# Numerical simulation of incompressible gusty flow past a circular cylinder

## Numerical simulation of incompressible gusty flow past a circular cylinder

Alexandria Engineering Journal (2018) 57, 3321–3332 H O S T E D BY Alexandria University Alexandria Engineering Journal www.elsevier.com/locate/aej...

Alexandria Engineering Journal (2018) 57, 3321–3332

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Numerical simulation of incompressible gusty flow past a circular cylinder Chirag J. Parekh a, Arnab Roy b, Atal Bihari Harichandan a,* a b

Department of Mechanical Engineering, Marwadi University Rajkot, 360003, India Department of Aerospace Engineering, IIT Kharagpur, 721302, India

Received 30 September 2016; revised 30 November 2017; accepted 4 December 2017 Available online 13 November 2018

KEYWORDS Circular cylinder; One dimensional sinusoidal gust; Transverse perturbation velocity; Vortex shedding; Frequency locking; Incompressible NavierStokes equations

Abstract Flow around a circular cylinder under the inﬂuence of a one dimensional gust is investigated in the present paper. In the present study the gusty ﬂow which impinges on the cylinder is produced by superimposing a single harmonic sinusoidal transverse perturbation velocity component on uniform ﬂow. A gust source region is deﬁned for creating the perturbation velocity. Streamlines, vorticity and iso-velocity contours are used to physically interpret the ﬂow ﬁeld features. Temporal variations of force coefﬁcients and stagnation-separation points on the cylinder surface are studied. Frequency spectrum of force coefﬁcients reveal interesting facts about the variation of energy content in the gust frequency peak in comparison with vortex shedding frequency peak as a function of Reynolds number (Re) and the angular frequency of the gust (x). Wherever possible, an attempt has been made to compare the ﬂow features between uniform and gusty ﬂow conditions. The present numerical work has been performed using ‘CFRUNS’ – a numerical scheme developed by Harichandan and Roy (2010) for solving incompressible two-dimensional NavierStokes equations using unstructured grid. Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

1. Introduction Circular cylinder has a simple geometry but the ﬂow around it is very complex. Due to its bluff geometry a large separated wake is formed behind it. It is nearly impossible to predict the ﬂow features within its separated wake analytically and hence experimentation or numerical simulations are performed to study them. Vortex shedding behind a cylinder and consequent periodic variation of the ﬂow ﬁeld above a threshold * Corresponding author. E-mail address: [email protected] (A.B. Harichandan). Peer review under responsibility of Faculty of Engineering, Alexandria University.

Reynolds number are very attractive ﬂow features which are worth studying. When Reynolds number increases beyond around 47, asymmetry in the vortex structure is observed in the rear portion of the cylinder surface due to growth of ﬂow instability. Subsequently alternate separation of vortices from upper and lower portions of the cylinder gets initiated. The shed vortices convect and diffuse into the cylinder wake forming the well-known Von-Karman vortex street. The study of ﬂow past circular cylinder is of great practical importance for many real life problems and industrial applications. The commonly used cylindrical structures are exposed to either air or water ﬂow, and therefore they experience ﬂowinduced vibration along with ﬂuid and thermal stresses which could lead to structural failure under severe conditions. Thus it

https://doi.org/10.1016/j.aej.2017.12.008 1110-0168 Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

