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Numerical study of wind pressures on a Domed roof and near wake flows
Journal of Wind Engineering and Industrial Aerodynamics, 36 ( 1990 ) 1001-1010
1001
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Numerical Study of Wind Pressures on a Domed Roof and Near Wake Flows Tetsuro Tamura*l, Kunio Kuwahara*2 and Masahiro Suzuki*3 ABSTRACT
Unsteady wind flows and pressures around a dome-shaped structure are investigated from the computer simulation. We obtain the three-dimensional numerical solutions by the direct integration based on the finite difference technique of the incompressible NavierStokes equations in the generalized coordinate system. No turbulence models have been incorporated and the third-order upwind scheme for the convection terms has been adopted in the present numerical procedure. In order to confirm the accuracy of the computational results, we have compared the numerically-computed aerodynamic characteristics obtained t h r o u g h the t i m e - a v e r a g i n g with the experimental data. We also bring into focus the pressure distributions acting on a domed roof and the near wake flows. Their fluctuating characters are discussed in detail. INTRODUCTION
Recent advancements of computational speed and storage capacity make it possible to simulate numerically complex flow fields, such as even separated shear flows around a bluff body[Tamura et al., 1987, 1988a]. The range of applications has spread remarkably among various kinds of aerodynamic problems. In wind engineering there also have existed many cases of numerical simulations, for example the wind flow over hills, the strong wind around a building, and the i n t e r n a l air flow in a v e n t i l a t e d room. U n f o r t u n a t e l y most computational examples, due to use of the turbulence transport model, cannot present unsteady characteristics, nevertheless which are essential for the high-Reynolds-number flow. Nowadays it has become increasingly important to clarify dynamic effects of the wind. Because various undesiable incidents, the damages to structures, gust to the pedestrians around tall buildings or wind-induced vibrations to the * 10RI of Shimizu Corporation, 2-2-2 Uchisaiwai-cho, Chiyoda-ku, Tokyo 100, Japan *2 ISAS, 3-1-I Yoshinodai, Sagamihara-shi, Kanagawa 229, Japan *3 ICFD, 1-22-3 Haramachi, Meguro-ku, Tokyo 152, Japan
0167-6105/90/$03.50
© 199{N--Elsevier Science Publishers B.V.
] 002
building occupants, have substantial relation to meticulous phenomena owing to unsteadiness of wind flows. Tamura et a1.[1987] calculated two-dimensional flows around a rectangular cylinder and succeeded in capturing unsteady flow patterns for various shapes. But in three-dimensional(3-D) problems it is not so easy to carry out the time dependent analysis. Since 3-D vortices are inherently stable because of their 3-D dissipations[Tamura et al., 1988b, c]. Therefore even a very little numerical diffusion is liable to make the solution steady. The 3-D flow structures seem to comprise a mechanism more delicate than it has been thought to. At present there are few cases to obtain unsteady solutions of the flow around a bluff body in the 3-D problem. In this p a p e r we c o m p u t e u n s t e a d y 3-D flows a r o u n d a hemisphere, as a typical shape of a dome, situated on the ground surface. The grid system is generated by the hybrid method of the hyperbolic and the parabolic numerical solutions[Nakamura & Suzuki, 1987]. In order to obtain unsteady solutions without much numerical diffusion, the third-order upwind scheme[Kawamura & K uw ahara, 1984] is employed for the n o n l i n e a r convection terms and any turbulence models are not used. The incident flow is assumed to be uniform in order to simplify the results for effective comparison with previous experimental observations. We concentrate on the near wake of the above surface-mounted structures and illuminate the unsteady flow patterns by the computer visualization technique. Concerning the flow around a hemisphere, experimental studies have been carried out by several investigators to the present date. Maher[1965] measured the pressure distributions on a roof in the flow at high Reynolds numbers beyond the critical range. This experiment was conducted in the thin boundary layer( the depth of boundary layer was a half of the height of a hemisphere). Blessmann[1971] carried out the experiment with several wind profiles. Savory and Toy[1986] investigated the flow around a hemisphere immersed in three different boundary layers. This study concentrated on the near wake region from separation to reattachment on the ground surface. In experiment a lot of labor and endurance is required from the viewpoint of systematic investigations for m any parameters and conditions. To date, 3-D unsteady flow patterns almost remains unknown.
