- Email: [email protected]
Flow around a cylindrical structure mounted in a plane turbulent boundary layer
J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Contents lists available at SciVerse ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Flow around a cylindrical structure mounted in a plane turbulent boundary layer Takayuki Tsutsui n Department of Mechanical Engineering, The National Defense Academy, 1-10-20 Hashirimizu, Yokosuka, Kanagawa 239-8686, Japan
a r t i c l e i n f o
abstract
Available online 24 May 2012
The wind force acting on a low-aspect-ratio (height to diameter) cylindrical structure placed in a turbulent boundary layer has been investigated. The diameters of the cylindrical structure models were D ¼40 and 80 mm, and the Reynolds numbers based on D were varied from 1.1 104 and 1.1 105. The turbulent boundary layer thickness at which the cylindrical structure was placed varied from 26–120 mm. The aspect ratio was varied from 0.125–1.0. Flow visualizations were performed based on the surface oil-flow pattern method. The surface pressure distributions on the models were measured, and the drag and lift coefficients were determined by integration of the surface pressure distributions. The correlations between the fluid force, the turbulent boundary layer thickness, and the aspect ratio of the cylindrical structure were clarified. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Cylindrical structure Three-dimensional flow Boundary layer Horseshoe vortex Flow visualization Pressure distribution Fluid force
1. Introduction Cylindrical structures with low aspect ratios (height/diameter, H/D r1) are used in various engineering applications, such as oil storage tanks for large structures, silos, and small parts of electrical equipment. However, few studies have examined fluid flow around low-aspect-ratio cylinders (Baker, 1980; Okamoto and Sunabashiri, 1992; Portela and Godoy, 2007; Sabransky, 1987; Sakamoto and Arie, 1983; Tsutsui and Kawahara, 2006; Tsutsui et al., 2000). Fig. 1 shows the experimental Reynolds numbers Re and H/D ranges for low-aspect-ratio cylinders reported in previous papers. Baker (1980) investigated the horseshoe vortex system around the base of a circular cylinder in a turbulent boundary layer for a wide range of aspect ratio. The variations of the position of the horseshoe vortex and the boundary layer separation were determined experimentally as a function of the main flow parameters. Okamoto and Sunabashiri (1992) and Sakamoto and Arie (1983) found that the flow pattern changes dramatically between H/D ¼2 and 4, while the shedding vortices change from the Karman type to the arch type for low-aspect-ratio cylindrical structures. The heat transfer characteristics of low-aspect-ratio (H/D r1) cylindrical structures were reported in previous studies by the present author (Tsutsui and Kawahara, 2006; Tsutsui et al., 2000), in which the top face (roof) of the cylinder was shown to have a
n
Tel.: þ81 46 841 3810x3412; fax: þ 81 46 844 5900. E-mail address: [email protected]
0167-6105/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jweia.2012.05.003
greater influence on the overall fluid flow and heat transfer characteristics of the cylindrical structure. Sabransky (1987) and Poltera and Godoy (2007) reported the pressure force acting on tanks and silos with conical roofs. They reported the wind load acting on structures under systematic changes in aspect ratio, number of structures, and arrangement of structures. Because the scope of these studies is limited, useful pressure load data for low-aspect-ratio cylindrical structures is needed for general use in engineering fields. The present paper examines the flow around a cylindrical structure and the fluid force acting on the structure in order to obtain strength design data for lowaspect-ratio cylindrical structures. The present paper clarifies the correlations among the drag, lift, aspect ratio, and turbulent boundary layer thickness. The pressure distribution on the cylindrical structure was measured under various Reynolds numbers, aspect ratios, and turbulent boundary layer thicknesses, and the time-averaged surface flow patterns were clarified by the oilfilm visualization method.
2. Experimental apparatus and method The flow geometry and coordinate system for the experiment are shown in Fig. 2(a). The diameters of the cylindrical structure D were 40 and 80 mm, and the aspect ratios H/D ( ¼height/ diameter) of the models were varied from 0.125–1.0. The experiments were performed in a low-speed wind tunnel with a working section having a height of 400 mm, a width of 300 mm, and a length of 1400 mm. The free stream velocity U0 was varied
240
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Nomenclature CD CL Cp D Drag dt H L Lift
Dag coefficient Lift coefficient Pressure coefficient¼ ðpp0 Þ=0:5 rU 20 Diameter of the cylindrical structure Drag Diameter of the trip wire Height of the cylindrical structure Length (basic dimension) Lift
between 4 and 20 m/s, resulting in Reynolds numbers Re in the range of 1.1 104–1.1 105. In this range, the turbulent intensity was approximately 0.4% for the free stream velocity. The turbulent boundary layer was induced by attaching a 5-, 10-, or 12 mm diameter trip wire (T.W.) to the plane at a position 800 mm upstream from the model. The boundary layer thickness d ranged from 26–120 mm at the center position of the model. Fig. 2(b) shows the variation of the normalized velocity, u þ , in the turbulent boundary layer as a function of the normalized height, z þ , for three different trip wire diameters and without trip wire for U0 ¼20 m/s. The data agree well with the experimental universal velocity distribution of the turbulent boundary layer on a flat-plate at zero pressure gradient (Schlichting, 1968) and can be expressed as: u ¼ un 5:85logðun z=nÞ þ5:56 ð1Þ
M p, p0 Re T U0, u un X, Y, Z
d
n, r t0 j
Mass (basic dimension) Pressure, static pressure Reynolds number ¼U0D/v Time (basic dimension) Free stream velocity, velocity ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Friction velocity¼ t0 =r ¼ 0:0225U 20 ðn=U 0 dÞ Direction Boundary layer thickness Kinematic viscosity, density of the fluid Shearing stress at the plane Angle
The oil film is white oil mixed with oleic acid, titanium oxide, and liquid paraffin. The surface of the model was painted black in order to provide the highest possible contrast with the white oil film.