3322 becomes very important to predict the stress and response of the cylindrical structures under such conditions. Flow past circular cylinders arranged in various geometric conﬁgurations has been studied experimentally and numerically by many researchers in the past. Investigators have studied various aspects of the problem including ﬂow interference between multiple cylinders placed at small gaps, vortex induced vibration, time evolution of pressure and force coefﬁcients acting on the cylinders, their structural stability under the action of such stresses, effect of wall proximity on the cylinder ﬂow, vortex shedding frequency etc. Uematsu  conducted a series of wind-tunnel tests on the aeroelastic behavior of a pair of thin circular cylindrical shells in staggered arrangements. Prebuckling and buckling behavior of the downstream cylinder in both smooth and turbulent ﬂows was measured. Gu  experimentally investigated the interference between multiple cylinders in uniform smooth ﬂow at supercritical Reynolds number in a wind-tunnel. Abrupt change in pressure distribution over the cylinders and effect of interference at a small spacing ratio between the cylinders in both staggered and tandem arrangements was investigated. Brika and Laneville , Brika and Laneville  experimentally investigated the vortex-induced oscillations of single as well as multiple cylinders in tandem and staggered arrangements. Gu and Sun [6,7] experimentally investigated the interference between multiple circular cylinders in staggered arrangements, placed in a uniform smooth ﬂow at high subcritical Reynolds number. They studied the differential lift forces on the cylinders based on different spacing ratios between them. Ozono  investigated the ﬂow around a circular cylinder with a few interference elements along the wake. He compared the spatial trace of the critical gaps with those of the experiments using splitter plates as an interference element. Sumner et al.  and Li and Sumner  experimentally studied the Strouhal number data for two staggered circular cylinders placed at different gap ratios and incidences. The two-dimensional (2D) and three-dimensional (3D) ﬂow features were studied in detail. Flow past two side-by-side identical circular cylinders was numerically investigated by Liu et al.  with unstructured spectral element method. They identiﬁed a total of nine different wake patterns based on various non-dimensional distances between the cylinder centres and studied the time evolution of lift and drag coefﬁcients corresponding to each unsteady wake pattern. Lee and Yang  numerically studied the ﬂow patterns past two circular cylinders in proximity using an immersed boundary method. Carmo et al.  and Kim et al.  investigated the ﬂow-induced vibration of a single circular cylinder and two circular cylinders in tandem arrangement respectively. They observed that the vibration amplitude of the downstream cylinder is strongly dependent on whether the upstream cylinder is vibrating or ﬁxed, whereas response of the upstream cylinder is weakly dependent on the downstream cylinder. Harichandan and Roy  have investigated the ﬂow past single and tandem cylindrical bodies in the vicinity of a plane wall. They discussed the interaction of the vorticity in the boundary layer formed over the plane wall with the vorticity associated with the shear layer emanating from the separation points on the cylinder surface. Flow ﬁeld around cylindrical bodies becomes more complex if there is an unsteady ﬂow impinging on it instead of a uniform one. Study of an unsteady gusty ﬂow past a bluff body

C.J. Parekh et al. has enormous practical importance due to its varied real life and industrial applications. Though many research works on one-dimensional gusty ﬂow  and two-dimensional gusty ﬂow  past airfoils have been reported, signiﬁcant work on gusty ﬂow past circular cylinders is scanty in open literature. In the present paper, authors have focused on the study of ﬂow past a circular cylinder placed in a one dimensional gusty ﬂow. The difference in ﬂow features and vortex shedding frequency between gusty ﬂow and uniform ﬂow conditions have been studied at low Reynolds number. The maximum Reynolds number used in the present study is 200 and the ﬂow has been assumed to be two dimensional. A two dimensional incompressible Navier Stokes solver developed based on the ‘CFRUNS’ scheme of Harichandan and Roy  has been used for performing the present computations. The solver uses unstructured grid comprising of triangular elements. This solver is an extension of the original ﬁnite volume scheme proposed by Roy and Bandyopadhyay  based on structured grid comprising of quadrilateral elements. 2. Governing equations and numerical scheme The physical problem considered in this study is a twodimensional unconﬁned incompressible ﬂow around a circular cylinder. The ﬂow past cylinder has been computed both for uniform ﬂow and gusty ﬂow condition. In case of gusty ﬂow simulations, a zone is deﬁned near the inﬂow boundary where a v-component of perturbation velocity is added to the uniform ﬂow. A suitable source term is included in the momentum equation to account for this perturbation and satisfy the incompressibility constraint. The Continuity equation and the two components of Momentum equation incorporating the previously deﬁned source terms are the governing equations for the present ﬂow investigation. These equations, in absence of body forces and heat transfer, can be expressed in the conservative non-dimensional primitive variable form as follows: ! !! @V 1 2! ! ! r: V ¼ 0 ð1Þ þ r: V V ¼ rp þ r VþS @t Re ! where V is the velocity vector in the ﬂow ﬁeld, p is ratio of pressure and density, Re is the Reynolds number and t is the ! non-dimensional time. S is the source term added to simulate the gust, which in the case of a two-dimensional gust would have two components and therefore can be represented as ! ! S ¼ S ðsu ; sv Þ. An actual gust can be represented in the form of a Fourier spectrum comprising of various perturbation frequencies and wave numbers. For the present numerical simulations, a single harmonic one-dimensional (1D) vcomponent perturbation velocity is used. The perturbation velocity component may be represented as follows: vg ¼ I cos ðkx  xtÞ