PROBLEM FORMULATION Numerical Simulation for Hi qh Reynolds Number Flows In order to overcome the numerical instability in the computation of flows at high Reynolds number, the upwind scheme for finite difference approximation has been widely employed. The first-order upwind scheme has a quite good stability, but it also has a strong diffusive effect similar to the effect of molecular viscosity. As highReynolds-number flows have the very small viscous diffusion, the numerical diffusion is not suitable for our computation. The second-
1003
order upwind scheme is better t h a n first-order one c o n c e r n i n g the numerical diffusive effect, b u t is not stable so m u c h and yield to undesirable propagation of errors. In the case of the third-order upwind scheme leading numerical error terms are the fourth-order-derivative terms. The numerical diffusion of fourth-order derivatives is of short r a n g e and m u s t not conceal the effect of m o l e c u l a r diffusion and stabilizes the computation very well.
Governinq Equations Non-dimensional governing equations, the continuity e q u a t i o n and the Navier-Stokes equations, are presented as follows: div u = 0, ..... (1) a u / a t + u • grad u = - g r a d p + 1 / R e A u, •.... (2) where u, p, t and R e stand for wind velocity vector, wind pressure, time and Reynolds n u m b e r . The Poisson e q u a t i o n for pressure can be derived by operating divergence ofeq. (2). zXp = - d i v ( u - g r a d ) u + R , •.... (3) R =-aD/at+l/ReAD, D =divu, ..... (4) where R should be equal to zero, but is retained as a corrective term in order to prevent the accumulation of numerical errors.
Numerical Scheme The Poisson equation for pressure can be treated on the basis of MAC method[Harlow & welch, 1965]. Second term in the right hand side ofeq. (4) is assumed to be zero. First term is discretized as: aD/at = (D ~+I-D ~)/At, ..... (5) where n is the time step and At is the time increment. D n is not to be equal to zero necessarily by the numerical error. As D -÷~ is assumed to be zero, the final form is obtained as follows: zxp =-div( u " grad )u + D . / At, ..... (6) In the p r e s e n t n u m e r i c a l s i m u l a t i o n , the f i n i t e d e f e r e n c e approximation is performed as follows. All spatial derivatives except those of the convection t e r m s are a p p r o x i m a t e d by c e n t r a l f i n i t e difference. Third-order upwind scheme to the convection t e r m s are presented as follows[Kawamura & K u w a h a r a , 1984]: au
( u -a-x) . ~ =u.~
-- u i + 2 + 8 (
Ui+l--Ui_l)+Ui_ 2 12h
ui+2--4Ui+l+6 lti--4Ui_l+Ui_ 2 +luil
4h
..... (7) where h is grid-scale. The second term in the right hand side of eq.(7) represent the numerical diffusion by a fourth-order derivative. The rid point of wind velocity and pressure is defined at identical place in nite difference mesh systems.
1004
The Poisson equation is solved using the SOR scheme. For the marching Navier-Stokes equation, the semi-implicit scheme which is equivalent to the Euler backward scheme except the n o n l i n e a r convection term is used to integrate temporally. The convection term is linearized as, u " gradu = u n + l • gradun+l -~ u n . g r a d u n + l . •.... (8) The Generalized Coordinate System and Grid Systems
From the practical point of view, we must occasionally deal with objects having curvilinear boundaries. For the wind flow at high Reynolds number, the estimation of the flow close to the boundary is so significant that the numerical analysis using some interpolation cannot yield to a solution with good accuracy. Further it is essentially required that computational discretized points are distributed for each place appropriately enough to express flow fields near the boundary. Then the governing equations have been represented by the generalized coordinate system[Thompson et al., 1985]. Nakamura and Suzuki[1987] proposed the hybrid grid generation scheme by combining the hyperbolic and parabolic type equations linearly. The hybrid scheme generates the grid system which satisfies the orthogonality near the b o u n d a r y and the prescribed outer boundary. Accordingly, the hybrid method is used for the hemisphere problem which has a smoothly changing shape and no sharp corner. COMPUTATIONAL MODEL AND BOUNDARY CONDITIONS
Figure i shows the computational model. A hemisphere-shaped dome is placed on the g r o u n d surface w i t h i n the box-shaped computational domain. The size of the box domain is 26D X 21D X 12.5D (D: the diameter of a hemisphere) and the number of grid points is 1 0 4 X 8 9 X 5 0 = 4 6 2 8 0 0 points. B o u n d a r y conditions a r o u n d the computational domain are treated as follows: The Neumann condition that the normal derivative of pressure equals zero is given. For the wind velocity, the constant value U0 is given at in-flow area and the Neumann condition is made at the other area. On the surface of a hemisphere and the ground, no-slip condition for velocity and Neumann condition for pressure are employed respectively. R e y n o l d s numbers(Re= U o D / v , v:kinematic viscosity) are 2X 103 and 2X 104. The depth of the approaching boundary layer is about a half of the dome height( It is confirmed by computations of the flow on a flat plate. ). VALIDATION OF COMPUTATION Horse-shoe and Arch-shaped Vortex
Figure 2 shows the 3-D streamlines around a hemisphere-shaped dome. In front of a dome the horse-shoe vortex is recognized and the arch-shaped vortex at the back side. These types of vortices are
1005
commonly seen in previous e x p e r i m e n t s . As a whole, g e n e r a l structures of the flow around a surface-mounted structure can be captured by computation.