3. Results and discussion 3.1. Flow visualization Fig. 4 describes the typical surface oil flow patterns on the cylindrical structure and flow schematic. On the top face, a steady separation bubble (S.B.), similar to a lunette, is evident just
Fig. 3(a) shows the 80 mm-diameter pressure measurement model. The model was fabricated from acrylic resin pipes, which have many pressure taps in a spiral arrangement at ten-degree intervals along the surface of the model to avoid influence of nearness of each taps. The model set up and the pressure measurement system are shown in Fig. 3(b) and (c). The aspect ratio can be changed by changing the height of the model. The surface pressures were measured by the pressure measurement system, which is the Net scanner system 9816/98RK-1(PSI, Inc.). All data were measured for 1 min in 100 Hz sampling rate, and were averaged. The measure distribution on the entire surface of the cylindrical structure can be determined by rotating the model on its axis. The same methods were applied for the 40 mm-diameter model. The pressure on the surface of the cylindrical structure was measured with respect to Reynolds number, aspect ratio, and turbulent boundary layer thickness. The fluid forces acting on the cylindrical structure were then obtained by integrating the pressure distribution. The surface flow visualization of the cylindrical structure and plane was performed using an oil-film method in order to allow for the observation of the time averaged flow behavior around the structure.
Fig. 1. Experimental Reynolds number ranges of previous studies.
Fig. 2. Flow geometry, coordinate system and turbulent boundary layer: (a) flow geometry, coordinate system and (b) variation of u þ in the turbulent boundary layer as a function of z þ .
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
241
Fig. 3. Experimental model, model setup and measurement system: (a) model (D ¼80 mm), (b) model setup and (c) measurement system.
behind the leading edge and an arc-shaped reattachment line (R.) is present behind the center cord. A reverse flow region (R.F.) forms between the lunette and the arc. A down wash (D.W.) exists
from the side edge of the top face to the rear face, a reattachment flow spread from the rear stagnation point (R.St.) and a horseshoe vortex (H.V.) is evident on the floor.
242
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 4. Flow around the cylindrical structure (D ¼ 80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, H/D ¼0.5, d ¼ 52 mm) (Tsutsui and Kawahara, 2006): (a) front view, (b) rear view and (c) flow schematic.
The surface oil flow patterns of the cylindrical structure with various aspect ratios are shown in Fig. 5. The oncoming flow separates in front of the cylindrical structure. The separated shear layer roles upto form a horseshoe vortex on the plane. The horseshoe vortex swept away the white oil film on the plane. The swept area became gradually small with decreasing the aspect ratio. The change of the oil flow pattern on the top face is plain. The reattachment line moves forward when the aspect ratio H/D is decreased. Because of this, the reverse flow region moves forward and is reduced. In addition, the separation bubble gradually shrinks in (d) and (e), at last, it disappeared in (f). On the front face, a line of white oil film remains on the front face in (a) and (b), and in (c)–(f), rather than lines, nodes appear, which indicate the front stagnation point. The front stagnation on the front face and the separation line on the side face exhibit two-dimensional characteristics, except for the upper and lower regions in (a) and (b). Decreasing the aspect ratio gradually reduces the amount of two-dimensional flow. On the rear face, the flow spreads radially from the rear stagnation point in (a)–(c). In (d)–(f), the radial spread patterns are not plain, the reattachment flow on the back of the cylindrical structure is weak.
the negative pressure area moves forward, and its area on the top face decreases. On the side of the front face, the higher pressure area (Cp Z0.8), which is the front stagnation point, is reduced with decrease in the aspect ratio. Fig. 7 shows the effect of the turbulent boundary layer thickness on the pressure coefficients. On the top face, the lower pressure area (Cp Z 1.0) moves forward and the higher pressure area (Cp Z 0.2) is extended with increasing the boundary layer thickness. On the side face, the higher pressure area (Cp Z0.8) at the front stagnation point becomes smaller when the boundary layer thickness increases. Comparison of Figs. 6 and 7 reveals that the aspect ratio affects the pressure distributions more than the boundary layer thickness. A previous study (Tsutsui and Kawahara, 2006) reported that the reattachment point on the top face depends on the velocity gradient du/dz. Decreasing height H and increasing the turbulent boundary layer thickness d are causes of the increased velocity gradient du/dz. The change in H was found to have a greater effect on du/dz than the change in d.