ð2Þ

where I ¼ Ig  U1 , Ig is the gust intensity relative to the mean ﬂow, k is the gust wave number, x is the angular frequency of the gust and U1 is the free-stream velocity. The linear frex quency of the gust has been considered as: f ¼ 2p and the xD imposed gust reduced frequency has been deﬁned as: fi ¼ 2U . 1 A gust source is provided in the rectangular box shaped region shown in Fig. 1(a) which is responsible for creating

Numerical simulation of incompressible gusty ﬂow past a circular cylinder the v-component of perturbation velocity. This gust source is placed in a region of uniform ﬂow and is oriented along the ﬂow direction. The gust source should satisfy the incompressibility constraint and it is added as a source term to the ymomentum equation within the box region. The gust source sv should satisfy the following equations: @vg @vg þ U1 ¼ sv @t @x @sv ¼0 and @y

ð3Þ

The solution of such a gust perturbation velocity is developed by Golubev et al.  as follows: sv ¼ Kv  g0 ðxÞ  kðyÞ sin ðxt  kxs Þ

where b satisﬁes the constraint: jx  xs j 6 pb (x being the coordinate of any point within the gust source box) and g0 ðxÞ is deﬁned as: (  b2 sin fbðx  xs Þg; jx  xs j 6 pb 0 g ðxÞ ¼ ð6Þ 0; jx  xs j > pb and kðyÞ is a smoothing function deﬁned as: ð7Þ

This approach is followed in order to minimize wave reﬂections and ﬂow distortion which might occur if the perturbation ﬁeld does not satisfy the incompressibility constraint. In the present calculations, free stream values for pressure and u-velocity are assigned as the initial values to each triangular cell in the domain. Free stream zero value for v-velocity is assigned to each cell in the domain except for the cells lying within the gust source region. Within the gust source region the cells are assigned with initial v-velocity component of vg ¼ I cos ðkx  xtÞ. Gust generation from this source region initiates at t = 0 and it convects downstream towards the cylinder with local wave velocity. Since the computational

Fig. 1

Table 1 Grid independence test carried out for ﬂow past a circular cylinder at Re = 100. Number of nodes on the body

Drag coeﬃcient (CD)

Lift coeﬃcient (CL)

Strouhal number (St)

80 (Grid 1: 14,278 cells) 120 (Grid 2: 23,057 cells) 160 (Grid 3: 29,464 cells) 200 (Grid 4: 36,015 cells)

1.112 ± 0.021

0 ± 0.181

0.168

1.185 ± 0.015

0 ± 0.21

0.164

1.352 ± 0.010

0 ± 0.278

0.161

1.358 ± 0.010

0 ± 0.281

0.161

ð4Þ

where xs is the x-coordinate of the gust source box centre.   Ig  U21 k2  b2   and Kv ¼ ð5Þ b2 sin Uxp 1b