Effects of Reynolds Number on the Near Wake Figure 3 illustrates the instantaneous pressure distributions on the roof of a hemisphere. The Reynolds-number variations of the flow patterns are investigated. In the case o£ Re=2X104, small and fine structures are shown and asymmetrical flow patterns appear distinctly. Figure 4 depicts instantaneous surface streamlines around a dome. In case of R e = 2 X 1 0 3 , the separation line is straightly formed at a leeward of the center of a dome. At Re= 2 X 104, the separations from the singular points occurs at several places and fine separating patterns due to secondary vortices are recognized in the wake. We confirm the effect of Reynolds number on the wake flow by this numerical method.
Time-Avera,qed Flow Field
-Comparison with Experimental Data-
F i g u r e s 5 ~ 8 p r e s e n t the t i m e - a v e r a g e d flow p a t t e r n s and pressure contours. The evaluation time for average is tUo/D = 15-45. Figure 5 illustrates the time-averaged streamlines. We can decide the separation point on the dome surface and reattachment point on the ground. The separation point is at the a little leeward of the center, X/D = 0.1. The reattachment point, X/D = 1.3, is good agreement with the the experimental result by Savory et a1.[1986]. In front of a hemisphere, the sectional reverse flow patterns of a horse-shoe vortex can be seen. The averaged pressure contours Cp(=p/1/2pUo 2, p:density) on the roof of a hemisphere are shown in Fig. 6. It can be seen that there is a positive value area on the windward face. But the other larger area is subject to suction. Figure 7 shows the contours of fluctuating pressure coefficients Cp'(=~/(p-p)2/1/2pUo 2, P:the averaged pressure). They become larger at a little leeward of the separation point in the wake and near the horse-shoe vortex region. We can also recognize large Cp' region along the separated shear layer. The Cp and Cp' distributions in the vertical section on the longitudinal center-line are presented in Fig. 8. At the stagnation point Cp is equal to 0.8 and the m axi m um negative value is about -0.7 near the top of a hemisphere. These values have some difference from Maher's result[1965] at the supercritical Re number, but are almost similar to the experimental result in the thin boundary layer(D/boundary layer depth = 2.21) by Savory et a1.[1986].
UNSTEADY FLOW STRUCTURES AROUND A HEMISPHERE Time Variations of Flow Field F i g u r e 9 i l l u s t r a t e s time v a r i a t i o n s of the i n s t a n t a n e o u s streamlines on the vertical section in the windward direction. We can
LO06
see vortices successively separated from the dome surface, flowing leeward. These figures have revealed clearly the dynamics and details of the vortex motion, such as merging, in the near wake. Figure 10 shows time variations of the i n s t a n t a n e o u s pressure distributions on a hemisphere. We can see unsteady patterns concerned with the s e p a r a t e d vortices. N e a r the s e p a r a t i o n region, the low negative zone are partially moving leeward and convective phenomena are recognized evidently. Wind Pressures and Wind Forces on a Hemisphere
Time histories of the pressure coefficient Cp is presented in Fig. 11. According to the Cp values at several positions on the roof along the windward direction, convective characteristics are confirmed near the separation point. The Cp values do not have strong inverse correlations on the opposite sides in the cross-wind direction. Figure 12 shows the time histories of the drag coefficient CD, the lift coefficient CL and the side force coefficient CS. These values are changing temporally but not so intensively. The averaged values of CD and CL equal about 0.40 and 0.33. These are similar to previous experimental results in the subcritical regime(Re = 5 × 104, D/boundary layer depth = 2.30) by Taniguchi et a1.[1982]. 3-D Flow Structures in the Wake