3.2. Pressure distribution
Fluid forces acting on the cylindrical structure were obtained by integrating the pressure distribution. To obtain the total drag coefficient of the cylindrical structure CD, the local drag coefficient CDz was calculated in advance, as follows: Z 1 2p C pz cos jdj ð2Þ C Dz ¼ 2 0
The pressure coefficient distributions on the center line and the top and side face contours are presented in Fig. 6. The reattachment point (R.), the reverse flow region (R.F.), and the separation bubble region (S.B.) are shown on the pressure distribution. Cpfz and Cpbz are the pressure coefficients on the front and rear faces at height z. CDz is the local drag coefficient at height z, as described in the following paragraph. The effect of the aspect ratio appears clearly on the top face. On the top face, the pressure decreases significantly in the area forward from the reattach point (R.). The minimum pressure area lies at the border between the reverse flow region (R.F.) and the separation bubble region (S.B.). Downward from the reattach point (R.), the change in the pressure is smooth. The reattachment point (R.) moves forward with decreasing aspect ratio. Therefore,
3.3. Fluid force
where Cpz is the pressure coefficient at height z. Then, CD is obtained as follows: Z 1 z ð3Þ CD ¼ C Dz d H 0 Next, the lift coefficient CL is considered. The lift force acting on the top face Lift is given by Lif t ¼ ðpp0 ÞpD2 =4
ð4Þ
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
243
Fig. 5. Surface oil-flow patterns of the cylindrical structure (D¼ 80 mm, U0 ¼20 m/s, Re ¼1.1 105, d ¼52 mm): (a) H/D ¼1.0, (b) H/D ¼0.75, (c) H/D ¼0.5, (d) H/D ¼ 0.25, (e) H/D ¼ 0.1875 and (f) H/D ¼0.125.
where p is the average pressure on the top face. Then, CL is obtained as follows: CL ¼
ðpp0 ÞpD2 =4 ð1=2ÞrU 20 pD2 =4
¼ Cp
where C p is average pressure coefficient on the top face.
ð5Þ
Dimensionless quantities are introduced for the purpose of discussing the drag and lift coefficients, while referencing Sakamoto and Arie (1983) and Good and Joubert (1968). The physical quantities that determine the drag and lift coefficients, CD and CL, are Drag (or Lift), D, H, U0, un, d, and nð ¼ m=rÞ These quantities are defined in items of Mass (M), Length (L) and Time (T).
244
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 6. Pressure coefficient distributions and contours as a function of aspect ratio effects (D ¼80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, d ¼52 mm): (a) H/D ¼1.0, (b) H/D ¼ 0.75, (c) H/D ¼ 0.5 and (d) H/D ¼0.25.
Fig. 7. Pressure coefficient distributions and contours as a function of boundary layer thickness effects (D ¼80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, HXD ¼ 1.0): (a) d ¼ 26 mm, (b) d ¼52 mm and (c) d ¼120 mm.
Reference quantities and dimensions are follows: Drag and lift: Drag, Lift ¼MLT 2 Diameter of the cylindrical structure: D ¼L Height of the cylindrical structure: H¼L
Boundary layer thickness: d ¼L Free stream velocity: U0 ¼LT 1 Friction velocity: un ¼LT 1 Viscosity: m ¼ML 1T 1 Density of the fluid: r ¼ML 3
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
245
For example, the drag coefficients, CD, is expressed as follows: d
C D ¼ K ðDrag or Lift Þa Db Hc d U e0 un f mg rh
ð6Þ
where K is the constant of proportionality. Substituting into the M.L.T. equation we have 0 ¼ ðMLT 2 Þa ðLÞb ðLÞc ðLÞd ðLT 1e ÞðLT 1f ÞðML1 T 1 Þg ðML3 Þh ¼ M a þ g þ h La þ b þ c þ d þ e þ f g3h T 2aef g
ð7Þ
Numbers of the reference and the basic dimensions are 8 and 3. The following five (8 3¼5) dimensionless quantities are obtained d f Drag r a U 0 D b U 0 H c U 0 d un , , , , : 2 m=r m=r m=r U0 m We can combine these dimensionless quantities each other and simplify them !
Drag r Drag U0 H U0D ð8Þ = ¼ m=r m=r m2 rU 0 2 HD
un U0
U0 d un d ¼ m=r m=r
U0H U0 d H = ¼ m=r m=r d U0 d U0 D d = ¼ m=r m=r D
ð9Þ
ð10Þ
ð11Þ
The following five dimensionless quantities are obtained ! Drag un un d H d , , , , U0 m=r d D rU 20 DH
Fig. 8. Drag and lift coefficient: (a) drag coefficient and (b) lift coefficient.
On the supposition that the cylindrical structure is always buried among the turbulent boundary layer, the correlation between un/U0 and undXn is well-known as U0 un d ¼ 5:85 log þ 5:56: un n
ð12Þ
We can eliminate the third dimensionless quantity ðun dÞ=ðm=rÞ. The references dimensionless quantities are the follows: ! Drag un H d , , , U0 d D rU 20 DH As a result, CD is given as a function of these dimensionless quantities as follows: Drag un H d ð13Þ ¼ f CD ¼ , , U0 d D ð1=2ÞrU 20 HD In the case of the lift coefficients CL, the reference area is the roof. The relation between dimensionless quantities is written as follows: Lif t un H d ð14Þ CL ¼ ¼ f , , U0 d D ð1=2ÞrU 20 pD2 =4 Fig. 8 shows the drag and lift coefficients of the cylindrical structure. Log of CD and CL are proportional to log of H/d. The gradients of CD and CL depend on un/U0. Correlations between CD and un/U0, and CL and un/U0 are shown in Fig. 9. The following relations were obtained: C D pðH=dÞ27ðun =U0 Þ0:75
ð15Þ
C L pðH=dÞ24ðun =U 0 Þ0:07
ð16Þ
Fig. 9. Correlation between CD, CL, and u*/U0.