1 kðyÞ ¼ ftanh ½3ðy þ ys Þ  tanh ½3ðy  ys Þg 2

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domain is in general more restricted than the physical domain, suitable boundary conditions need to be implemented for accurate solution of the governing equations. Zero normal pressure gradient has been applied at the inlet, top and bottom boundaries. Dirichlet free stream pressure has been applied at the outlet. The gust source box is situated at a small distance downstream of the inlet. On top and bottom boundaries zero shear boundary condition: v = 0 and @u = 0 is imposed. At @y

@u @v outlet boundary, @x ¼ @x ¼ 0 is maintained. The inlet, outlet, top and bottom boundaries of the rectangular domain are kept far away from the body surface as shown in Fig. 1. No slip condition is applied on the body surface. ‘CFRUNS’ scheme has been used to solve the incompressible Navier-Stokes equations based on the above initial and boundary conditions with suitable incorporation of the gust source. In this solver, the full Navier-Stokes equations have been solved numerically in the physical plane itself without using any transformation to the computational plane. The ﬂux reconstruction cell on the face of the control volume is placed centrally between the two cells which share that face. The cell face center velocities are reconstructed explicitly by solving the momentum equations on ﬂux reconstruction control volumes deﬁned judiciously around the respective cell face centers. This is followed by solution of the cell centre pressure ﬁeld using a discrete Poisson equation developed from the reconstructed velocities and updating the cell centre velocities by using an explicit scheme.

(a) Geometrical description of the problem. (b) Close-up view of the triangular mesh around a single circular cylinder.

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3. Results and discussions Numerical simulations have been performed for Re = 100 and 200 with gust intensity, I = 0.8, and a range of angular frequencies of the gust, x = 0.25p, 0.5p, 0.75p and 1.0p respectively. This gives linear frequencies of 0.125, 0.25, 0.375 and 0.5 Hz, respectively. A few simulations have been performed with two other values of gust intensity, namely, I = 0.4 and

Fig. 2

Fig. 3

1.2 respectively, for x = 0.5p. The results for these two gust intensities have been used only for qualitative comparison of ﬂow ﬁeld features and not reported in detail in the present paper. The grid independence test has been carried out for uniform ﬂow past a single circular cylinder at Re = 100 to optimize the computational effort as shown in Table 1. Based on the grid independence test carried out for ﬂow past an unconﬁned circular cylinder in uniform ﬂow, a grid

Instantaneous streamlines and vorticity contours for uniform ﬂow at Re = 100, t = 200.

Instantaneous streamlines and vorticity contours with unsteady inlet ﬂow for different gust frequencies at Re = 100, t = 200.

Numerical simulation of incompressible gusty ﬂow past a circular cylinder

Fig. 4

Instantaneous streamlines and vorticity contours for uniform ﬂow at Re = 200, t = 200.

with 160 nodes on the cylinder surface is used for all the simulations. This grid contains approximately 30,000 cells in the entire domain. A close-up view of the mesh generated for the present simulation work has been presented in Fig. 1(b). Figs. 2 and 3 represents the instantaneous streamlines and vorticity contours for circular cylinder placed in an uniform ﬂow and a gusty ﬂow respectively with different gust frequencies at Re = 100 and non-dimensional time t = 200. It is observed that the shear layers from the top and bottom surfaces of the cylinder curl up into an alternating sequence of vortices of differing strengths due to difference in gust frequency and shed vortices frequency. Under gusty ﬂow, the vortices are substantially elongated both in stream-wise and crossstream directions at lower gust frequencies. The negative vortices which are shown with dotted lines are more elongated and branched as compared to the positive vortices which are pre-

Fig. 5

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sented with solid lines. With increase in gust frequency the vortex structures gradually show enhanced resemblance with those of uniform ﬂow past a circular cylinder as shown in Fig. 2. Figs. 4 and 5 are similar to those of Figs. 2 and 3 respectively, except for the Reynolds number, which is 200 in the latter case. Comparing the vorticity contours at two Reynolds numbers it can be seen that the vortex structures show resemblance with that of uniform ﬂow at higher values of gust frequency as Reynolds number increases. Due to the asymmetry in the vortex structures between the negative vortices (shown with dotted lines) and positive vortices (shown with solid lines), a small non zero mean lift force acts on the cylinder at lower gust frequencies which is later mentioned in Table 2. For the range of gust intensity and angular frequencies considered in the present study, the negative and positive

Instantaneous streamlines and vorticity contours for different gust frequencies at Re = 200, t = 200.

3326 Table 2

C.J. Parekh et al. Mean values of force coefﬁcients.

Angular frequency of gust (x)

0.25 p 0.5 p 0.75 p 1.0 p Uniform ﬂow 

Lift coeﬃcient

Drag coeﬃcient

Re = 100

Re = 200

Re = 100

Re = 200

0.083 ± 1.3806 0.0267 ± 0.8684 0.0224 ± 0.3814 0.022 ± 0.2697 0 ± 0.278

0.1851 ± 1.4911 0.0638 ± 0.8945 0.0410 ± 0.5534 0.0403 ± 0.4024 ±0.602

1.2712 ± 0.1434 1.3376 ± 0.1765 1.3598 ± 0.0226 1.3637 ± 0.0216 1.352 ± 0.010

1.2158 ± 0.8333 1.2476 ± 0.4318 1.2872 ± 0.2143 1.4385 ± 0.2017 1.32 ± 0.05

vortices are observed to originate from the upper and lower surfaces of the cylinder respectively similar to that of the case with uniform ﬂow. It is interesting to study the stream-wise and cross-stream distances between the vortex cores as they convect downstream along the cylinder wake under the combined inﬂuence of the freestream and gust perturbation. Fig. 6 presents the schematic diagram for mean vortex core distance (VCD). The mean vortex core distances in stream-wise and cross-stream directions are deﬁned as: Fig. 6

Schematic diagram to show mean vortex core distance.

Fig. 7

Mean vortex core distance at Re = 100 and 200 for different gust frequencies.

Re = 100 Fig. 8

Re = 200

Temporal behavior of lift coefﬁcient for uniform ﬂow over a circular cylinder at Re = 100 and Re = 200.

Numerical simulation of incompressible gusty ﬂow past a circular cylinder VCDðstream  wiseÞ ¼

N X Dxi

N

i

VCDðcross  streamÞ ¼

N X Dy

ð8Þ

i

i

N

where Dxi and Dyi are the distances between vortex cores in stream-wise and cross-stream directions respectively for N samples as shown in Fig. 6. DxiU and DxiB are the streamwise distances between vortex cores in upper and lower layer of vortices respectively as indicated in the ﬁgure. Dxi is calculated as net summation of DxiU and DxiB . Fig. 7 presents the distribution of mean vortex core distances (stream-wise and cross-stream) as a function of Reynolds number and angular frequency of gust. It is observed that the mean vortex core distances decrease with increase in gust frequencies. Also, the mean values are lower at higher Reynolds number. Therefore, highly stretched vortex structures which exist at the lower frequency range become signiﬁcantly narrower with increase in gust frequency and marginally more so due to increase in Reynolds number. This observation is also clearly evident from the contour plots in Figs. 2 and 4 respectively. Fig. 8 shows the temporal variation of lift coefﬁcients for uniform ﬂow over a circular cylinder at Re = 100 and Re = 200. The periodic variation of lift coefﬁcient about

Fig. 9

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zero-mean value are experienced due to the alternate shedding of vortices from the upper and lower surfaces of the cylinder in the downstream side. However, the variation of lift coefﬁcient in case of gusty ﬂow over a circular cylinder will vary considerably from the case of uniform ﬂow. Figs. 9 and 10 show the time history of lift and drag coefﬁcients at Re = 100 and 200 for different gust frequencies. Unlike the case of uniform ﬂow, lift coefﬁcient varies periodically about a non-zero mean value. The mean value varies with gust frequency and Reynolds number. It decreases with increase of frequency and increases with increase in Reynolds number as shown in Table 2. The ﬂuctuating component of the lift exhibits similar dependence on frequency and Reynolds number as well. The mean drag seems to gradually increase with increase in frequency. The ﬂuctuating component of drag decreases with increase in frequency. Figs. 11 and 12 represents the frequency spectrum of the lift coefﬁcient at Re = 100 and 200 respectively for the four different gust frequencies. This has been obtained by performing FFT of lift coefﬁcient time history. FFT of time history of front stagnation point angle also yielded identical results for the dominant frequencies. At the lowest gust frequency of x = 0.125 Hz, there is evidence of vortex locking between the gust and cylinder vortex shedding frequency. The vortex shedding frequency merges with gust frequency to form a sin-

Temporal behavior of force coefﬁcients for different gust frequencies at Re = 100.

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C.J. Parekh et al.

Fig. 10

Temporal behavior of force coefﬁcients for different gust frequencies at Re = 200.

Fig. 11

Frequency spectrum of lift for different gust frequencies at Re = 100.

Numerical simulation of incompressible gusty ﬂow past a circular cylinder

Fig. 12

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Frequency spectrum of lift for different gust frequencies at Re = 200.

gle dominant peak of large amplitude and power content. This is observed at both the Reynolds numbers as is evident from their spectra. The higher harmonics at this gust frequency are visible at 0.25 Hz and 0.375 Hz for both Re = 100 and 200. As the gust frequency increases, amplitude of the gust frequency peak reduces gradually. A comparatively weak vortex shedding frequency peak is also produced at all the four gust frequencies and at both Reynolds number. Its amplitude increases with increase in gust frequency and Reynolds number. As far as the other frequency peaks and harmonics are concerned, for example, at x = 0.25 Hz, secondary frequency peaks for Re = 100 are observed at 0.29 Hz, 0.5 Hz and 0.75 Hz and for Re = 200 at 0.28 Hz and 0.5 Hz respectively. Therefore, several secondary frequencies and harmonics are spread across the spectra. As the gust frequency increases, the vortex shedding frequency tends to asymptotically approach the value for uniform ﬂow. This signiﬁes that at higher gust frequency there is a gradual decoupling between the two. At x = 0.375 Hz and higher, the vortex shedding frequency is nearly restored to its uniform ﬂow value. The results are briefed in Table 3.

Unsteady separation of shear layers and their complex interaction in the wake region have profound impact on the time history of stagnation point as well as lower and upper separation point locations on the cylinder surface. The stagnation point angle is identiﬁed on the basis of highest value of pressure coefﬁcient on the cylinder surface. The location of the pair of separation points is identiﬁed on the basis of vanishing wall shear stress. The angles ‘h1’ and ‘h2’ are the angular position of upper and lower separation point measured clockwise from the front stagnation point along the cylinder surface in degrees as shown in Fig. 13. The angles oscillate periodically and the frequency of oscillation is found to be identical to that of the Strouhal frequency which is obtained from FFT of the time history of the lift coefﬁcient as mentioned earlier.

Table 3 Strouhal number values in gusty ﬂow at Re = 100 and 200. Angular frequency of gust (x)

Strouhal number Re = 100

Re = 200

0.125 0.25 0.375 0.5 Uniform ﬂow 

0.125 0.13 0.15 0.157 0.161

0.125 0.13 0.185 0.188 0.192

Fig. 13

Stagnation point and separation points.

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Fig. 14

Table 4 ﬂow.

Temporal behavior of stagnation angle for gusty ﬂow and uniform ﬂow.

Amplitude of variation of stagnation angles in gusty

Angular frequency of gust (x)

Stagnation angle Re = 100

Re = 200

x = 0.25 p x = 0.5 p x = 0.75 p x = 1.0 p Uniform ﬂow 

32.7° 16.15° 4.98° 1.8° 1.32°

35.88° 18.75° 6.19° 4.63° 2.13°

Fig. 14 shows the time variation of stagnation point angle (in degrees) for x = 0.25 p at Re = 100 and 200. The results for uniform ﬂow are superimposed in the same ﬁgure. The amplitude of variation of the stagnation angle is much higher for gusty ﬂow as compared to uniform ﬂow for obvious reasons. The amplitude of variation is higher at Re = 200 and decreases as the value of the gust frequency increases. The numerical values are summarized in Table 4 for different gust frequencies. Fig. 15 shows the time history of upper-lower separation point angles (in degrees) for both uniform and gusty ﬂow condition at Re = 100 and x = 0.25 p. Upper and lower separa-

Fig. 15

tion points are almost symmetrically disposed about the horizontal axis of symmetry for uniform ﬂow. However, for gusty ﬂow, the variations of respective separation point angles are asymmetrical about the horizontal axis of symmetry. Consequently, the included angle between the two points varies with time, making it sometimes higher and sometimes lower than that of uniform ﬂow. This is another characteristic of highly unsteady wide wake structure in gusty ﬂow, especially at lower gust frequencies. However, as gust frequency increases, the included angle between upper and lower separation points decreases, thereby leading to narrower wake. The variations of upper-lower separation angles for gusty ﬂow with different frequencies are summarized in Table 5. All the angles are expressed in the form of abd , where a represents the mean value of upper-lower separation angle for uniform ﬂow. b and d represent the variation of these separation angles for gusty ﬂow from mean value (a). Difference between the upper-lower separation angles becomes considerably less for the cylinder with higher gust frequency. Also, the separated ﬂow zones are displaced symmetrically above and below the horizontal axis of symmetry which passes through the centre of the cylinder. This is also evident from the vortex patterns of the ﬂow shown in Figs. 3 and 5 for Re = 100 and 200 respectively at higher gust frequencies.

Temporal behavior of upper-lower separation angles for gusty ﬂow and uniform ﬂow.

Numerical simulation of incompressible gusty ﬂow past a circular cylinder Table 5

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Variation of upper-lower separation angles in gusty ﬂow.

Gust frequencies (x)

x = 0.25 p x = 0.5 p x = 0.75 p x = 1.0 p Uniform ﬂow 

Fig. 16

Upper separation angle

Lower separation angle

Re = 100

Re = 200

Re = 100

Re = 200

116:841:14 148:91 116:826:02 14:04 116:81:44 5:87 116:82:21 2:86 116:81:70 1:79

111:841:02 147:20 111:835:65 14:93 111:81:73 9:04 111:86:09 6:19 111:84:45 4:49

243:228:21 275:87 243:219:69 8:30 243:212:04 0:21 243:22:21 1:95 243:22:19 2:22

247:922:82 283:03 247:914:99 13:0 7:34 247:94:48 5:84 247:95:11 4:49 247:94:41

Iso v –velocity contours with superimposed streamlines for Re = 100, x = 0.5p at t = 30.

Fig. 16 shows the iso v-velocity contours for three different gust intensities, namely, I = 0.4, 0.8 and 1.2 respectively at x = 0.5p, t = 30. The ﬁgures show that the gust front on impinging the cylinder clearly disintegrates into three different segments. Two outer segments seem to get convected with the freestream, whereas the central segment interacts in a complex way with the cylinder wake. The gust waves seem to retain their identity more strongly in the near wake with increase in gust intensity, thereby making the interaction between them and the shed vortices more intense. This interference has a strong bearing on the ﬂow around the cylinder as well as in its near wake. A more detailed study on the effect of change of amplitude is therefore warranted. 4. Conclusions The ‘CFRUNS’ scheme has been used to study the ﬂow features for low Reynolds number two dimensional uniform ﬂow and one dimensional sinusoidal gusty ﬂow past a circular cylinder. Streamlines, vorticity and iso-velocity contours were used to physically interpret the ﬂow ﬁeld features. The interesting ﬂow characteristics arising due to gust-vortex interactions were well captured by these ