Steady Streaming around a Circular Cylinder near a

Steady Streaming around a Circular Cylinder near a Plane Boundary due to Oscillatory Flow

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Hongwei An1; Liang Cheng2; and Ming Zhao, M.ASCE3 Abstract: Steady streaming due to an oscillatory flow around a circular cylinder close to and sitting on a plane boundary is investigated numerically. Two-dimensional 共2D兲 Reynolds-averaged Navier-Stokes equations are solved using a finite element method with a k-␻ turbulent model. The flow direction is perpendicular to the axis of the cylinder. The steady streaming around a circular cylinder is investigated for Keulegan-Carpenter 共KC兲 number of 2 ⱕ KCⱕ 30 with a constant value of Stokes number 共␤兲 of 196. The gap 共between the cylinder and the plane boundary兲 to diameter ratio 共e / D兲 investigated is in the range of 0.0–3.0. The steady streaming structures and velocity distribution around the cylinder are analyzed in detail. It is found that the structures of steady streaming are closely correlated to KC regimes. The gap to diameter ratio 共e / D兲 has a significant effect on the steady streaming structure when e / D ⬍ 1.0. The magnitude of the steady streaming velocity around the cylinder can be up to about 70% of the velocity amplitude of the oscillatory flow. One three-dimensional 共3D兲 simulation 共KC= 10, ␤ = 196, and e / D = ⬁兲 is carried out to examine the effect of three dimensionality of the flow on the steady streaming. Although strong 3D vortices are found around the cylinder, the steady streaming in a cross section of the cylinder span is in good agreement with the 2D results. DOI: 10.1061/共ASCE兲HY.1943-7900.0000258 CE Database subject headings: Oscillatory flow; Finite element method; Streamflow; Cross sections. Author keywords: Steady streaming; Oscillatory flow; Reynolds-averaged Navier-Stokes equations; Finite-element method.

Introduction Steady streaming is a flow phenomenon observed in oscillatory flow around a circular cylinder where a nonzero period-averaged flow field exists under sinusoidal approaching flow conditions. Such a nonzero mean flow is induced by the nonlinear Reynolds stress in the unsteady boundary layer 共Wang 1968兲. Oscillatory flow around a circular cylinder is governed by KeuleganCarpenter number 共KC= UmT / D兲 and Stokes number 共␤ = D2 / T␯兲, or alternatively Reynolds number 共R = KC· ␤ = UmD / ␯兲, where Um = amplitude of oscillatory velocity; T = period of velocity oscillation; D = diameter of the cylinder; and ␯ = kinematic viscosity of fluid. Extensive research work concerning steady streaming around a circular cylinder for small KC 共KC⬍ 1兲 has been carried out. Analytical work of steady streaming has been published by Holtsmark et al. 共1954兲, Stuart 共1966兲, and Wang 共1968兲. A sketch of steady streaming structure around an isolated cylinder under 1

Research Associate, School of Civil and Resource Engineering, The Univ. of Western Australia, 35 Stirling Hwy., Crawley, Western Australia 6009, Australia 共corresponding author兲. E-mail: [email protected] 2 Winthrop Professor, School of Civil and Resource Engineering, The Univ. of Western Australia, 35 Stirling Hwy., Crawley, Western Australia 6009, Australia. E-mail: [email protected] 3 Research Assistant Professor, School of Civil and Resource Engineering, The Univ. of Western Australia, 35 Stirling Hwy., Crawley, Western Australia 6009, Australia. E-mail: [email protected] Note. This manuscript was submitted on February 4, 2009; approved on April 12, 2010; published online on April 24, 2010. Discussion period open until June 1, 2011; separate discussions must be submitted for individual papers. This paper is part of the Journal of Hydraulic Engineering, Vol. 137, No. 1, January 1, 2011. ©ASCE, ISSN 0733-9429/2011/123–33/$25.00.

small oscillatory amplitude is depicted in Fig. 1 共Wang 1968兲, which consists of four small recirculating cells attached on the cylinder surface and four large ones further away from the cylinder. An impressive picture of such steady streaming taken from an experiment by Masakazu Tatsuno can be found in the book of van Dyke 共1982兲. Review of previous work about steady streaming for small KC was given by Riley 共1967, 2001兲. For higher KC 共KC⬎ 1兲, the flow field and force coefficients of an isolated cylinder in oscillatory flow have been well documented experimentally in the range of 0 ⱕ KCⱕ 50 共Williamson 1985; Sarpkaya 1986; Obasaju et al. 1988; Tatsuno and Bearman 1990, etc.兲. This topic has also been investigated with twodimensional 共2D兲 numerical simulation by Justesen 共1991兲 and Saghafian et al. 共2003兲. Some three-dimensional 共3D兲 simulation results were presented at low KC and ␤ conditions 共Zhang and Dalton 1999; Nehari et al. 2004; Elston et al. 2006兲. A numerical investigation by Badr et al. 共1995兲 demonstrated the existence of steady streaming under separated flow condition 共KC= 2 and 4 and R = 1 , 000兲. An et al. 共2009兲 studied the steady streaming around an isolated circular cylinder in the range of 2 ⱕ KCⱕ 40 with a constant ␤ = 196. Six typical steady streaming structures were found corresponding to the six vortex shedding regimes classified by Williamson 共1985兲. Sumer et al. 共1991兲 investigated the effect of a plane boundary on oscillatory flow around a circular cylinder in the range of 0 ⬍ KC⬍ 60 using flow visualization, and 4 ⬍ KC⬍ 65 by pressure measurement. The gap-to-diameter ratio 共e / D兲 has significant effect on the flow structure and hydrodynamic force coefficients. Wybrow et al. 共1996兲 studied the steady streaming around a circular cylinder close to a plane boundary in oscillatory flow condition 共KCⱕ 1兲 by means of analytical, computational, and experimental methods. They indicated that steady streaming plays an important role for the sediment transport beneath the cylinder. JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2011 / 23

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y

x

D

oscillatory direction

u=Umsin(2πt/T) e


plane boundary

Fig. 2. Definition sketch of the oscillatory flow around a cylinder near a plane boundary

Fig. 1. Steady streaming around a circular cylinder in oscillatory flow under small oscillatory amplitude condition

Steady streaming produces a nonzero net sediment transport rate on mobile sandy bed, which is the main reason of scour around the cylinder. Liang and Cheng 共2005兲 investigated the waveinduced scour below a submarine pipeline and reported the presence of steady streaming in the vicinity of the pipeline throughout the whole scour process 共KC= 11, R = 7 , 000兲. The results of Wybrow et al. 共1996兲 and Liang and Cheng 共2005兲 indicated that the existence of steady streaming when the cylinder is located close to a plane boundary in oscillatory flow. However, there is lack of systematic research work with regard to the influence of the plane boundary on the steady streaming. This is the motivation of present work. In this study, the oscillatory flow around a circular cylinder close to a plane boundary is simulated using a 2D FEM model. This study covers the range of 2 ⱕ KCⱕ 30, 0 ⱕ e / D ⱕ 3.0, with a constant ␤ = 196. The steady streaming around the cylinder is calculated by period-averaged method. The steady steaming structure and velocity distribution of steady streaming are analyzed. It is believed that the oscillatory flow examined in the present study is largely 3D. Scandura et al. 共2009兲 studied the oscillatory flow around a circular cylinder close to a plane boundary by 2D and 3D direct numerical simulation of the Navier Stokes equations 共KC= 10 and R = 200 and 500兲. Scandura et al. 共2009兲 stated that the correlation of sectional forces increase in e / D due to different reasons. At R = 200, the reduction of the gap leads to a reduction of 3D effects. At R = 500, the reduction of the gap leads to the generation of a large number of 3D vortices with a wide range of spatial scales, which produces the homogenization of the flow structures along the spanwise direction. To examine the effect of 3D flow on steady streaming, a 3D flow of KC= 10, ␤ = 196, and e / D = ⬁ is simulated in this study. The steady streaming structures in cross sections of the cylinder are compared with the 2D result of KC= 10, ␤ = 196, and e / D = 3.0.

and the equation of k and ␻ are solved using a streamline upwind finite-element method. Details of the numerical model are not provided in the present paper but can be found in Zhao et al. 共2007兲. The boundary conditions employed in the computation are described. On the cylinder surface and the plane boundary, the velocity and the turbulent kinetic energy k are specified as zero. The boundary condition on ␻ at the first node away from the wall is specified by ␻ = 6␯ / ⌬21, with ⌬1 is the distance between the first node and the wall. This boundary condition for ␻ requires that the nondimensional distance y + = u f ⌬1 / ␯ 共where u f = friction velocity at cylinder surface兲 is smaller than 2.5 共Wilcox 1994兲. Symmetric boundary condition 共vertical velocity and gradients of other quantities in the normal direction of the boundary are set as zero兲 is employed at the top boundary. The vertical velocity component at the left and right boundary is specified as zero and the horizontal velocity component is specified as u共y , t兲 = U共y兲sin共2␲t / T兲, where U共y兲 is calculated as U共y兲 yuⴱ , = uⴱ ␯

冉

yuⴱ ⱕ 11.26 ␯

冊

U共y兲 1 yuⴱ = ln 9.0 , uⴱ ␬ ␯

yuⴱ ⬎ 11.26 ␯

共1兲

where uⴱ = friction velocity and ␬ = von Kármán constant 共=0.42兲. The Reynolds number is defined based on the velocity at the level of the cylinder top, i.e., Um = U共0.5D兲. Eq. 共1兲 is the velocity distribution of fully-developed steady boundary layer flow. In the simulation, the length of the computational domain is 50D and the cylinder is located at the center of the domain in horizontal direction. Preliminary computation showed that the length of 50D is sufficient to allow the oscillatory boundary layer flow to fully develop. Further increase in length does not make change on velocity distribution at the center of the domain. The gradients of the turbulence quantities in the horizontal direction are set to be zero at the left and right boundaries. The initial value of velocity in the domain is set zero and the initial values of turbulent quantities k and ␻ are specified as k = 0.0001Um2 and ␻ = k / ␯.

Mathematical Model and Boundary Conditions Fig. 2 shows the sketch of a circular cylinder in a sinusoidal oscillatory flow. A Cartesian coordinate system is fixed at the center of the circular cylinder and y-axis is perpendicular to the oscillatory flow direction. 2D Reynolds-averaged Navier-Stokes 共RANS兲 equations are solved with a low Reynolds number k-␻ turbulence model closure 共Wilcox 1994兲. The RANS equations

Model Validation The convergence of the numerical results are examined under the flow condition of KC= 30 and ␤ = 196, which corresponds to the highest R number of 5,880 calculated in this study. The case with e / D of 0.2 is examined. A rectangular domain with 50D in width

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and 10D + e in height is employed. The cylinder location in the x-direction is located in the middle of computational domain. Fig. 3 shows a typical finite-element mesh near the cylinder 共e / D = 0.2兲. Structured four-node quadrilateral elements are used. The cylinder perimeter is discretized by 160 nodes. The minimum element size in the radial direction next to the cylinder surface is 0.001D. The flow oscillates in the horizontal 共x兲 direction. The maximum nondimensional mesh size y + for this case is 2.3. The numerical results do not show substantial differences with further increase of the density of computational mesh. Although the present numerical model has been validated against both steady and oscillatory flows past an isolated circular cylinder 共Zhao et al. 2007; An et al. 2009兲, it is validated against available experimental data of oscillatory flow around a circular cylinder. First, the numerical model is validated against the experimental data of Dütsch et al. 共1998兲 at KC= 5 and ␤ = 20 共isolated cylinder, e / D = ⬁兲. The flow of KC= 5, ␤ = 20, and e / D = ⬁ is characterized by absolutely stable 2D flow 共Tatsuno and Bearman 1990兲. The velocity distribution measured by Dütsch et al. 共1998兲 has been broadly used to validate a variety of numerical models, such as finite volume model 共Dütsch et al. 1998兲, boundary element model 共Uzunoðlu et al. 2001兲, immersed boundary method model 共Choi et al. 2007; Wang et al. 2009兲, collocated finite volume embedding method 共Ng 2009兲, and the present finite-element model.

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/□ comp./exp. x/D=-0.6 /∆ comp./exp. x/D= 0.0 /◊ comp./exp. x/D= 0.6 /○ comp./exp. x/D= 1.2

-0.5

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1.5 1.0

y/D

(a)

y/D


Fig. 3. Typical finite-element mesh around cylinders 共e / D = 0.2兲

The turbulent model is switched off to simulate this case of low R number. The calculated velocity distribution at four sections 共x / D = −0.6, 0.0, 0.6, and 1.2, t / T = 10.5兲 are compared with experimental results of Dütsch et al. 共1998兲 in Fig. 4. The experimental data were recorded in a coordinate system fixed with still water, while the coordinate system employed in the present work is fixed with the cylinder. A coordinate transformation has been done before the numerical results are compared with the experimental data. Fig. 4 shows the numerical results agree with the experimental data reasonably well, but with some noticeable differences between numerical and experimental data. Similar differences were found in the numerical results of other numerical models mentioned in previous paragraph. Second, the calculated pressure distribution is compared with the data measured by Sumer et al. 共1991兲 at KC= 10, R = 1.0 ⫻ 105, and e / D = 0.2. For a case with such a high R number, the computational mesh is refined. The minimum element size in the radial direction next to the cylinder surface is reduced to 0.0002D and 200 elements are used to discretize cylinder surface. The k-␻ turbulent model is switched on. Fig. 5 shows the comparison of the calculated and measured pressure coefficient distributions along the cylinder surface at six instants in a half period of flow. The pressure coefficient is defined as C p = 共p-p0兲 / 共␳Um2 / 2兲, where p0 = pressure at the reference point 关␣ = 0 on the cylinder surface, the definition of angle ␣ is given in Fig. 5共a兲兴. A strong variation of C p in this half period exists due to flow reversal and vortex shedding. Obvious difference between the numerical results and the experimental data exists in Fig. 5共e兲, corresponding to the instant of flow reversal. The reason for this is unclear, although this may be due to many factors such as 3D effects, use of turbulence models, and experimental uncertainties. The overall comparison between present numerical results and experimental results is considered to be satisfactory for engineering applications. Third, the calculated lift coefficients of the cylinder are also compared with the experimental results of Sumer et al. 共1991兲 as shown in Fig. 6. The lift coefficient is defined as CL = FL / 共0.5␳DUm2 兲 while FL is the force component in the direction perpendicular to the plane boundary. From Fig. 6, that the overall agreement between numerical results and experimental data are reasonable, although there exist obvious differences. The measured time histories 共Sumer et al. 1991兲 are obviously aperiodic, which indicates variation of flow structures exist among different

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Fig. 4. Comparison of the calculated velocity components and the experimental data of Dütsch et al. 共1998兲 at four cross sections with constant x values 共KC= 5, R = 100, e / D = ⬁, and t / T = 10.5兲 JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2011 / 25

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(a) φ/π =0

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Num. Exp. (Sumer et al., 1991)

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Fig. 5. Pressure distribution along the cylinder surface compared with the experimental results of Sumer et al. 共1991兲. 共R = 1.0⫻ 105 and KC = 10, e / D = 0.2兲.