In addition, based on Fig. 10, the following relations are obtained: C D ¼ 0:468ðd=DÞ0:154 ðH=dÞ27ðun =U 0 Þ0:75
ð17Þ
C L ¼ 0:581ðd=DÞ0:592 ðH=dÞ24ðun =U 0 Þ0:07
ð18Þ
It is difficult to explain that the exponential of HXd is equation of un/U0. For the drag coefficient, Eq. (17) is good agree to previous result (Tsutsui et al., 2000) as shown in Fig. 11.
246
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 12 shows the vectors of the resultant fluid force acting on ~ and L~ are the dimensionthe cylindrical structure. The symbols D less drag and lift forces, respectively, given by D ¼ CD
L ¼ CL
HD H ¼ CD DD D
p 4
D2 ¼ C L
p 4
ð19Þ
where D ¼ 1
ð20Þ
The distance from center line and height from ground of the fluid force acting point xc/D and Zc/H are obtained as follows: xc =D ¼
Z
1=2
C Lx
1=2
x x Z 1=2 x d = C Lx d D D D 1=2
ð21Þ
where CLx is the local lift coefficient at x, and Z c =H ¼
Z 0
1
z z Z 1 z d = CDz C Dz H F H 0
ð22Þ
The magnitude and position xc/D of the lift force depend on the pressure distribution on the top area. The pressure distribution is affected by the aspect ratio and the boundary layer thickness. However, the height Zc/H is constant for any aspect ratio or boundary layer thickness.
Fig. 10. Correlations between among u*/U0, H/d, and d/D: (a) for drag coefficient and (b) for lift coefficient.
4. Conclusions Experimental studies of the flow around a low-aspect-ratio cylindrical structure were performed under the Reynolds numbers were varied from 1.1 104–1.1 105. The aspect ratio was varied from 0.125–1.0. The variation of the turbulent boundary layer thickness at which the cylindrical structure was placed were 26 to 120 mm. The change of the flow pattern by the changes of the aspect ratio and the turbulent boundary layer thickness appears on the top face well. The reattachment point, the reverse flow region and the separation bubble region move forward with decreasing the aspect ratio or increasing the turbulent boundary layer thickness. The aspect ratio affects the pressure distributions more than the boundary layer thickness. The minimum pressure area lies at the border between the reverse flow region and the separation bubble region. Fluid forces acting on the cylindrical structure change depending on the aspect ratio and the turbulent boundary layer thickness. The experimental results revealed that the drag and lift coefficients are functions of un/U0, H/d, and d/D, as follows: C D ¼ 0:468ðd=DÞ0:154 ðH=dÞ27ðun =U 0 Þ0:75
Fig. 11. Comparison of drag coefficients.
C L ¼ 0:581ðd=DÞ0:592 ðH=dÞ24ðun =U 0 Þ0:07 :
Fig. 12. Direction of the fluid force acting on the cylindrical structure: (a) H/D¼ 1.0, (b) H/D ¼0.5 and (c) H/D ¼ 0.25.
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
References Baker, C.J., 1980. The turbulent horseshoe vortex. Journal of Wind Engineering and Industrial Aerodynamics 6, 9–23. Good, M.C., Joubert, P.N., 1968. The form drag of two-dimensional buff-plates immersed in turbulent boundary layers. Journal of Fluid Mechanics 31 (part 3), 547–582. Okamoto, S., Sunabashiri, Y., 1992. Vortex shedding from a circular cylinder of finite length placed on a ground plane. Transactions ASME, Journal of Fluids Engineering 114, 512–521. Portela, G., Godoy, L.A., 2007. Wind pressure and buckling of grouped steel tanks. Wind and Structures 10 (1), 23–44.
247
Sabransky, I.J., 1987. Design pressure distribution on circular silos with conical roofs. Journal of Wind Engineering and Industrial Aerodynamics 26, 65–84. Sakamoto, H., Arie, M., 1983. Vortex shedding from a rectangular prism and a circular cylinder placed vertically in a turbulent boundary layer. Journal of Fluid Mechanics 126, 147–165. Schlichting, H., 1968. Boundary Layer Theory, sixth ed. McGraw-Hill, New York (pp. 534 and 601–602). Tsutsui, T., Igarashi, T., Nakamura, H., 2000. Fluid flow and heat transfer around a cylindrical structure mounted on a flat plate boundary layer. JSME International Journal B 43–2, 279–287. Tsutsui, T., Kawahara, M., 2006. Heat transfer around a cylindrical structure mounted in a plate turbulent boundary layer. Transactions ASME, Journal of Heat Transfer 161, 153–161.
Contents lists available at SciVerse ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Flow around a cylindrical structure mounted in a plane turbulent boundary layer Takayuki Tsutsui n Department of Mechanical Engineering, The National Defense Academy, 1-10-20 Hashirimizu, Yokosuka, Kanagawa 239-8686, Japan
a r t i c l e i n f o
abstract
Available online 24 May 2012
The wind force acting on a low-aspect-ratio (height to diameter) cylindrical structure placed in a turbulent boundary layer has been investigated. The diameters of the cylindrical structure models were D ¼40 and 80 mm, and the Reynolds numbers based on D were varied from 1.1 104 and 1.1 105. The turbulent boundary layer thickness at which the cylindrical structure was placed varied from 26–120 mm. The aspect ratio was varied from 0.125–1.0. Flow visualizations were performed based on the surface oil-flow pattern method. The surface pressure distributions on the models were measured, and the drag and lift coefficients were determined by integration of the surface pressure distributions. The correlations between the fluid force, the turbulent boundary layer thickness, and the aspect ratio of the cylindrical structure were clarified. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Cylindrical structure Three-dimensional flow Boundary layer Horseshoe vortex Flow visualization Pressure distribution Fluid force
1. Introduction Cylindrical structures with low aspect ratios (height/diameter, H/D r1) are used in various engineering applications, such as oil storage tanks for large structures, silos, and small parts of electrical equipment. However, few studies have examined fluid flow around low-aspect-ratio cylinders (Baker, 1980; Okamoto and Sunabashiri, 1992; Portela and Godoy, 2007; Sabransky, 1987; Sakamoto and Arie, 1983; Tsutsui and Kawahara, 2006; Tsutsui et al., 2000). Fig. 1 shows the experimental Reynolds numbers Re and H/D ranges for low-aspect-ratio cylinders reported in previous papers. Baker (1980) investigated the horseshoe vortex system around the base of a circular cylinder in a turbulent boundary layer for a wide range of aspect ratio. The variations of the position of the horseshoe vortex and the boundary layer separation were determined experimentally as a function of the main flow parameters. Okamoto and Sunabashiri (1992) and Sakamoto and Arie (1983) found that the flow pattern changes dramatically between H/D ¼2 and 4, while the shedding vortices change from the Karman type to the arch type for low-aspect-ratio cylindrical structures. The heat transfer characteristics of low-aspect-ratio (H/D r1) cylindrical structures were reported in previous studies by the present author (Tsutsui and Kawahara, 2006; Tsutsui et al., 2000), in which the top face (roof) of the cylinder was shown to have a
n
Tel.: þ81 46 841 3810x3412; fax: þ 81 46 844 5900. E-mail address: [email protected]
0167-6105/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jweia.2012.05.003
greater influence on the overall fluid flow and heat transfer characteristics of the cylindrical structure. Sabransky (1987) and Poltera and Godoy (2007) reported the pressure force acting on tanks and silos with conical roofs. They reported the wind load acting on structures under systematic changes in aspect ratio, number of structures, and arrangement of structures. Because the scope of these studies is limited, useful pressure load data for low-aspect-ratio cylindrical structures is needed for general use in engineering fields. The present paper examines the flow around a cylindrical structure and the fluid force acting on the structure in order to obtain strength design data for lowaspect-ratio cylindrical structures. The present paper clarifies the correlations among the drag, lift, aspect ratio, and turbulent boundary layer thickness. The pressure distribution on the cylindrical structure was measured under various Reynolds numbers, aspect ratios, and turbulent boundary layer thicknesses, and the time-averaged surface flow patterns were clarified by the oilfilm visualization method.
2. Experimental apparatus and method The flow geometry and coordinate system for the experiment are shown in Fig. 2(a). The diameters of the cylindrical structure D were 40 and 80 mm, and the aspect ratios H/D ( ¼height/ diameter) of the models were varied from 0.125–1.0. The experiments were performed in a low-speed wind tunnel with a working section having a height of 400 mm, a width of 300 mm, and a length of 1400 mm. The free stream velocity U0 was varied
240
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Nomenclature CD CL Cp D Drag dt H L Lift
Dag coefficient Lift coefficient Pressure coefficient¼ ðpp0 Þ=0:5 rU 20 Diameter of the cylindrical structure Drag Diameter of the trip wire Height of the cylindrical structure Length (basic dimension) Lift
between 4 and 20 m/s, resulting in Reynolds numbers Re in the range of 1.1 104–1.1 105. In this range, the turbulent intensity was approximately 0.4% for the free stream velocity. The turbulent boundary layer was induced by attaching a 5-, 10-, or 12 mm diameter trip wire (T.W.) to the plane at a position 800 mm upstream from the model. The boundary layer thickness d ranged from 26–120 mm at the center position of the model. Fig. 2(b) shows the variation of the normalized velocity, u þ , in the turbulent boundary layer as a function of the normalized height, z þ , for three different trip wire diameters and without trip wire for U0 ¼20 m/s. The data agree well with the experimental universal velocity distribution of the turbulent boundary layer on a flat-plate at zero pressure gradient (Schlichting, 1968) and can be expressed as: u ¼ un 5:85logðun z=nÞ þ5:56 ð1Þ
M p, p0 Re T U0, u un X, Y, Z
d
n, r t0 j
Mass (basic dimension) Pressure, static pressure Reynolds number ¼U0D/v Time (basic dimension) Free stream velocity, velocity ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Friction velocity¼ t0 =r ¼ 0:0225U 20 ðn=U 0 dÞ Direction Boundary layer thickness Kinematic viscosity, density of the fluid Shearing stress at the plane Angle
The oil film is white oil mixed with oleic acid, titanium oxide, and liquid paraffin. The surface of the model was painted black in order to provide the highest possible contrast with the white oil film.