Figure 13 presents the instantaneous streamlines on the vertical section in the transverse direction. There exist longitudinal vortices in the wake. The 3-D structures have appeared at immediate leeward from the separation line. CONCLUSION
The unsteady flow past a hemisphere at high Reynolds numbers is analysed by direct integration of the three-dimensional Navier-Stokes equations in the generalized c u r v i l i n e a r coordinate system. It is confirmed t h a t the numerical results presented here agree with the previous experimental results about a v e r a g e d data. We h a v e also obtained new information concerning 3-D unsteady flow patterns with the separation for wind engineering related problems. REFERENCES
Blessmann, J., "Pressures on Domes with Several Wind Profiles", Proc. Wind Effects on Buildings and Structures, 1971 ttarlow, F. It. and Welch, J. E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface",Phys. Fluids, Vol. 8, 1965 Kawamura, T. and Kuwahara, K., "Computation of High Reynolds Number Flow around a Circular Cylinder with Surface Roughness", AIAA Paper, 84-0340, 1984
1007
Maher, F. J., "Wind Loads on Basic Dome Shapes", J. Structural Div., Proc. ASCE, Vol. 91, ST3, 1965 Nakamura, S. and Suzuki, M., "Non-iterative Three-dimensionalGrid Generation Using a Parabolic-Hyperbolic Hybrid Scheme", AIAA Paper, 87-0277, 1987 Savory, E. and Toy, N., "Hemispheres and Hemisphere-cylinders in Turbulent Boundary Layers", J. Wind Engineering, Vol. 23, 1986 Tamura, T.,Kuwahara, K. and Shirayama, S., "Numerical Study of Unsteady Flow Patterns and Pressure Distributions on a Rectangular Cylinder", 7th International Conference on Wind Engineering, 1987 Tamura, T., Tsuboi, K. and Kuwahara, K., "Numerical Simulation of Unseady Flow Patterns around a Vibrating Circular Cylinder", AIAA Paper, 88-0128, 1988a Tamura, T., Krause E., Shirayama, E. Ishii, K. and Kuwahara, K., "Threedimensional Computation of Unsteady Flows around a Square Cylinder", 11th International Conference on Numerical Methods in Fluid Dynamics, 1988b Tamura, T. and Kuwahara, K., "Numerical Study of Aerodynamic Behavior of a Square Cylinder", International Colloquium on Bluff Body Aerodynamics and Its Applications, 1988c Taniguchi, S., Sakamoto, H., Kiya, M. and Arie, M., "Time-averaged Aerodynamic Forces Acting on a Hemisphere Immersed in a Turbulent BoundaryLayer", J. Wind Engineering, Vol. 9, 1982 Thompson, J. E., Warsi, Z. U. A. and Mastin, C. W., Numerical Grid Generation; Foundation and Application, North-Holland, 1985
Fig. 2 The 3-D flow patterns around a hemisphere-shaped dome Fig. 1 Grid systems for a hemisphere (Re = 2 x 103). on the g r o u n d ( 1 0 4 x 8 9 x 50 points).
Fig. 3
(a) Re = 2 x 1 0 3 (b) Re = 2 x 1 0 4 Effect of Reynolds number on instantaneous pressure distributions.
I(){)~
(a) Re=2x103 (b) Re=2x104 Fig. 4 Effect of Reynolds number on instantaneous surface streamlines. -
-
,
~,~
(a)
Vertical section
] i ~
" ~
(b)
Surface of a hemisphere # and the ground. Fig. 5 Time-averaged streamlines(Re=2x104)
" ~
Exf A
1.0~
o.o 10
',-. - ~ ~ ' ~ - - - - ~ = 30
'
60
~
90
~
120
150
(b) Surface of a hemisphere Fig. 7 Contours of fluctuating pressu re coefficients Fig. 8 Pressuredistributions along the windward center-line of a hemisphere(Re = 2 x 104).
1009
~..~"~---~"'-~"--.-~U 0/D=21.7 5
•
{~:,
=22.50 Fig. 9 Time variations o f streamlines Fig. 10 Time variations o pressure contours on a hemisphere on t h e w i n d w a r d vertical section (Re -- 2 x 104). (Re = 2 x I04).
lOlO
Z
Surface
1='50
i =50 ;"
iI (a)
A(01=122),B(01=128),C(01=139°)Point
~o ~.
(b)
.
.
. ~o .
~o,o
° ~
(a)
Surface(l=50
~
C(01 = 139°), D(01 = 148°),E(01 = 158 °) Point
(b)
Surface(I = 60)
0.0 -1.0 S0 30 tUoID 40 (c) F(02 = 36°), G(02 = 144 °) Point Fig. 11 Time histories of pressure coefficients (c) Surface(l =70) (Re = 2 x 104). Fig. 13 Longitudinal vortices LIFt ANO ~ A G in the wake(Re = 2 x 104).
