wind Eng. Ind. Aerodyn.

coordinates of s(B or SW). And the circulation according to Kelvin’s theorem,

65 (1996)

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311

in the calculated region is constant

NS ~s(t,

v)ds

+

ys([, q)ds = 0. m=l P Sm

(5)

C

To determine the separation point, the integral boundary solved between stagnation point and separation point:

U2~+(20+S*)U~=~.

layer equation (6) is

(6)

P

The explanation of the symbols used in Eq. (6) follows: s is the distance from stagnation point along body surface, U the flow velocity at periphery of boundary layer on body surface, 0 the momentum thickness of boundary layer on the body surface, 6* the displacement thickness of boundary layer on the body surface, r. the wall surface shear stress, and p the density. 2.2. Discretization In the discrete vortex method, the distribution of circulation of the separated shear layer is approximated by the distribution of point vortices. Then Eqs. (2) (3) and (5)

Fig. 2. The outline

of numerical

procedure

in the discrete

vortex

method

Fig.

3. Wind

tunnel

[ 1

..

r E =E i 1-

l-

K. Kushioka Table 1 Conditions Case

1

employed

for experimental

Cross section Section

et al/J.

Wind Eng. Ind. Aerodyn.

and theoretical

2

Section

I

3

Section

I

4

Section

II

5

Section

II

Experiment (Gottingen-type wind Experiment (boundarylayered wind tunnel)

309-319

313

studies

Method

I

65 ll99b)

tunnel)

Calculation (discrete vortex method) Experiment (Gottingen-type wind tunnel) Calculation (discrete vortex method)

Dimension (aspect ratio)

Approachmg

2D (22.2) with end plate 3D (9.56)

Uniform

flOW

How

2D

Turbulent Ilo\+ average velocity I/ = V,(::‘;,)” ‘5 Uniform how

3D (4.45)

Uniform

tlo\l,

2D

Uniform

How

are discretized as expressed by the following equation:

ysi and jlsmi represent the circulation positions of point vortices.

of point vortices, (rsi, rlei) and (lsm;. ~I.\-~,Jthe

Fig. 2 outlines the numerical calculation procedure. To obtain the velocity potential Eqs. (7))(9) are solved. In these equations the unknowns are yBi and Y~,,,~.ysmi is determined from flow velocity at the separation point, and the point vortices are introduced into the flow field at every step of progressive shift in time. hi, on the other hand, is determined from the body surface boundary condition indicated by the following equation: n,u + n,v = 0.

(10)

Here, (rr,, ylY)represents the normal vector relative to the body surface and (~1,c) the flow velocity. In other words, since Eqs. (7)-(10) constitute simultaneous linear equations with “;Sias the unknown, l’si is obtained by solving these equations. The progressive shift in time is factored by moving point vortices, introducing point vortices at the separation point, and calculating :jsi at each succeeding step of shift in time.

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(4

Fig. 4. The cross

sections

of the smokestacks

examined

here: (a) Section

I; (b) Section

II

The integral boundary Eq. (6) is solved using Polhausen’s method [3]. The use of Polhausen’s method assumes laminar separation, which means that Re is in the subcritical region.

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315

In the Polhausen method, Eq. (6) is rewritten into the following equations:

K(A) =zg, dZ ds-

J(A) u’

where

is the Polhausen shape factor, 315

945

9072

’

These equations are integrated starting from the stagnation point with the velocity U at the periphery of boundary layer obtained from the flow calculation by the discrete vortex method. The condition for separation is A = - 12 in the Polhausen method, which means that the gradient of flow velocity on the wall is zero.

3. Wind tunnel testing Two different types of wind tunnels were used for the model test for smokestack aerodynamic stability; a Gottingen-type wind tunnel and a large boundary-layered wind tunnel (see Fig. 3). The test section of the former, type I, wind tunnel was 1.5 m in width, 1.5 m in height, and 2.3 m in length. That of the latter, type II, wind tunnel was 6 m in width, 5 m in height, and 30 m in length. The type I wind tunnel was used to measure the aerodynamic force acting on the model in uniform air flow, for comparison of data thereby obtained with those given by calculation. The type II wind tunnel, being capable of producing the turbulent flow within the boundary layer of air flow by means of spires and blocks, was used to determine the smokestack aerodynamic characteristics through the model test closely simulating the real-life situation.

4. Test condition The configurations of the smokestacks of noncircular cross section selected for study are shown in Fig. 4. The cross section resembles a square with round corners. Case 1 represents the model test performed to obtain data for comparison with those obtained by calculation, and cases 2 and 4 the model tests performed under the conditions approximating those applicable to the real smokestack.