3. Results and discussion 3.1. Flow visualization Fig. 4 describes the typical surface oil flow patterns on the cylindrical structure and flow schematic. On the top face, a steady separation bubble (S.B.), similar to a lunette, is evident just
Fig. 3(a) shows the 80 mm-diameter pressure measurement model. The model was fabricated from acrylic resin pipes, which have many pressure taps in a spiral arrangement at ten-degree intervals along the surface of the model to avoid influence of nearness of each taps. The model set up and the pressure measurement system are shown in Fig. 3(b) and (c). The aspect ratio can be changed by changing the height of the model. The surface pressures were measured by the pressure measurement system, which is the Net scanner system 9816/98RK-1(PSI, Inc.). All data were measured for 1 min in 100 Hz sampling rate, and were averaged. The measure distribution on the entire surface of the cylindrical structure can be determined by rotating the model on its axis. The same methods were applied for the 40 mm-diameter model. The pressure on the surface of the cylindrical structure was measured with respect to Reynolds number, aspect ratio, and turbulent boundary layer thickness. The fluid forces acting on the cylindrical structure were then obtained by integrating the pressure distribution. The surface flow visualization of the cylindrical structure and plane was performed using an oil-film method in order to allow for the observation of the time averaged flow behavior around the structure.
Fig. 1. Experimental Reynolds number ranges of previous studies.
Fig. 2. Flow geometry, coordinate system and turbulent boundary layer: (a) flow geometry, coordinate system and (b) variation of u þ in the turbulent boundary layer as a function of z þ .
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
241
Fig. 3. Experimental model, model setup and measurement system: (a) model (D ¼80 mm), (b) model setup and (c) measurement system.
behind the leading edge and an arc-shaped reattachment line (R.) is present behind the center cord. A reverse flow region (R.F.) forms between the lunette and the arc. A down wash (D.W.) exists
from the side edge of the top face to the rear face, a reattachment flow spread from the rear stagnation point (R.St.) and a horseshoe vortex (H.V.) is evident on the floor.
242
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 4. Flow around the cylindrical structure (D ¼ 80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, H/D ¼0.5, d ¼ 52 mm) (Tsutsui and Kawahara, 2006): (a) front view, (b) rear view and (c) flow schematic.
The surface oil flow patterns of the cylindrical structure with various aspect ratios are shown in Fig. 5. The oncoming flow separates in front of the cylindrical structure. The separated shear layer roles upto form a horseshoe vortex on the plane. The horseshoe vortex swept away the white oil film on the plane. The swept area became gradually small with decreasing the aspect ratio. The change of the oil flow pattern on the top face is plain. The reattachment line moves forward when the aspect ratio H/D is decreased. Because of this, the reverse flow region moves forward and is reduced. In addition, the separation bubble gradually shrinks in (d) and (e), at last, it disappeared in (f). On the front face, a line of white oil film remains on the front face in (a) and (b), and in (c)–(f), rather than lines, nodes appear, which indicate the front stagnation point. The front stagnation on the front face and the separation line on the side face exhibit two-dimensional characteristics, except for the upper and lower regions in (a) and (b). Decreasing the aspect ratio gradually reduces the amount of two-dimensional flow. On the rear face, the flow spreads radially from the rear stagnation point in (a)–(c). In (d)–(f), the radial spread patterns are not plain, the reattachment flow on the back of the cylindrical structure is weak.
the negative pressure area moves forward, and its area on the top face decreases. On the side of the front face, the higher pressure area (Cp Z0.8), which is the front stagnation point, is reduced with decrease in the aspect ratio. Fig. 7 shows the effect of the turbulent boundary layer thickness on the pressure coefficients. On the top face, the lower pressure area (Cp Z 1.0) moves forward and the higher pressure area (Cp Z 0.2) is extended with increasing the boundary layer thickness. On the side face, the higher pressure area (Cp Z0.8) at the front stagnation point becomes smaller when the boundary layer thickness increases. Comparison of Figs. 6 and 7 reveals that the aspect ratio affects the pressure distributions more than the boundary layer thickness. A previous study (Tsutsui and Kawahara, 2006) reported that the reattachment point on the top face depends on the velocity gradient du/dz. Decreasing height H and increasing the turbulent boundary layer thickness d are causes of the increased velocity gradient du/dz. The change in H was found to have a greater effect on du/dz than the change in d.