20
30
tUo/D
40
CD = Dragl( ll2pUo 2 uD2/8) CL = Lift/( ll2pUo 2 uD2/4) Cs = Side force/(1/2pUo 2 uD2/8) Fig. 12 Time histories of w i n d force(Re = 2 x 104).
1001
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Numerical Study of Wind Pressures on a Domed Roof and Near Wake Flows Tetsuro Tamura*l, Kunio Kuwahara*2 and Masahiro Suzuki*3 ABSTRACT
Unsteady wind flows and pressures around a dome-shaped structure are investigated from the computer simulation. We obtain the three-dimensional numerical solutions by the direct integration based on the finite difference technique of the incompressible NavierStokes equations in the generalized coordinate system. No turbulence models have been incorporated and the third-order upwind scheme for the convection terms has been adopted in the present numerical procedure. In order to confirm the accuracy of the computational results, we have compared the numerically-computed aerodynamic characteristics obtained t h r o u g h the t i m e - a v e r a g i n g with the experimental data. We also bring into focus the pressure distributions acting on a domed roof and the near wake flows. Their fluctuating characters are discussed in detail. INTRODUCTION
Recent advancements of computational speed and storage capacity make it possible to simulate numerically complex flow fields, such as even separated shear flows around a bluff body[Tamura et al., 1987, 1988a]. The range of applications has spread remarkably among various kinds of aerodynamic problems. In wind engineering there also have existed many cases of numerical simulations, for example the wind flow over hills, the strong wind around a building, and the i n t e r n a l air flow in a v e n t i l a t e d room. U n f o r t u n a t e l y most computational examples, due to use of the turbulence transport model, cannot present unsteady characteristics, nevertheless which are essential for the high-Reynolds-number flow. Nowadays it has become increasingly important to clarify dynamic effects of the wind. Because various undesiable incidents, the damages to structures, gust to the pedestrians around tall buildings or wind-induced vibrations to the * 10RI of Shimizu Corporation, 2-2-2 Uchisaiwai-cho, Chiyoda-ku, Tokyo 100, Japan *2 ISAS, 3-1-I Yoshinodai, Sagamihara-shi, Kanagawa 229, Japan *3 ICFD, 1-22-3 Haramachi, Meguro-ku, Tokyo 152, Japan
0167-6105/90/$03.50
© 199{N--Elsevier Science Publishers B.V.
] 002
building occupants, have substantial relation to meticulous phenomena owing to unsteadiness of wind flows. Tamura et a1.[1987] calculated two-dimensional flows around a rectangular cylinder and succeeded in capturing unsteady flow patterns for various shapes. But in three-dimensional(3-D) problems it is not so easy to carry out the time dependent analysis. Since 3-D vortices are inherently stable because of their 3-D dissipations[Tamura et al., 1988b, c]. Therefore even a very little numerical diffusion is liable to make the solution steady. The 3-D flow structures seem to comprise a mechanism more delicate than it has been thought to. At present there are few cases to obtain unsteady solutions of the flow around a bluff body in the 3-D problem. In this p a p e r we c o m p u t e u n s t e a d y 3-D flows a r o u n d a hemisphere, as a typical shape of a dome, situated on the ground surface. The grid system is generated by the hybrid method of the hyperbolic and the parabolic numerical solutions[Nakamura & Suzuki, 1987]. In order to obtain unsteady solutions without much numerical diffusion, the third-order upwind scheme[Kawamura & K uw ahara, 1984] is employed for the n o n l i n e a r convection terms and any turbulence models are not used. The incident flow is assumed to be uniform in order to simplify the results for effective comparison with previous experimental observations. We concentrate on the near wake of the above surface-mounted structures and illuminate the unsteady flow patterns by the computer visualization technique. Concerning the flow around a hemisphere, experimental studies have been carried out by several investigators to the present date. Maher[1965] measured the pressure distributions on a roof in the flow at high Reynolds numbers beyond the critical range. This experiment was conducted in the thin boundary layer( the depth of boundary layer was a half of the height of a hemisphere). Blessmann[1971] carried out the experiment with several wind profiles. Savory and Toy[1986] investigated the flow around a hemisphere immersed in three different boundary layers. This study concentrated on the near wake region from separation to reattachment on the ground surface. In experiment a lot of labor and endurance is required from the viewpoint of systematic investigations for m any parameters and conditions. To date, 3-D unsteady flow patterns almost remains unknown.