(a)KC=4, e/D=0.0 — Numerical results 8 (1)KC=4,e/D=0.0 ○ Sumer et al (1991)

(b)KC=10, e/D=0.0 — Numerical results 8 (2)KC=10,e/D=0.0 ○ Sumer et al (1991)

6

6 CLl

CL

periods of flow. The calculated time histories of the lift show strong periodicity. Last, the simulated flow structure around the cylinder is qualitatively compared with the experimental observation of Sumer et al. 共1991兲. Fig. 7 shows the typical vorticity contours at six instants in a half period for KC= 10, ␤ = 196, and e / D = 0.1. The vortices rotating in clockwise direction are defined as negative and plotted with dashed lines, and the vortices rotating in counter clockwise direction are defined as positive and plotted with solid lines. The vortex motion on the free stream side is examined first. At the beginning of this flow period 关Fig. 7共a兲兴, a positive vortex is shed in previous period and an attached negative vortex is found on the left side of the cylinder 关Fig. 7共a兲兴. Both vortices are carried toward the right side of the cylinder by the flow. The

4 2 0 0

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1 t/T

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(c)KC=4, e/D=2.0 — Numerical results (3)KC=4,e/D=2.0

○ Sumer et al (1991)

CL

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φ = 40°

φ = 75°

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1 t/T

1.5

φ = 120°

φ = 150°

2

(d)KC=10, e/D=1.0 — Numerical results (4)KC=10,e/D=1.0

○ Sumer et al (1991)

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0 0

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attached negative vortex is swept over the cylinder and develops from the right side 关Figs. 7共b–e兲兴. It is then shed from the cylinder at the instant of ␾ = 150° 关Fig. 7共f兲兴. In the gap side, a flat positive vortex is generated and grows up from the lower side of the cylinder due to a strong jetlike flow through the gap 关Figs. 7共b and c兲兴. At the same time, the boundary layer on the plane boundary rolls up to from a negative vortex the in the gap. When the phase angle ␾ = 90°, the flat vortex generated from the lower side of the cylinder splits into two parts 关Fig. 7共d兲兴. One part is convected away by flow together with the negative vortex generated on the plane boundary, and the other part rolls up and attaches on the right side of the cylinder 关Fig. 7共e兲兴. This vortex shedding

4 2

2 CL


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Fig. 6. Lift force coefficient compared with the experimental data 关— numerical results; 䊊, experimental data 共Sumer et al. 1991兲兴

Fig. 7. Vortex shedding process in half period of oscillatory flow 共KC= 10, ␤ = 196, and e / D = 0.1兲


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process shown in Fig. 7 agrees qualitatively with the experimental observation of Sumer et al. 共1991兲关please see Fig. 21共c兲 in Sumer et al. 共1991兲 for comparison兴.

(a) e/D=3.0

(b) e/D=1.0 Us/Um:

Us/Um:


Steady Streaming Structure Representative flow conditions corresponding to the following five vortex shedding regimes: 共1兲 2 ⱕ KCⱕ 4; 共2兲 4 ⱕ KCⱕ 7; 共3兲 7 ⱕ KCⱕ 15; 共4兲 15ⱕ KCⱕ 24; and 共5兲 24ⱕ KCⱕ 30 关classified by Williamson 共1985兲; Sumer et al. 共1991兲兴 are simulated in this study. Distinct flow features were found in each of the vortex shedding regimes 共Williamson 1985; Sumer et al. 1991兲. The steady streaming structures corresponding to five KC numbers 共KC= 2, 5, 10, 16, and 26兲 within the five vortex shedding regimes are presented. For each KC number, the effect of e / D is investigated in the range of 0 ⱕ e / D ⱕ 3.0. A k-␻ turbulence model is employed for cases with R number larger than 500. ¯ , ¯v兲 is obtained by time-averaging Steady streaming velocity 共u flow velocity in the 20th flow period of each simulation, which can be calculated as ¯u =

冕

20T

19T

¯v =

冕

20T

19T

(c) e/D=0.5 Us/Um:

Us/Um:

(e) e/D=0.1

u dt T v dt T

(d) e/D=0.2

Us/Um:

(f) e/D=0.0 Us/Um:

共2兲

The structures of steady streaming are represented by means of steady streaming streamlines and contours of steady streaming ¯ 2 + ¯v2兲1/2. velocity magnitude Us, which is defined as Us = 共u Fig. 8. Steady streaming structures for KC= 2 and ␤ = 196

2 ⱕ KCⱕ 4 In this range of KC numbers, the oscillatory flow around unbounded cylinder is characterized by symmetry with respect to the x axis and attached 共nonshedding兲 vortices 共Williamson 1985兲. Fig. 8 shows the steady streaming structures at different e / D values 共0 ⱕ e / D ⱕ 3.0兲.The amplitude of steady streaming is generally small 共less than 10% of Um兲 in the far field of the cylinder. Therefore, the focus of investigation is the near field of the cylinder. For the case of e / D = 3.0 关Fig. 8共a兲兴, the basic structure of steady streaming is similar to that shown in Fig. 1. It is also nearly identical to the steady streaming structure around an unbounded cylinder in the same oscillatory flow condition 共An et al. 2009兲. The high velocity zone overlaps with the four inner recirculating cells. The four big cells generate two jetlike flows on the left and right sides of the cylinder. The steady streaming structure does not vary substantially in the range of 1.0⬍ e / D ⱕ 3.0. This indicates the effect of the plane on the steady streaming around the cylinder is insignificant when the gap ratio is larger than one in this vortex shedding regime. When e / D is reduced to 1.0, two more recirculating cells are generated between the cylinder and the plane boundary 关Fig. 8共b兲兴. At the same time the two jetlike flows tilt toward the plane boundary. The distribution of contours of Us / Um around the cylinder changes only slightly for cases with e / D ⱖ 0.5 关see Figs. 8共a–c兲兴. With further decreasing of e / D, the magnitude of Us / Um in the gap jumps up sharply to 0.4 at e / D = 0.2 关Fig. 8共d兲兴, and then to 0.6 at e / D = 0.1 关Fig. 8共e兲兴, while free stream side velocity just changes slightly. When the cylinder is mounted on the plane boundary 共e / D = 0.0兲, only the four cells on the free stream side

exist and the maximum value of Us / Um drops back to about 0.15. A common feature of all the steady streaming structures shown in Fig. 8 is symmetry with respect to the y axis. This is because the oscillatory flow structures at two instants with a time difference of T / 2 are symmetric with respect to the y axis. 4 ⱕ KCⱕ 7 In this range of KC, the flow around an isolated cylinder is characterized by asymmetric, nonshedding vortices 共Williamson 1985兲. A group of typical results of this vortex shedding regime is given in Fig. 9 共KC= 5兲. For the cases with e / D ⱖ 1.0, the structures of steady streaming are similar to that around an isolated cylinder under same flow conditions 共An et al. 2009兲. The streamlines around the cylinder form four small recirculating cells attached to the cylinder surface and another four bigger ones in the outer area 关Figs. 9共a and b兲兴. The effect of plane boundary on streaming structure becomes obvious when e / D ⬍ 1.0. The streaming structure becomes asymmetric with respect to the y axis 关Fig. 9共c兲兴, while the main structure of contours of Us is still symmetric in this case. The degree of asymmetry is further enhanced as e / D further decreases as shown in Figs. 9共d and e兲. Instantaneous flow structures are examined to find out the reason for the asymmetry. Fig. 10 shows the typical vorticity contours of KC= 5 and e / D = 0.2 at four instants in the 15th period. Vorticity contours shown in Figs. 10共a and c兲共corresponding to the two instants of flow reversal in one period兲 are asymmetric with reJOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2011 / 27

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(a) e/D=3.0 Us/Um:


(c) e/D=0.5 Us/Um:

(e) e/D=0.1 Us/Um:

(b) e/D=1.0 Us/Um:

(d) e/D=0.2

(a) e/D=3.0 Us/Um:

(b) e/D=1.5 Us/Um:

(c) e/D=0.5

(d) e/D=0.2 Us/Um:

Us/Um:

(f) e/D=0.0

(e) e/D=0.1

(f) e/D=0.0 Us/Um:

Us/Um:

Fig. 9. Steady streaming structures for KC= 5 and ␤ = 196

spect to y axis. Similarly vorticity contours shown in Figs. 10共b and d兲 are also asymmetric. The half-period asymmetry of flow structure is the reason for the asymmetry of steady streaming. The breakup of flow symmetry was also observed by Sumer et al. 共1991兲. When e / D is reduced to zero, the steady streaming switches back to symmetry 关Fig. 9共f兲兴. 7 ⱕ KCⱕ 15 In the range of 7 ⱕ KCⱕ 15, one vortex shedding cycle happens in each half a period of flow. At KC= 10, the vortices are shed from the same side of the cylinder for isolated cylinder and form

Fig. 10. Vorticity contours at four instants with a time interval of 0.25T 共KC= 5 and e / D = 0.2兲

Us/Um:

Us/Um:


a transverse vortex street 共Williamson 1985兲. The transverse vortex street has also been found by Sumer et al. 共1991兲 under the condition of cylinder in oscillatory flow and close to a plane boundary 共e / D ⬎ 1.7兲. The steady streaming structures of KC= 10 with six e / D values are discussed here as an example of this vortex shedding regime. Fig. 11共a兲 shows the steady streaming around a cylinder with e / D = 3.0. The steady streaming structure consists of two small recirculating cells attached on the top side of the cylinder and an intensive jetlike flow flowing out from the bottom side of the cylinder. Such a jetlike flow is induced by the transverse vortex street observed by Williamson 共1985兲 and Sumer et al. 共1991兲. The steady streaming structure shown in Fig. 11共a兲 is almost identical to that of an isolated cylinder at KC= 10 共An et al. 2009兲. The strength of the jetlike flow decreases with the decrease in e / D. The effects of plane boundary on the main structure of steady streaming are rather weak in the range of e / D ⱖ 1.5 as shown in Figs. 11共a and b兲. When the cylinder is moved toward the plane boundary further, the size of the two cells on the top side of the cylinder increase gradually 关see Figs. 11共c–f兲兴. This is because of the increase of vorticity on the free stream side. The location where the maximum value of Us / Um is observed switches to the free stream side from the gap side when e / D is reduced to 0.2. Again, all the steady streaming structures shown in Fig. 11 are symmetric with respect to the y axis.


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(a) e/D=3.0 Us/Um:

(b) e/D=1.0

(c) e/D=0.5

(d) e/D=0.2

Us/Um:

(e) e/D=0.1 Us/Um:

(a) e/D=3.0

Us/Um:

Us/Um:

(c) e/D=0.5

Us/Um:

Us/Um:

(f) e/D=0.0

(e) e/D=0.1 Us/Um:

Us/Um:


15ⱕ KCⱕ 24 One of the most obvious features of flow around an isolated cylinder with KC number in this range is the diagonal vortex street 共Williamson 1985; Obasaju et al. 1988兲. Two vortices shed in each half a period of flow and all the vortices shed from the cylinder are convected away in the direction of about 45° angle with respect to the flow direction. The oscillatory flow is asymmetric between two halves of a flow period. Fig. 12 shows the features of steady streaming of the present KC regime. For e / D = 3.0 关Fig. 12共a兲兴, the steady streaming is asymmetric since there is no half-period symmetry in the vortex motion 共Williamson 1985; Sumer et al. 1991兲. With the narrowing of the gap, the arrangement of recirculating cells around the cylinder changes gradually and the degree of asymmetry reduces 关see Figs. 12共b–d兲兴. When e / D is reduced to 0.2, the steady streaming becomes symmetric as shown in Fig. 12共d兲. Steady streaming structure remains symmetric with further decrease of e / D 关Figs. 12共e and f兲兴. The streaming structures shown in Figs. 12共d–f兲 are almost identical to those shown in Figs. 11共d–f兲, correspondingly. 24ⱕ KCⱕ 30 In this vortex shedding regime, two more vortices shed from the free cylinder in each period and this regime is referred to as three-pair regime 共Williamson 1985兲. Fig. 13 shows the steady streaming structures of KC= 26 at six different values of e / D. All

(b) e/D=1.0 Us/Um:

(d) e/D=0.2 Us/Um:

(f) e/D=0.0 Us/Um:


of the six figures in Fig. 13 contain four recirculating cells: two small cells in the gap side and two larger ones on the free stream side. The size of the two cells in the gap side is squeezed with the reducing of ratio of e / D. Based on the steady streaming structures show for different vortex shedding regimes, the streaming structure is not sensitive to KC in the range of 7 ⱕ KCⱕ 30 when e / D ⬍ 0.2 关see Figs. 11共d–f兲, 12共d–f兲, and 13共d–f兲兴.

Steady Streaming Velocity Fig. 14 summarizes the change of Usmax with KC and e / D, where Usmax is defined as the maximum value of Us of each case calculated in this study. For the cases with KC= 2, the Usmax / Um increases dramatically from 0.16 to 0.6 when the cylinder is raised upwards by 0.1D from e / D = 0, and then drops back to about 0.2 when e / D increases to 0.5. Due to the dramatic change of Usmax in the range of 0 ⬍ e / D ⬍ 0.5, four more cases with KC= 2, e / D = 0.03, 0.07, 0.15, and 0.3 were calculated. The sharp increase of Usmax / Um was because the strong jetlike flow is generated on the gap side when the cylinder is elevated from the plane boundary by a small distance. The value of Usmax / Um stabilizes at 0.16 for cases with e / D ⱖ 1.0. For the case with KC= 5 and e / D = 0.0, a value of Usmax / Um = 0.48 is found. The value of Usmax / Um reaches the peak value of 0.64 with e / D the increasing to 0.5. For further increase of the e / D, the value of Usmax / Um stabilizes at about 0.53. For cases with KC= 10, the value of Usmax / Um experiences JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2011 / 29

J. Hydraul. Eng. 2011.137:23-33.

(b) KC=2,θ=-45°

0.8

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.4 Uθ/Um

0.6

Usmax/Um

(a) KC=2,θ= 45°

KC= 2 KC= 5 KC=10 KC=16 KC=26

0

θ= 45

-0.4

0

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.4 0

θ=−45 0

-0.4

0.4 -0.80.5

1

(c) KC=5,θ= 45°

L/D

1.5

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

1

e/D

2

3

0

-0.4

Fig. 14. Values of Usmax / Um at different KC numbers and gap to diameter ratio

-0.80.5

1

(e) KC=10,θ= 45°

L/D

1.5

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

-0.4 1

L/D

1.5

2

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0

-0.80.5

1

L/D

1.5

0.8

2

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.4 0

-0.80.5

1

0

-0.4

L/D

1.5

0.8

2

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.4 Uθ/Um

0.4 Uθ/Um

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

(h) KC=16,θ=-45°

0.8

0

-0.4 1

(i) KC=26,θ= 45°

L/D

1.5

2

-0.80.5

1

(j) KC=26,θ=-45°

0.8

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0

-0.4

L/D

1.5

0.8

2

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.4 Uθ/Um

0.4 Uθ/Um

2

-0.4

(g) KC=16,θ= 45°

-0.80.5

1.5

0.4

Uθ/Um

Uθ/Um

2

0

-0.80.5

L/D

0.8

(f) KC=10,θ=-45°

0.4

-0.80.5

1

-0.4

0.8

slight oscillation in the range of 0 ⱕ e / D ⱕ 1.0 and then stabilizes at about 0.5. For cases with KC= 16, the maximum value of Us / Um drops from 0.71 to 0.5 when the gap ratio increases from 0 to 1.0, and then no obvious change is observed. Identical trend is observed for the cases with KC= 26. For the five KC numbers studied, the influence from gap ratio on the Usmax / Um is insignificant for e / D ⱖ 1.5. The location of high steady streaming velocity is affected by KC and e / D. At KC= 2, the high velocity zone is found close to the cylinder in each quarter of the cylinder surface when e / D ⬎ 0.5 关Figs. 8共a–c兲兴, then the high velocity zone switches to the gap side at e / D = 0.2 and 0.1关Figs. 8共d and e兲兴. High velocity is found on the two shoulders when the cylinder is mounted on the plane boundary 关Fig. 8共f兲兴. Similar features to those found at KC= 2 共Fig. 8兲 are also found at KC= 5, 16, and 26, but, high velocity zone is found only on the lower side of the cylinder at e / D = 3.0 and 1.5 and KC= 10 关Figs. 11共a and b兲兴, because the vortex shedding only happens on the lower side of the cylinder in these two cases. Fig. 15 shows the circumferential velocity 共U␪兲 against L / D at ␪ = 45° and ␪ = −45°, where L represents the distance from a point to the center of the cylinder. The sign of U␪ in clockwise direction is defined as positive. The positions of ␪ = 45° and ⫺45° are shown in Figs. 15共a and b兲, respectively. Fig. 15共a兲 shows the distribution of U␪ in the direction of ␪ = 45° of KC= 2 for five e / D values. The velocity profiles of the five cases almost collapse together, indicating that e / D does not influence the distribution of U␪ on the free stream side of the cylinder. While the peak value of U␪ at ␪ = −45° increases with the decrease of e / D in the range of 0.1ⱕ e / D ⱕ 1.0 as shown in Fig. 15共b兲. When the cylinder is mounted on the plane boundary 共e / D = 0, results are not plotted here兲, the U␪ is close to zero at ␪ = −45°, and similar to that of e / D = 0.1 at ␪ = 45°. For all the other results plotted in Fig. 15, a significant effect of gap ratio on the distribution of U␪ in the two directions is observed. The distribution of U␪ is not sensitive to the change of KC when e / D ⱕ 0.2 and KC ⬎ 7. This agrees with the corresponding steady streaming structures shown in Figs. 11–13. Fig. 16 shows the influence of KC and e / D on the normal velocity 共Ur兲 along a concentric circle 共L / D = 0.7兲 of the cylinder surface. Velocity flowing out of the concentric circle is defined as positive. For small KC 共KC⬍ 4兲, e / D only has effect on normal velocity distribution on the gap side 共−␲ ⬍ ␪ ⬍ 0兲 as shown in Fig. 16共a兲. For all the other four KC values, e / D appears to have significant influence on the distribution of Ur on both the gap and free stream sides. The maximum value of Ur / Um happens at

-0.80.5

Uθ/Um

0

Uθ/Um

0.4

0

2

(d) KC=5,θ=-45°

0.8

0.2


0.8

Uθ/Um

0.8

0

-0.4 1

L/D

1.5

2

-0.80.5

1

L/D

1.5

2

Fig. 15. Distribution of circumferential velocity 共U␪兲 of the steady streaming at ␪ = 45° and ␪ = −45°

e / D = 0.1 except when KC= 10 关Fig. 16共c兲兴, where the maximum value of Ur / Um is observed at e / D = 3 corresponding to the strong jetlike flow 关Fig. 10共a兲兴.

3D Effect on Steady Streaming It has been well known that the oscillatory flow remains 2D only under very low KC and low ␤. Honji instability, which is characterized by mushroom shape 3D vortex structures, starts to emerge when KC number exceeds a certain value as observed by Honji 共1981兲, Sarpkaya 共1986, 2002兲, and Tatsuno and Bearman 共1990兲. Hall 共1984兲 carried out a stability analysis of oscillatory flow around a circular cylinder for high ␤ and found the critical KC 共Kh兲 of the onset of Honji instability can be calculated as Kh = 5.788␤−0.25共1 + 0.21␤−0.25 + ¯兲

共3兲

Recently, Sarpkaya 共2002兲 found that 3D instability emerges before the Honji instability happens. The critical KC for the onset of 3D instability 共defined as Kcr兲 was given by Sarpkaya 共2002兲 as


J. Hydraul. Eng. 2011.137:23-33.

(a) KC=2

0.4

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0

-0.2 -0.4-1

-0.5

0

θ/π

0.5

0.4

0

-0.2

(a) z/D=-3.0 Us/Um:

(b) z/D=0.0 Us/Um:

0

-0.4-1

-0.5

0

θ/π

0.5

(d) KC=16

0.4

1 e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.2 0

(c) z/D=3.0 Us/Um:

-0.2 -0.5

0

θ/π

0.5

(e) KC=26

0.4

1

-0.4-1

-0.5

0

θ/π

0.5

1

e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.2 Ur/Um

0.2

Ur/Um

Ur/Um

1 e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

0.2


e/D=0.1 e/D=0.2 e/D=0.5 e/D=1.0 e/D=3.0

-0.2

(c) KC=10

-0.4-1

0.4

Ur/Um

Ur/Um

0.2

(b) KC=5

0

-0.2 -0.4-1

-0.5

0

θ/π

0.5

1

Fig. 16. Distribution of normal velocity 共Ur兲 along a concentric circle of the cylinder surface

Kcr = 12.5␤−2/5

共4兲

According to Eq. 共4兲, Kcr = 1.51 corresponds to ␤ = 196. So all the flows simulated in this work should be 3D. Scandura et al. 共2009兲 observed strong three-dimensionality in their 3D numerical results of oscillatory flow around a circular cylinder near a plane boundary at KC= 10, R = 200 and 500, and e / D = 0.25⬃ 1.5. To examine the effect of 3D flow structures on steady streaming, a 3D simulation is carried out at KC= 10, ␤ = 196, and e / D = ⬁. The 3D model developed by Zhao et al. 共2009兲 is used to simulate the flow. The 3D results are compared with the 2D results of KC= 10, ␤ = 196, and e / D = 3.0. In the 2D results, when e / D ⬎ 3, the effect of the plane boundary on the steady streaming around the cylinder is negligible. A cylinder of length of 12D is used in the simulation. The center of the coordinate system is located at the center of the cylinder. The x axis is parallel to the flow direction and the z axis is along the spanwise direction. The steady streaming structures in the planes of z / D = −3.0, 0.0, and 3.0 are plotted in Fig. 17. Basic structures of the streaming in the three diagrams are similar. Two recirculating cells sitting on the shoulders of the cylinder and a jetlike flow flowing out from the bottom side of the cylinder are similar to the 2D results discussed before. However, variation of the streaming structure in spanwise direction of the cylinder can be clearly seen. In Figs. 17共b and c兲 the streamlines above the cylinder bias to the right hand side of the cylinder. In the 3D model results, there is not exact repeatability of the flow structure between two successive flow periods. So, the steady streaming obtained by averaging flow over one flow period is different from that in the succeeding period. The steady streaming results shown in Fig. 17 only represent a snapshot of the 3D steady streaming process because of the aperiodic of the flow. It is expected that the steady streaming should converge to a constant value if it is calculated by averaging the flow over a long time scale. The time scale 共Ts兲 dependence of the steady streaming are investigated. Fig. 18 shows the isosurfaces of spanwise vorticity 共␻z = ⫾ 1.5兲 of the steady streaming calculated by different time

Fig. 17. Steady streaming structures at z / D = −3, 0 and 3 calculated by the average of the 15th period of oscillatory flow 共KC= 10, ␤ = 196, and e / D = ⬁兲

scales 共started from the 15th period兲.The structures of vorticity isosurfaces by averaging velocity over 20 and 40T 关Figs. 18共d and e兲兴 are very close to each other. The steady streaming structures averaged over a time scale of 20T appear to be converged for KC= 10. Further increase of Ts does not result in significant difference on streaming structures. The flow structures in Figs. 18共d and e兲 show very weak variation along the spanwise direction. The convergence of the sectional steady streaming structure is examined. Fig. 19 shows the steady streaming at the middle cross section 共z / D = 0兲 calculated by different Ts values. Steady streaming converges when Ts reaches 20T. The steady streaming structure shown in Fig. 19 is asymmetric with respect to the y axis. This indicates that the structures of oscillatory flow in two succeeding halves of flow period are asymmetric with respect to the y axis at the cross section of z / D = 0.0. Similar convergence process can be found at other cross sections. For KC= 10, although the flow is aperiodic, the major structure of steady streaming obtained by averaging the flow velocity over a long time scale varies little along the cylinder’s spanwise direction as shown in Fig. 18共f兲. A converged steady streaming structure is achieved if the averaging time scale exceeds 20 flow periods. The converging time scale may be larger for some KC numbers, at which poor correlation of instantaneous flow structure along the spanwise direction was found, i.e., KC= 22 共Obasaju et al. 1988兲 and KC= 20 共Zhao et al. 2009兲. At these KC values, because the flow structures within two successive periods differ from each other dramatically 共Zhao et al. 2009兲, the 3D numerical simulation has to be run for an extreme long time to capture all the possible flow structures; however this is out of reach with currently available computational facility.

Conclusions In this paper, RANS equations with a k-␻ turbulent model are solved to simulate the oscillatory flow around a circular cylinder close to or sitting on a plane boundary by using finite-element JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2011 / 31

J. Hydraul. Eng. 2011.137:23-33.


(a) Ts = T

(b) Ts = 5T

(c) Ts = 10T

(d) Ts = 15T

(e) Ts = 20T

(f) Ts = 40T

Fig. 18. Isosurfaces of averaged spanwise vorticity 共␻z = ⫾ 1.5兲 calculated by different time intervals 共Ts / T = 1, 5, 10, 15, 20, and 40兲 at KC = 10, ␤ = 196, and e / D = ⬁

method. The major findings of this study can be summarized as following: 1. For KC number in the range of 2 ⱕ KCⱕ 4, 7 ⱕ KCⱕ 15, and 24ⱕ KCⱕ 30, the steady streaming structures are symmetric with respect to the y axis. In the range of 4 ⱕ KCⱕ 7, the streaming structure is symmetric except the gap range of 0 ⬍ e / D ⱕ 0.5. In the range of 15ⱕ KCⱕ 24, the streaming structure are asymmetric for e / D ⬎ 0.2 and then switches to symmetric for e / D ⱕ 0.2. 2. The steady streaming structure is not sensitive to the change of KC number in the range of 7 ⱕ KCⱕ 30 when e / D ⱕ 0.2. 3. For a certain KC number, significant effect of the plane boundary on the steady streaming structure is found when e / D ⱕ 1.0.

(a) Ts = T Us/Um:

4.

5.

6.

The amplitude of velocity of steady streaming is up to about 70% of the amplitude of Um at KC= 10 and e / D = 0. The location of high steady streaming velocity is affected by the KC number and gap ratio. The 3D results of KC= 10, ␤ = 196, and e / D = ⬁ show strong 3D flow structures around the cylinder. The flow structures are aperiodic and a converged steady streaming is achieved by the average of 20 successive flow periods. The comparison between 2D and 3D results shows the 2D simulation captures the major features of the steady streaming.

Acknowledgments The writers would like to acknowledge the support from Australia Research Council through ARC Discovery Project Program 共Grant No. DP0557060兲 and the National Natural Science Foundation of China 共Grant No. 50428908兲.

(b) Ts = 5T Us/Um:

References

(c) Ts = 10T Us/Um:

(e) Ts = 20T Us/Um:

(d) Ts = 15T Us/Um:

(f) Ts = 40T Us/Um:

Fig. 19. Steady streaming structures at z / D = 0 calculated with different Ts values 共KC= 10, ␤ = 196, and e / D = ⬁兲

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