3.2. Pressure distribution
Fluid forces acting on the cylindrical structure were obtained by integrating the pressure distribution. To obtain the total drag coefficient of the cylindrical structure CD, the local drag coefficient CDz was calculated in advance, as follows: Z 1 2p C pz cos jdj ð2Þ C Dz ¼ 2 0
The pressure coefficient distributions on the center line and the top and side face contours are presented in Fig. 6. The reattachment point (R.), the reverse flow region (R.F.), and the separation bubble region (S.B.) are shown on the pressure distribution. Cpfz and Cpbz are the pressure coefficients on the front and rear faces at height z. CDz is the local drag coefficient at height z, as described in the following paragraph. The effect of the aspect ratio appears clearly on the top face. On the top face, the pressure decreases significantly in the area forward from the reattach point (R.). The minimum pressure area lies at the border between the reverse flow region (R.F.) and the separation bubble region (S.B.). Downward from the reattach point (R.), the change in the pressure is smooth. The reattachment point (R.) moves forward with decreasing aspect ratio. Therefore,
3.3. Fluid force
where Cpz is the pressure coefficient at height z. Then, CD is obtained as follows: Z 1 z ð3Þ CD ¼ C Dz d H 0 Next, the lift coefficient CL is considered. The lift force acting on the top face Lift is given by Lif t ¼ ðpp0 ÞpD2 =4
ð4Þ
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
243
Fig. 5. Surface oil-flow patterns of the cylindrical structure (D¼ 80 mm, U0 ¼20 m/s, Re ¼1.1 105, d ¼52 mm): (a) H/D ¼1.0, (b) H/D ¼0.75, (c) H/D ¼0.5, (d) H/D ¼ 0.25, (e) H/D ¼ 0.1875 and (f) H/D ¼0.125.
where p is the average pressure on the top face. Then, CL is obtained as follows: CL ¼
ðpp0 ÞpD2 =4 ð1=2ÞrU 20 pD2 =4
¼ Cp
where C p is average pressure coefficient on the top face.
ð5Þ
Dimensionless quantities are introduced for the purpose of discussing the drag and lift coefficients, while referencing Sakamoto and Arie (1983) and Good and Joubert (1968). The physical quantities that determine the drag and lift coefficients, CD and CL, are Drag (or Lift), D, H, U0, un, d, and nð ¼ m=rÞ These quantities are defined in items of Mass (M), Length (L) and Time (T).
244
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 6. Pressure coefficient distributions and contours as a function of aspect ratio effects (D ¼80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, d ¼52 mm): (a) H/D ¼1.0, (b) H/D ¼ 0.75, (c) H/D ¼ 0.5 and (d) H/D ¼0.25.
Fig. 7. Pressure coefficient distributions and contours as a function of boundary layer thickness effects (D ¼80 mm, U0 ¼ 20 m/s, Re¼ 1.1 105, HXD ¼ 1.0): (a) d ¼ 26 mm, (b) d ¼52 mm and (c) d ¼120 mm.
Reference quantities and dimensions are follows: Drag and lift: Drag, Lift ¼MLT 2 Diameter of the cylindrical structure: D ¼L Height of the cylindrical structure: H¼L
Boundary layer thickness: d ¼L Free stream velocity: U0 ¼LT 1 Friction velocity: un ¼LT 1 Viscosity: m ¼ML 1T 1 Density of the fluid: r ¼ML 3
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
245
For example, the drag coefficients, CD, is expressed as follows: d
C D ¼ K ðDrag or Lift Þa Db Hc d U e0 un f mg rh
ð6Þ
where K is the constant of proportionality. Substituting into the M.L.T. equation we have 0 ¼ ðMLT 2 Þa ðLÞb ðLÞc ðLÞd ðLT 1e ÞðLT 1f ÞðML1 T 1 Þg ðML3 Þh ¼ M a þ g þ h La þ b þ c þ d þ e þ f g3h T 2aef g
ð7Þ
Numbers of the reference and the basic dimensions are 8 and 3. The following five (8 3¼5) dimensionless quantities are obtained d f Drag r a U 0 D b U 0 H c U 0 d un , , , , : 2 m=r m=r m=r U0 m We can combine these dimensionless quantities each other and simplify them !
Drag r Drag U0 H U0D ð8Þ = ¼ m=r m=r m2 rU 0 2 HD
un U0
U0 d un d ¼ m=r m=r
U0H U0 d H = ¼ m=r m=r d U0 d U0 D d = ¼ m=r m=r D
ð9Þ
ð10Þ
ð11Þ
The following five dimensionless quantities are obtained ! Drag un un d H d , , , , U0 m=r d D rU 20 DH
Fig. 8. Drag and lift coefficient: (a) drag coefficient and (b) lift coefficient.
On the supposition that the cylindrical structure is always buried among the turbulent boundary layer, the correlation between un/U0 and undXn is well-known as U0 un d ¼ 5:85 log þ 5:56: un n
ð12Þ
We can eliminate the third dimensionless quantity ðun dÞ=ðm=rÞ. The references dimensionless quantities are the follows: ! Drag un H d , , , U0 d D rU 20 DH As a result, CD is given as a function of these dimensionless quantities as follows: Drag un H d ð13Þ ¼ f CD ¼ , , U0 d D ð1=2ÞrU 20 HD In the case of the lift coefficients CL, the reference area is the roof. The relation between dimensionless quantities is written as follows: Lif t un H d ð14Þ CL ¼ ¼ f , , U0 d D ð1=2ÞrU 20 pD2 =4 Fig. 8 shows the drag and lift coefficients of the cylindrical structure. Log of CD and CL are proportional to log of H/d. The gradients of CD and CL depend on un/U0. Correlations between CD and un/U0, and CL and un/U0 are shown in Fig. 9. The following relations were obtained: C D pðH=dÞ27ðun =U0 Þ0:75
ð15Þ
C L pðH=dÞ24ðun =U 0 Þ0:07
ð16Þ
Fig. 9. Correlation between CD, CL, and u*/U0.
In addition, based on Fig. 10, the following relations are obtained: C D ¼ 0:468ðd=DÞ0:154 ðH=dÞ27ðun =U 0 Þ0:75
ð17Þ
C L ¼ 0:581ðd=DÞ0:592 ðH=dÞ24ðun =U 0 Þ0:07
ð18Þ
It is difficult to explain that the exponential of HXd is equation of un/U0. For the drag coefficient, Eq. (17) is good agree to previous result (Tsutsui et al., 2000) as shown in Fig. 11.
246
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
Fig. 12 shows the vectors of the resultant fluid force acting on ~ and L~ are the dimensionthe cylindrical structure. The symbols D less drag and lift forces, respectively, given by D ¼ CD
L ¼ CL
HD H ¼ CD DD D
p 4
D2 ¼ C L
p 4
ð19Þ
where D ¼ 1
ð20Þ
The distance from center line and height from ground of the fluid force acting point xc/D and Zc/H are obtained as follows: xc =D ¼
Z
1=2
C Lx
1=2
x x Z 1=2 x d = C Lx d D D D 1=2
ð21Þ
where CLx is the local lift coefficient at x, and Z c =H ¼
Z 0
1
z z Z 1 z d = CDz C Dz H F H 0
ð22Þ
The magnitude and position xc/D of the lift force depend on the pressure distribution on the top area. The pressure distribution is affected by the aspect ratio and the boundary layer thickness. However, the height Zc/H is constant for any aspect ratio or boundary layer thickness.
Fig. 10. Correlations between among u*/U0, H/d, and d/D: (a) for drag coefficient and (b) for lift coefficient.
4. Conclusions Experimental studies of the flow around a low-aspect-ratio cylindrical structure were performed under the Reynolds numbers were varied from 1.1 104–1.1 105. The aspect ratio was varied from 0.125–1.0. The variation of the turbulent boundary layer thickness at which the cylindrical structure was placed were 26 to 120 mm. The change of the flow pattern by the changes of the aspect ratio and the turbulent boundary layer thickness appears on the top face well. The reattachment point, the reverse flow region and the separation bubble region move forward with decreasing the aspect ratio or increasing the turbulent boundary layer thickness. The aspect ratio affects the pressure distributions more than the boundary layer thickness. The minimum pressure area lies at the border between the reverse flow region and the separation bubble region. Fluid forces acting on the cylindrical structure change depending on the aspect ratio and the turbulent boundary layer thickness. The experimental results revealed that the drag and lift coefficients are functions of un/U0, H/d, and d/D, as follows: C D ¼ 0:468ðd=DÞ0:154 ðH=dÞ27ðun =U 0 Þ0:75
Fig. 11. Comparison of drag coefficients.
C L ¼ 0:581ðd=DÞ0:592 ðH=dÞ24ðun =U 0 Þ0:07 :
Fig. 12. Direction of the fluid force acting on the cylindrical structure: (a) H/D¼ 1.0, (b) H/D ¼0.5 and (c) H/D ¼ 0.25.
T. Tsutsui / J. Wind Eng. Ind. Aerodyn. 104-106 (2012) 239–247
References Baker, C.J., 1980. The turbulent horseshoe vortex. Journal of Wind Engineering and Industrial Aerodynamics 6, 9–23. Good, M.C., Joubert, P.N., 1968. The form drag of two-dimensional buff-plates immersed in turbulent boundary layers. Journal of Fluid Mechanics 31 (part 3), 547–582. Okamoto, S., Sunabashiri, Y., 1992. Vortex shedding from a circular cylinder of finite length placed on a ground plane. Transactions ASME, Journal of Fluids Engineering 114, 512–521. Portela, G., Godoy, L.A., 2007. Wind pressure and buckling of grouped steel tanks. Wind and Structures 10 (1), 23–44.
247
Sabransky, I.J., 1987. Design pressure distribution on circular silos with conical roofs. Journal of Wind Engineering and Industrial Aerodynamics 26, 65–84. Sakamoto, H., Arie, M., 1983. Vortex shedding from a rectangular prism and a circular cylinder placed vertically in a turbulent boundary layer. Journal of Fluid Mechanics 126, 147–165. Schlichting, H., 1968. Boundary Layer Theory, sixth ed. McGraw-Hill, New York (pp. 534 and 601–602). Tsutsui, T., Igarashi, T., Nakamura, H., 2000. Fluid flow and heat transfer around a cylindrical structure mounted on a flat plate boundary layer. JSME International Journal B 43–2, 279–287. Tsutsui, T., Kawahara, M., 2006. Heat transfer around a cylindrical structure mounted in a plate turbulent boundary layer. Transactions ASME, Journal of Heat Transfer 161, 153–161.