PROBLEM FORMULATION Numerical Simulation for Hi qh Reynolds Number Flows In order to overcome the numerical instability in the computation of flows at high Reynolds number, the upwind scheme for finite difference approximation has been widely employed. The first-order upwind scheme has a quite good stability, but it also has a strong diffusive effect similar to the effect of molecular viscosity. As highReynolds-number flows have the very small viscous diffusion, the numerical diffusion is not suitable for our computation. The second-
1003
order upwind scheme is better t h a n first-order one c o n c e r n i n g the numerical diffusive effect, b u t is not stable so m u c h and yield to undesirable propagation of errors. In the case of the third-order upwind scheme leading numerical error terms are the fourth-order-derivative terms. The numerical diffusion of fourth-order derivatives is of short r a n g e and m u s t not conceal the effect of m o l e c u l a r diffusion and stabilizes the computation very well.
Governinq Equations Non-dimensional governing equations, the continuity e q u a t i o n and the Navier-Stokes equations, are presented as follows: div u = 0, ..... (1) a u / a t + u • grad u = - g r a d p + 1 / R e A u, •.... (2) where u, p, t and R e stand for wind velocity vector, wind pressure, time and Reynolds n u m b e r . The Poisson e q u a t i o n for pressure can be derived by operating divergence ofeq. (2). zXp = - d i v ( u - g r a d ) u + R , •.... (3) R =-aD/at+l/ReAD, D =divu, ..... (4) where R should be equal to zero, but is retained as a corrective term in order to prevent the accumulation of numerical errors.
Numerical Scheme The Poisson equation for pressure can be treated on the basis of MAC method[Harlow & welch, 1965]. Second term in the right hand side ofeq. (4) is assumed to be zero. First term is discretized as: aD/at = (D ~+I-D ~)/At, ..... (5) where n is the time step and At is the time increment. D n is not to be equal to zero necessarily by the numerical error. As D -÷~ is assumed to be zero, the final form is obtained as follows: zxp =-div( u " grad )u + D . / At, ..... (6) In the p r e s e n t n u m e r i c a l s i m u l a t i o n , the f i n i t e d e f e r e n c e approximation is performed as follows. All spatial derivatives except those of the convection t e r m s are a p p r o x i m a t e d by c e n t r a l f i n i t e difference. Third-order upwind scheme to the convection t e r m s are presented as follows[Kawamura & K u w a h a r a , 1984]: au
( u -a-x) . ~ =u.~
-- u i + 2 + 8 (
Ui+l--Ui_l)+Ui_ 2 12h
ui+2--4Ui+l+6 lti--4Ui_l+Ui_ 2 +luil
4h
..... (7) where h is grid-scale. The second term in the right hand side of eq.(7) represent the numerical diffusion by a fourth-order derivative. The rid point of wind velocity and pressure is defined at identical place in nite difference mesh systems.
1004
The Poisson equation is solved using the SOR scheme. For the marching Navier-Stokes equation, the semi-implicit scheme which is equivalent to the Euler backward scheme except the n o n l i n e a r convection term is used to integrate temporally. The convection term is linearized as, u " gradu = u n + l • gradun+l -~ u n . g r a d u n + l . •.... (8) The Generalized Coordinate System and Grid Systems
From the practical point of view, we must occasionally deal with objects having curvilinear boundaries. For the wind flow at high Reynolds number, the estimation of the flow close to the boundary is so significant that the numerical analysis using some interpolation cannot yield to a solution with good accuracy. Further it is essentially required that computational discretized points are distributed for each place appropriately enough to express flow fields near the boundary. Then the governing equations have been represented by the generalized coordinate system[Thompson et al., 1985]. Nakamura and Suzuki[1987] proposed the hybrid grid generation scheme by combining the hyperbolic and parabolic type equations linearly. The hybrid scheme generates the grid system which satisfies the orthogonality near the b o u n d a r y and the prescribed outer boundary. Accordingly, the hybrid method is used for the hemisphere problem which has a smoothly changing shape and no sharp corner. COMPUTATIONAL MODEL AND BOUNDARY CONDITIONS
Figure i shows the computational model. A hemisphere-shaped dome is placed on the g r o u n d surface w i t h i n the box-shaped computational domain. The size of the box domain is 26D X 21D X 12.5D (D: the diameter of a hemisphere) and the number of grid points is 1 0 4 X 8 9 X 5 0 = 4 6 2 8 0 0 points. B o u n d a r y conditions a r o u n d the computational domain are treated as follows: The Neumann condition that the normal derivative of pressure equals zero is given. For the wind velocity, the constant value U0 is given at in-flow area and the Neumann condition is made at the other area. On the surface of a hemisphere and the ground, no-slip condition for velocity and Neumann condition for pressure are employed respectively. R e y n o l d s numbers(Re= U o D / v , v:kinematic viscosity) are 2X 103 and 2X 104. The depth of the approaching boundary layer is about a half of the dome height( It is confirmed by computations of the flow on a flat plate. ). VALIDATION OF COMPUTATION Horse-shoe and Arch-shaped Vortex
Figure 2 shows the 3-D streamlines around a hemisphere-shaped dome. In front of a dome the horse-shoe vortex is recognized and the arch-shaped vortex at the back side. These types of vortices are
1005
commonly seen in previous e x p e r i m e n t s . As a whole, g e n e r a l structures of the flow around a surface-mounted structure can be captured by computation.
Effects of Reynolds Number on the Near Wake Figure 3 illustrates the instantaneous pressure distributions on the roof of a hemisphere. The Reynolds-number variations of the flow patterns are investigated. In the case o£ Re=2X104, small and fine structures are shown and asymmetrical flow patterns appear distinctly. Figure 4 depicts instantaneous surface streamlines around a dome. In case of R e = 2 X 1 0 3 , the separation line is straightly formed at a leeward of the center of a dome. At Re= 2 X 104, the separations from the singular points occurs at several places and fine separating patterns due to secondary vortices are recognized in the wake. We confirm the effect of Reynolds number on the wake flow by this numerical method.
Time-Avera,qed Flow Field
-Comparison with Experimental Data-
F i g u r e s 5 ~ 8 p r e s e n t the t i m e - a v e r a g e d flow p a t t e r n s and pressure contours. The evaluation time for average is tUo/D = 15-45. Figure 5 illustrates the time-averaged streamlines. We can decide the separation point on the dome surface and reattachment point on the ground. The separation point is at the a little leeward of the center, X/D = 0.1. The reattachment point, X/D = 1.3, is good agreement with the the experimental result by Savory et a1.[1986]. In front of a hemisphere, the sectional reverse flow patterns of a horse-shoe vortex can be seen. The averaged pressure contours Cp(=p/1/2pUo 2, p:density) on the roof of a hemisphere are shown in Fig. 6. It can be seen that there is a positive value area on the windward face. But the other larger area is subject to suction. Figure 7 shows the contours of fluctuating pressure coefficients Cp'(=~/(p-p)2/1/2pUo 2, P:the averaged pressure). They become larger at a little leeward of the separation point in the wake and near the horse-shoe vortex region. We can also recognize large Cp' region along the separated shear layer. The Cp and Cp' distributions in the vertical section on the longitudinal center-line are presented in Fig. 8. At the stagnation point Cp is equal to 0.8 and the m axi m um negative value is about -0.7 near the top of a hemisphere. These values have some difference from Maher's result[1965] at the supercritical Re number, but are almost similar to the experimental result in the thin boundary layer(D/boundary layer depth = 2.21) by Savory et a1.[1986].
UNSTEADY FLOW STRUCTURES AROUND A HEMISPHERE Time Variations of Flow Field F i g u r e 9 i l l u s t r a t e s time v a r i a t i o n s of the i n s t a n t a n e o u s streamlines on the vertical section in the windward direction. We can
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see vortices successively separated from the dome surface, flowing leeward. These figures have revealed clearly the dynamics and details of the vortex motion, such as merging, in the near wake. Figure 10 shows time variations of the i n s t a n t a n e o u s pressure distributions on a hemisphere. We can see unsteady patterns concerned with the s e p a r a t e d vortices. N e a r the s e p a r a t i o n region, the low negative zone are partially moving leeward and convective phenomena are recognized evidently. Wind Pressures and Wind Forces on a Hemisphere
Time histories of the pressure coefficient Cp is presented in Fig. 11. According to the Cp values at several positions on the roof along the windward direction, convective characteristics are confirmed near the separation point. The Cp values do not have strong inverse correlations on the opposite sides in the cross-wind direction. Figure 12 shows the time histories of the drag coefficient CD, the lift coefficient CL and the side force coefficient CS. These values are changing temporally but not so intensively. The averaged values of CD and CL equal about 0.40 and 0.33. These are similar to previous experimental results in the subcritical regime(Re = 5 × 104, D/boundary layer depth = 2.30) by Taniguchi et a1.[1982]. 3-D Flow Structures in the Wake
Figure 13 presents the instantaneous streamlines on the vertical section in the transverse direction. There exist longitudinal vortices in the wake. The 3-D structures have appeared at immediate leeward from the separation line. CONCLUSION
