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Application of 3D numerical analysis on the prediction of aerodynamic forces of a tall stack cross section
Journal of Wind Engineering and Industrial Aerodynamics 81 (1999) 391}401
Application of 3D numerical analysis on the prediction of aerodynamic forces of a tall stack cross section Kenji Shimada *, Kunio Takei, Shougo Yamamoto, Hajime Sugimoto Institute of Technology, SHIMIZU Corporation, 3-4-17 Etchujima, Koto-ku, Tokyo 135-8530, Japan Electric Power Development Co. Ltd, 6-15-1 Ginza, Chuo-ku, Tokyo 104-8165, Japan Kaihatsu Architects & Engineers, INC., 6-4-10 Tsukiji, Chuo-ku, Tokyo 104-0045, Japan
Abstract One of the recent design issues concerned with a tall stack is its landscape. Recently, con"gurations other than conventional circles have been often chosen as a cross section. In these cases, wind loads sometimes become critical in the structural design and therefore selection of its con"guration becomes important. In this paper, an attempt was made by applying direct numerical simulation to predict aerodynamic forces acting on the section models of a 200 m tall stack. A modi"ed elliptical cross section and a recessed corner cross section were compared from the aerodynamic viewpoints. The analysis could reproduce the strong unsteadiness inherent to each cross section. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Computational wind engineering; Aerodynamic force; Stack
1. Introduction One of the recent important issues concerned with the designing of a tall stack is its landscape. From environmental point of view, the height of the stack should be as tall as possible. Consequently, its con"guration has to be designed from landscape point of view. A 200 m tall power station stack whose construction is planned in Yokohama
* Corresponding author. E-mail address: [email protected] (K. Shimada) 0167-6105/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 0 3 2 - X
392
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
is designed to have a very slender con"guration to satisfy the above needs [1,2]. In general, when this slender structure is designed, aerodynamic design such as selection of the con"guration of its cross section becomes important for its reasonable structural design. Conventionally, a circular cross section is the most popular choice for a stack cross section. It is partly because of easiness in construction and partly because of convenience in structural design against wind loads since the wind load is not dependent on the wind direction. In this case there also exists some aerodynamical advantages. The Strouhal number is relatively large hence the vortex-induced vibration of the fundamental mode occurs at relatively low windspeed and therefore it is not critical. Furthermore, other kinds of aeroelastic instability such as galloping do not occur. However, if the more architectural oriented design described above had to be chosen, many of the points described above woulds have to be kept in mind. However, recent progress in material and construction methods have enabled the realization of such stack whose cross section is more sensitive to wind. In such a situation, wind tunnel experiment is the most reliable method and is necessary in the "nal stage of its design to assure structural safety against wind loads. On the other hand, it is also an e!ective way to employ an aerodynamically favorable cross section from its early stage of design process. By the way, recent advancement in computational #uid dynamics contributed many bene"ts to wind engineering, although further e!orts should be devoted for the realization as an ideal assessment tool. In wind engineering "eld so far, in particular great success has been achieved in the assessment of pedestrian comfort. However, application to the assessment of aerodynamic forces is not well established. In this paper, an attempt was made by applying direct numerical simulation to predict aerodynamic forces acting on the section models of a 200 m tall stack and its applicability was examined.
2. Con5guration of the stack At the early stage of the present design process, the following two con"gurations are proposed. These con"gurations are illustrated in Fig. 1. One of them is a stack which has a modi"ed elliptical cross section and the other is a stack with a rectangular cross section which has recessed corners in each corner. These sections will be denoted as `E-typea and `R-typea hereafter, respectively. Both of the stacks have tapper along their height and are made of reinforced concrete. Final con"guration of the stack is somewhat di!erent from the E-type cross section and was examined by wind tunnel experiments [2].
3. Numerical procedure The present numerical study is based on the following considerations: (1) Calculations are performed in three dimensions. Since calculations are limited to Reynolds number lower than Re(10 where three dimensionality is not predominant
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
393
and two-dimensional computations can be e!ective, three-dimensional calculation is performed in the present study. (2) The cross section for the analysis is the upper portion of the stacks and is considered as to be a two-dimensional cylinder which is supposed to have an in"nite
Fig. 1. Elevation and cross sections of a stack with E-type cross section. (i) Cross section at the top of the stack; (ii) cross section at the base of the stack. (b). Elevation and cross sections of a stack with R-type cross section.