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5. Results review Fig. 5 shows instantaneous stream lines obtained by the calculation (cases 3 and 5), from which it can be seen that there are Karman vortices produced downstream of the smokestack. Fig. 6 shows the time-average aerodynamic forces versus the wind directions, or attack angles, of cases l-3. Here, CD and CL are drag and lift coefficients defined by the

Fig. 5. Stream

lines: (a) case 3 (Sectlon

I); (b) cast

5 (Section

II).

-&CASE 1 (Experiment 2D) 4 CASE 2 (Experiment 3D) 0 CASE 3 (Numerical solution)

o~~--y

0

10

20

30

40

Attack angle a (Deg.) Fig. 6 Time-averaged

force (comparison

between

calculation

and experiment).

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following equations: c, =

Fll 0.5pV2DL.

CL =

FL

0.5pV’DL’

in which F,, and FL are drag and lift, p the density of air, V the approaching velocity, and DL the reference area. The conditions employed for case 3 (calculation) are nearly the same as those for case 1 (2D experiment). The C, and C, values determined in both cases show a high degree of agreement, witnessing the validity of the numerical analysis. The C, value obtained in case 2 is smaller than that obtained in case 1, most probably, due to the difference between the two in the approaching flow employed as well as to the presence and absence of the top-end effect. This top-end effect was recognizable with smokestacks of other cross sections as well [4]. The change in C, attributable to the approaching flow and top-end effect is smaller than it is in C,, probably because the entrainment at the top end of the smokestack in the direction of approaching flow is greater than in the direction normal to the approaching flow. Fig. 7 shows Strouhal number St, from which it can be seen that St obtained by calculation shows a good agreement with that obtained by wind tunnel test. This fact demonstrates that the unsteady flow was successfully simulated in the analysis by discrete vortex method. Generally, C, for smokestacks with top end is smaller than that without the top end. Figs. 8 and 9 show data corrected by the effect of the top end k = (C,),,/(C,),,. k is assumed to be 0.75 and 0.67 for the aspect ratio of 9.56 and 4.45. These values are obtained from the experimental result for a circular cylinder. In these figures, the

+A%. =*=.._ 0 rh0 $ 0.15- d ___-.---*2 “a- -________ -_0 z’ 2 : R

0.1 A l

O

Fig. 7. Strouhal

I 0

Number

I

I

CASE 2 (Experiment 2D) CASE 3 (NumericaLSolution)

I

IO St (comparison

I.

I

I

I

I

20 30 40 Attack angle a (Deg.) between

calculation

and experiment)

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8 -0 l

tt

-

0

65 (19%)

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CASE 1 Experiment(Original data) CASE 1 Experiment (Corrected data) CASE3 Calculation

I.

I.

I

10

20

30

I

I.

40

Attack angle a (Deg.) Fig. 8. The

corrected

C, of case 1 (Section

I).

8 CASE 4 Experiment (Original data) -El- CASE 4 Experiment (Corrected data) 0 CASE 5 Calculation

d 0” l-

OI

f 0

I

I

I

10

20

30

Attack angle a (Deg.) Fig. 9. The

corrected

CD of case 4 (Section

II).

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calculated C, show a good agreement with C, corrected by the top-end effect. Therefore, this numerical method with correction is almost applicable to the prediction of aerodynamic force on a smokestack with finite aspect ratio.

6. Conclusions The authors applied the discrete vortex method for predicting the aerodynamic stability of the smokestack of noncircular cross section, with the results as presented in the following: (1) The comparison of aerodynamic characteristics of the smokestack with noncircular cross section determined through 2-D numerical analysis by the discrete vortex method and those established by wind-tunnel testing showed a good agreement, witnessing the validity of the analysis by the discrete vortex method. (2) However, the results obtained through 2-D numerical analysis by the discrete vortex method require to be corrected for the effects of the smokestack top end and the air boundary layer on the smokestack wall surface used when dealing with the 3-D smokestack problem. The calculated C, corrected by the data of circular cylinder are almost valid.

References [l] [2] [3] [4]

T. Inamuro, T. Adachi. A numerical analysis of unsteady separated flow by vortex shedding model (2nd report flow around a circular cylinder), Trans Japan Sot. Mech. Eng. B 52 (476) (1986) 1600. K. Kushioka et al., Simulation of turbulent flow by discrete vortex approximation, J. Wind Eng. Ind. Aerodyn. 46/47 (1993) 371. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1960). R.W.E. Gould, W.G. Raymer, P.J. Ponsford, Wind tunnel tests on chimneys of circular section at high Reynolds numbers, Proc. Symp. on Wind Effects on Buildings and Structures, Loughborough University of Technology, Loughborough, UK, 1968.