The unsteady flow past a hemisphere at high Reynolds numbers is analysed by direct integration of the three-dimensional Navier-Stokes equations in the generalized c u r v i l i n e a r coordinate system. It is confirmed t h a t the numerical results presented here agree with the previous experimental results about a v e r a g e d data. We h a v e also obtained new information concerning 3-D unsteady flow patterns with the separation for wind engineering related problems. REFERENCES
Blessmann, J., "Pressures on Domes with Several Wind Profiles", Proc. Wind Effects on Buildings and Structures, 1971 ttarlow, F. It. and Welch, J. E., "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface",Phys. Fluids, Vol. 8, 1965 Kawamura, T. and Kuwahara, K., "Computation of High Reynolds Number Flow around a Circular Cylinder with Surface Roughness", AIAA Paper, 84-0340, 1984
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Maher, F. J., "Wind Loads on Basic Dome Shapes", J. Structural Div., Proc. ASCE, Vol. 91, ST3, 1965 Nakamura, S. and Suzuki, M., "Non-iterative Three-dimensionalGrid Generation Using a Parabolic-Hyperbolic Hybrid Scheme", AIAA Paper, 87-0277, 1987 Savory, E. and Toy, N., "Hemispheres and Hemisphere-cylinders in Turbulent Boundary Layers", J. Wind Engineering, Vol. 23, 1986 Tamura, T.,Kuwahara, K. and Shirayama, S., "Numerical Study of Unsteady Flow Patterns and Pressure Distributions on a Rectangular Cylinder", 7th International Conference on Wind Engineering, 1987 Tamura, T., Tsuboi, K. and Kuwahara, K., "Numerical Simulation of Unseady Flow Patterns around a Vibrating Circular Cylinder", AIAA Paper, 88-0128, 1988a Tamura, T., Krause E., Shirayama, E. Ishii, K. and Kuwahara, K., "Threedimensional Computation of Unsteady Flows around a Square Cylinder", 11th International Conference on Numerical Methods in Fluid Dynamics, 1988b Tamura, T. and Kuwahara, K., "Numerical Study of Aerodynamic Behavior of a Square Cylinder", International Colloquium on Bluff Body Aerodynamics and Its Applications, 1988c Taniguchi, S., Sakamoto, H., Kiya, M. and Arie, M., "Time-averaged Aerodynamic Forces Acting on a Hemisphere Immersed in a Turbulent BoundaryLayer", J. Wind Engineering, Vol. 9, 1982 Thompson, J. E., Warsi, Z. U. A. and Mastin, C. W., Numerical Grid Generation; Foundation and Application, North-Holland, 1985
Fig. 2 The 3-D flow patterns around a hemisphere-shaped dome Fig. 1 Grid systems for a hemisphere (Re = 2 x 103). on the g r o u n d ( 1 0 4 x 8 9 x 50 points).
Fig. 3
(a) Re = 2 x 1 0 3 (b) Re = 2 x 1 0 4 Effect of Reynolds number on instantaneous pressure distributions.
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(a) Re=2x103 (b) Re=2x104 Fig. 4 Effect of Reynolds number on instantaneous surface streamlines. -
-
,
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(a)
Vertical section
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(b)
Surface of a hemisphere # and the ground. Fig. 5 Time-averaged streamlines(Re=2x104)
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1.0~
o.o 10
',-. - ~ ~ ' ~ - - - - ~ = 30
'
60
~
90
~
120
150
(b) Surface of a hemisphere Fig. 7 Contours of fluctuating pressu re coefficients Fig. 8 Pressuredistributions along the windward center-line of a hemisphere(Re = 2 x 104).
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~..~"~---~"'-~"--.-~U 0/D=21.7 5
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{~:,
=22.50 Fig. 9 Time variations o f streamlines Fig. 10 Time variations o pressure contours on a hemisphere on t h e w i n d w a r d vertical section (Re -- 2 x 104). (Re = 2 x I04).
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Z
Surface
1='50
i =50 ;"
iI (a)
A(01=122),B(01=128),C(01=139°)Point
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(b)
.
.
. ~o .
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° ~
(a)
Surface(l=50
~
C(01 = 139°), D(01 = 148°),E(01 = 158 °) Point
(b)
Surface(I = 60)
0.0 -1.0 S0 30 tUoID 40 (c) F(02 = 36°), G(02 = 144 °) Point Fig. 11 Time histories of pressure coefficients (c) Surface(l =70) (Re = 2 x 104). Fig. 13 Longitudinal vortices LIFt ANO ~ A G in the wake(Re = 2 x 104).
20
30
tUo/D
40
CD = Dragl( ll2pUo 2 uD2/8) CL = Lift/( ll2pUo 2 uD2/4) Cs = Side force/(1/2pUo 2 uD2/8) Fig. 12 Time histories of w i n d force(Re = 2 x 104).