394
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
Fig. 1. Continued.
length in spanwise direction. The reason why the upper portion is chosen is that the local wind forces acting on the almost upper half portion of the stack make a predominant contribution in the generalized wind force. (3) Approaching #ow is assumed to be smooth uniform #ow. Since the structures are immersed in the atmospheric turbulent boundary layer, free stream turbulence is
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
395
Fig. 2. Numerical grid system: (a) E-type cross section, (b) R-type cross section.
a very important factor for aerodynamic assessment. However, its incorporation has not been established yet and is still in progress. Numerical analysis is based on solving the incompressible time-dependent threedimensional Navier}Stokes equations (1) and (2) by the "nite di!erence method, u #u ) u"! p#Re\u, R ' u"0.
(1) (2)
396
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
In order to evaluate the aerodynamic force as accurately as possible, the #ow structure in the immediate vicinity of the cylinder where steep gradients exist in the physical quantities must be "nely resolved and the behavior of the separated shear layer should be captured as correctly as possible. To achieve this, the generalized curvilinear coordinate system (body "tted coordinate system) is incorporated here to enable to concentrate the su$cient number of grid points near the boundary (Fig. 2). The physical analytical domain is a circle of 30d in radius ;1d in spanwise direction, where d is the cross streamwise length of the section. The cylinder itself is located slightly upstream (!5d ) from the center of the domain. In the present study, a total amount of 240;100;10"240 000 and 320;100;10"320 000 nodes are distributed around the E-type and R-type cross section, respectively. All primitive variables are de"ned on the same grid point. Time-marching algorithm is followed by Marker and Cell method, i.e., the continuum equation is incorporated as solving the Poisson equation for pressure and subsequently the momentum conservation equations are solved for the respective velocity components using the updated pressure, D
p"! ) (u ) u)# , D" ) u. *t
(3)
The third-order upwind scheme proposed by Kawamura and Kuwahara [3] is employed to stabilize the high Reynolds number numerical instability which arises from the non-linear e!ect of the convective term, c
*u (!u #8u !8u #u ) G> G> G\ G\ Kc 12*x *x #"c"
(u !4u #6u !4u #u ) G> G> G G\ G\ . 4*x
(4)
Other spatial derivatives are discretized by the second-order central di!erence. For time advancement, the "rst-order Euler implicit scheme is utilized. However, the convective term is treated by linearization as follows: u ) uKuL ) uL>.
(5)
The pressure boundary condition on the solid boundary is the Neumann boundary condition, i.e. *p/*n"0 is imposed. On the surface of the solid boundary, no-slip boundary condition is imposed for each velocity component. Dirichret boundary condition of p"0 is imposed on the remote boundary. For the spanwise direction, periodic boundary condition is imposed for every quantity. Reynolds number is chosen to be Re"; D/l"10 so as to be consistent with the order of the Reynolds number at which wind tunnel experiments are commonly carried out. In order to verify the validity of the present numerical code, in Fig. 3 comparisons are made with respect to the aerodynamic properties of some rectangular cylinders in smooth uniform #ow. At b/d"3.0, where b is the streamwise length of the cross section, a discontinuity in Strouhal number is found to be successfuly simulated by the present calculation.
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
397
Fig. 3. Validation of the present numerical code by rectangular cross sections with various B/D ratios [4}7]. (a) Drag coe$cient, (b) Strouhal number.
4. Results Mean drag coe$cient and RMS drag and lift coe$cients are de"ned as C "D/(qd), C "p /(qd) and C "p /(qd), respectively, where D is the mean drag " " " * * force acting on unit length, p and p are standard deviation of drag and lift force and " * q"0.5o;. Calculations are carried out with vertical and horizontal placement for each cross section. These aerodynamic drag, lift and Strouhal number which are obtained by the present numerical simulation are summarized in Table 1.
398
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
In both vertical and horizontal placement, aerodynamic coe$cients of the E-type cross section are smaller compared with those of R-type section. In Fig. 4, instantaneous vorticity u isosurface contours and time variations of drag and lift forces are X illustrated. In the case of vertical placement, comparison of the magnitudes of the Table 1 Aerodynamic coe$cients C "
Vertical Horizontal
R-type cross section E-type cross section R-type cross section E-type cross section
C
*
Mean
RMS
RMS
2.35 1.94 1.34 0.47
0.80 0.51 0.17 0.09
1.07 0.47 0.96 0.24
St
0.142 0.149 0.094 0.088
Fig. 4. Instantaneous vorticity contour around the cross section and time variation of aerodynamic coe$cients. (a) E-type cross section (vertical) t; /d"320.8. (b) R-type cross section (vertical). (c) E-type cross section (horizontal). (d) R-type cross section (horizontal).
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
399
Fig. 4. Continued.
coe$cients is also evident from the time histories. In particular, the lift #uctuation of the R-type cross section is twice as large as for the E-type cross section. At the placement, vortex is formed just behind the cross section, consequently extremely high drag is recognized instantaneously. Furthermore, in the case of vertical placement, it is noteworthy that both cross sections exhibit remarkable `intermittencya in the aerodynamic forces, i.e. in a certain range of time #uctuation ceases. In particular, E-type cross section exhibits strong intermittency as can be found in its time history (Fig. 4a). Fig. 5 exhibits vorticity contours when the #uctuation ceased. In this mode, a twin-vortex is found to be formed in the wake of the cross section instead of the Karman vortex. Subsequently, the Karman vortex is formed in the downstream apart from the section and gradually approaches the cross section again (Fig. 5b). In consequence, in this mode the base suction recovers and the drag force becomes small (low drag mode). Since the total amount of time of this intermittency is longer in the case of E-type cross section, its mean drag is smaller than for R-type cross section. This phenomenon is typical in rectangular cross sections with relatively
400
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
small b/d ratio [8,9] and some three-dimensional e!ect inherent to this range of cross sections is supposed to be involved with, however, the mechanism of the phenomenon has not been clari"ed in detail yet. In the horizontal placement, remarkable di!erence can be recognized in the mean drag coe$cient. This is because suction appears at the windward portion of the side surfaces as can be found in Fig. 6. In the R-type cross section, suction is also found on the windward surface near the recessed corners. Since the size of the area where suction acts on is narrower compared with E-type cross section, its drag is larger than for E-type cross section. However, because of this suction at leading recessed corners, mean drag is slightly reduced compared with a plain rectangular section with the same b/d ratio of 1.64 (Fig. 3a). In the E-type cross section, #ow separates from the midst of the side surface and therefore, the area which is subject to the #uctuation of the shear layer is limited to the
Fig. 5. Instantaneous vorticity u contour of low drag mode of the E-type cross section. (a) t; /d"355. X (b) t; /d"360.
Fig. 6. Instantaneous pressure contour around the cross sections. (a) E-type cross section. (b) R-type cross section.
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
401
leeward portion. However, in the R-type cross section, its overall side surface is immersed in the separated shear layer. This causes the di!erence in the magnitude of the dynamic lift. From the comparison of aerodynamic drag and lift of the stationary sections, as a result the E-type cross section was concluded to be aerodynamically more favorable than the R-type cross section.
5. Concluding remarks Direct numerical simulation was applied to the prediction of aerodynamic forces acting on the section models of a 200 m tall stack. A modi"ed elliptical cross section and a recessed corner cross section were compared from the aerodynamic viewpoint. The analysis could simulate the strong unsteadiness inherent to each cross section. A primal advantage of this kind of direct simulation is applicability to those cases which exhibit strong unsteadiness. Furthermore, a direct visual recognition of the #ow pattern formed around the cross section is also appreciated since it is di$cult experimentally and it will serve designers and engineers with useful suggestions for aerodynamic design. However, in order to obtain statistically converged data, a su$ciently long range of time trace should be required. For instance in the present calculation, by using a Convex C3840 vectorized parallel computer, it took about at least a couple of weeks to obtain converged results for aerodynamic coe$cients. In this sense, if more quick operation could be available, this kind of strategy could be more e!ective.
References [1] T. Yamamoto, S. Tamamoto, H. Sugimoto, K. Shimada, Wind and earthquake-resistant design for a tall RC Stack, Part 3. Structural design for wind and earthquake for a tall RC Stack with elliptical plan, Proc. AIJ Annual Conference, 1998. [2] K. Hibi, K. Takei, H. Sugimoto, Y. Seko, T. Wakahara, Wind and earthquake-resistant design for a tall RC Stack, Part 4. Evaluation of wind loads by wind tunnel tests for a stack with elliptic plan, Proc. AIJ Annual Conference, 1998. [3] T. Kawamura, K. Kuwahara, Computation of high Reynolds number #ow around circular cylinder with surface roughness, AIAA Paper 84-0340. [4] H. Nakaguchi, K. Hashimoto, S. Muto, An experimental study on aerodynamic drag of rectangular cylinders, J. Jpn. Soc. Aeronaut. Space Sci. 16 (168) (1968) 1}5. [5] H. Sakamoto, H. Haniu, Y. Kobayasi, Fluctuating force acting on rectangular cylinders in uniform #ow (On rectangular cylinders with fully separated #ow), Trans. JSME Ser. B 55 (516) (1989) 2310}2317. [6] T. Tamura, Y. Ito, Three-dimensional simulations of #ow and pressure around an elongated rectangular cylinder, J. Struct. Constr. Eng. AIJ (474) (1995) 41}48. [7] A. Okajima, Flow around a rectangular cylinder with a section of various width/height ratios, J. Wind Eng. (17) (1983) 1}19. [8] A. Roshko, Perspectives on blu! body aerodynamics, J. Wind Eng. Ind. Aerodyn. 49 (1993) 79}100. [9] T. Tamura, Y. Ito, Unstable aerodynamic phenomena of a rectangular cylinder with critical section, Volume of abstracts of the Fourth Asia}Paci"c Symposium on Wind Engineering, 1997, pp. 69}72.
Application of 3D numerical analysis on the prediction of aerodynamic forces of a tall stack cross section Kenji Shimada *, Kunio Takei, Shougo Yamamoto, Hajime Sugimoto Institute of Technology, SHIMIZU Corporation, 3-4-17 Etchujima, Koto-ku, Tokyo 135-8530, Japan Electric Power Development Co. Ltd, 6-15-1 Ginza, Chuo-ku, Tokyo 104-8165, Japan Kaihatsu Architects & Engineers, INC., 6-4-10 Tsukiji, Chuo-ku, Tokyo 104-0045, Japan
Abstract One of the recent design issues concerned with a tall stack is its landscape. Recently, con"gurations other than conventional circles have been often chosen as a cross section. In these cases, wind loads sometimes become critical in the structural design and therefore selection of its con"guration becomes important. In this paper, an attempt was made by applying direct numerical simulation to predict aerodynamic forces acting on the section models of a 200 m tall stack. A modi"ed elliptical cross section and a recessed corner cross section were compared from the aerodynamic viewpoints. The analysis could reproduce the strong unsteadiness inherent to each cross section. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Computational wind engineering; Aerodynamic force; Stack
1. Introduction One of the recent important issues concerned with the designing of a tall stack is its landscape. From environmental point of view, the height of the stack should be as tall as possible. Consequently, its con"guration has to be designed from landscape point of view. A 200 m tall power station stack whose construction is planned in Yokohama
* Corresponding author. E-mail address: [email protected] (K. Shimada) 0167-6105/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 9 ) 0 0 0 3 2 - X
392
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
is designed to have a very slender con"guration to satisfy the above needs [1,2]. In general, when this slender structure is designed, aerodynamic design such as selection of the con"guration of its cross section becomes important for its reasonable structural design. Conventionally, a circular cross section is the most popular choice for a stack cross section. It is partly because of easiness in construction and partly because of convenience in structural design against wind loads since the wind load is not dependent on the wind direction. In this case there also exists some aerodynamical advantages. The Strouhal number is relatively large hence the vortex-induced vibration of the fundamental mode occurs at relatively low windspeed and therefore it is not critical. Furthermore, other kinds of aeroelastic instability such as galloping do not occur. However, if the more architectural oriented design described above had to be chosen, many of the points described above woulds have to be kept in mind. However, recent progress in material and construction methods have enabled the realization of such stack whose cross section is more sensitive to wind. In such a situation, wind tunnel experiment is the most reliable method and is necessary in the "nal stage of its design to assure structural safety against wind loads. On the other hand, it is also an e!ective way to employ an aerodynamically favorable cross section from its early stage of design process. By the way, recent advancement in computational #uid dynamics contributed many bene"ts to wind engineering, although further e!orts should be devoted for the realization as an ideal assessment tool. In wind engineering "eld so far, in particular great success has been achieved in the assessment of pedestrian comfort. However, application to the assessment of aerodynamic forces is not well established. In this paper, an attempt was made by applying direct numerical simulation to predict aerodynamic forces acting on the section models of a 200 m tall stack and its applicability was examined.
2. Con5guration of the stack At the early stage of the present design process, the following two con"gurations are proposed. These con"gurations are illustrated in Fig. 1. One of them is a stack which has a modi"ed elliptical cross section and the other is a stack with a rectangular cross section which has recessed corners in each corner. These sections will be denoted as `E-typea and `R-typea hereafter, respectively. Both of the stacks have tapper along their height and are made of reinforced concrete. Final con"guration of the stack is somewhat di!erent from the E-type cross section and was examined by wind tunnel experiments [2].
3. Numerical procedure The present numerical study is based on the following considerations: (1) Calculations are performed in three dimensions. Since calculations are limited to Reynolds number lower than Re(10 where three dimensionality is not predominant
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
393
and two-dimensional computations can be e!ective, three-dimensional calculation is performed in the present study. (2) The cross section for the analysis is the upper portion of the stacks and is considered as to be a two-dimensional cylinder which is supposed to have an in"nite
Fig. 1. Elevation and cross sections of a stack with E-type cross section. (i) Cross section at the top of the stack; (ii) cross section at the base of the stack. (b). Elevation and cross sections of a stack with R-type cross section.
394
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
Fig. 1. Continued.
length in spanwise direction. The reason why the upper portion is chosen is that the local wind forces acting on the almost upper half portion of the stack make a predominant contribution in the generalized wind force. (3) Approaching #ow is assumed to be smooth uniform #ow. Since the structures are immersed in the atmospheric turbulent boundary layer, free stream turbulence is
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
395
Fig. 2. Numerical grid system: (a) E-type cross section, (b) R-type cross section.
a very important factor for aerodynamic assessment. However, its incorporation has not been established yet and is still in progress. Numerical analysis is based on solving the incompressible time-dependent threedimensional Navier}Stokes equations (1) and (2) by the "nite di!erence method, u #u ) u"! p#Re\u, R ' u"0.
(1) (2)
396
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
In order to evaluate the aerodynamic force as accurately as possible, the #ow structure in the immediate vicinity of the cylinder where steep gradients exist in the physical quantities must be "nely resolved and the behavior of the separated shear layer should be captured as correctly as possible. To achieve this, the generalized curvilinear coordinate system (body "tted coordinate system) is incorporated here to enable to concentrate the su$cient number of grid points near the boundary (Fig. 2). The physical analytical domain is a circle of 30d in radius ;1d in spanwise direction, where d is the cross streamwise length of the section. The cylinder itself is located slightly upstream (!5d ) from the center of the domain. In the present study, a total amount of 240;100;10"240 000 and 320;100;10"320 000 nodes are distributed around the E-type and R-type cross section, respectively. All primitive variables are de"ned on the same grid point. Time-marching algorithm is followed by Marker and Cell method, i.e., the continuum equation is incorporated as solving the Poisson equation for pressure and subsequently the momentum conservation equations are solved for the respective velocity components using the updated pressure, D
p"! ) (u ) u)# , D" ) u. *t
(3)
The third-order upwind scheme proposed by Kawamura and Kuwahara [3] is employed to stabilize the high Reynolds number numerical instability which arises from the non-linear e!ect of the convective term, c
*u (!u #8u !8u #u ) G> G> G\ G\ Kc 12*x *x #"c"
(u !4u #6u !4u #u ) G> G> G G\ G\ . 4*x
(4)
Other spatial derivatives are discretized by the second-order central di!erence. For time advancement, the "rst-order Euler implicit scheme is utilized. However, the convective term is treated by linearization as follows: u ) uKuL ) uL>.
(5)
The pressure boundary condition on the solid boundary is the Neumann boundary condition, i.e. *p/*n"0 is imposed. On the surface of the solid boundary, no-slip boundary condition is imposed for each velocity component. Dirichret boundary condition of p"0 is imposed on the remote boundary. For the spanwise direction, periodic boundary condition is imposed for every quantity. Reynolds number is chosen to be Re"; D/l"10 so as to be consistent with the order of the Reynolds number at which wind tunnel experiments are commonly carried out. In order to verify the validity of the present numerical code, in Fig. 3 comparisons are made with respect to the aerodynamic properties of some rectangular cylinders in smooth uniform #ow. At b/d"3.0, where b is the streamwise length of the cross section, a discontinuity in Strouhal number is found to be successfuly simulated by the present calculation.
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
397
Fig. 3. Validation of the present numerical code by rectangular cross sections with various B/D ratios [4}7]. (a) Drag coe$cient, (b) Strouhal number.
4. Results Mean drag coe$cient and RMS drag and lift coe$cients are de"ned as C "D/(qd), C "p /(qd) and C "p /(qd), respectively, where D is the mean drag " " " * * force acting on unit length, p and p are standard deviation of drag and lift force and " * q"0.5o;. Calculations are carried out with vertical and horizontal placement for each cross section. These aerodynamic drag, lift and Strouhal number which are obtained by the present numerical simulation are summarized in Table 1.
398
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
In both vertical and horizontal placement, aerodynamic coe$cients of the E-type cross section are smaller compared with those of R-type section. In Fig. 4, instantaneous vorticity u isosurface contours and time variations of drag and lift forces are X illustrated. In the case of vertical placement, comparison of the magnitudes of the Table 1 Aerodynamic coe$cients C "
Vertical Horizontal
R-type cross section E-type cross section R-type cross section E-type cross section
C
*
Mean
RMS
RMS
2.35 1.94 1.34 0.47
0.80 0.51 0.17 0.09
1.07 0.47 0.96 0.24
St
0.142 0.149 0.094 0.088
Fig. 4. Instantaneous vorticity contour around the cross section and time variation of aerodynamic coe$cients. (a) E-type cross section (vertical) t; /d"320.8. (b) R-type cross section (vertical). (c) E-type cross section (horizontal). (d) R-type cross section (horizontal).
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
399
Fig. 4. Continued.
coe$cients is also evident from the time histories. In particular, the lift #uctuation of the R-type cross section is twice as large as for the E-type cross section. At the placement, vortex is formed just behind the cross section, consequently extremely high drag is recognized instantaneously. Furthermore, in the case of vertical placement, it is noteworthy that both cross sections exhibit remarkable `intermittencya in the aerodynamic forces, i.e. in a certain range of time #uctuation ceases. In particular, E-type cross section exhibits strong intermittency as can be found in its time history (Fig. 4a). Fig. 5 exhibits vorticity contours when the #uctuation ceased. In this mode, a twin-vortex is found to be formed in the wake of the cross section instead of the Karman vortex. Subsequently, the Karman vortex is formed in the downstream apart from the section and gradually approaches the cross section again (Fig. 5b). In consequence, in this mode the base suction recovers and the drag force becomes small (low drag mode). Since the total amount of time of this intermittency is longer in the case of E-type cross section, its mean drag is smaller than for R-type cross section. This phenomenon is typical in rectangular cross sections with relatively
400
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
small b/d ratio [8,9] and some three-dimensional e!ect inherent to this range of cross sections is supposed to be involved with, however, the mechanism of the phenomenon has not been clari"ed in detail yet. In the horizontal placement, remarkable di!erence can be recognized in the mean drag coe$cient. This is because suction appears at the windward portion of the side surfaces as can be found in Fig. 6. In the R-type cross section, suction is also found on the windward surface near the recessed corners. Since the size of the area where suction acts on is narrower compared with E-type cross section, its drag is larger than for E-type cross section. However, because of this suction at leading recessed corners, mean drag is slightly reduced compared with a plain rectangular section with the same b/d ratio of 1.64 (Fig. 3a). In the E-type cross section, #ow separates from the midst of the side surface and therefore, the area which is subject to the #uctuation of the shear layer is limited to the
Fig. 5. Instantaneous vorticity u contour of low drag mode of the E-type cross section. (a) t; /d"355. X (b) t; /d"360.
Fig. 6. Instantaneous pressure contour around the cross sections. (a) E-type cross section. (b) R-type cross section.
K. Shimada et al. / J. Wind Eng. Ind. Aerodyn. 81 (1999) 391}401
401
leeward portion. However, in the R-type cross section, its overall side surface is immersed in the separated shear layer. This causes the di!erence in the magnitude of the dynamic lift. From the comparison of aerodynamic drag and lift of the stationary sections, as a result the E-type cross section was concluded to be aerodynamically more favorable than the R-type cross section.
5. Concluding remarks Direct numerical simulation was applied to the prediction of aerodynamic forces acting on the section models of a 200 m tall stack. A modi"ed elliptical cross section and a recessed corner cross section were compared from the aerodynamic viewpoint. The analysis could simulate the strong unsteadiness inherent to each cross section. A primal advantage of this kind of direct simulation is applicability to those cases which exhibit strong unsteadiness. Furthermore, a direct visual recognition of the #ow pattern formed around the cross section is also appreciated since it is di$cult experimentally and it will serve designers and engineers with useful suggestions for aerodynamic design. However, in order to obtain statistically converged data, a su$ciently long range of time trace should be required. For instance in the present calculation, by using a Convex C3840 vectorized parallel computer, it took about at least a couple of weeks to obtain converged results for aerodynamic coe$cients. In this sense, if more quick operation could be available, this kind of strategy could be more e!ective.
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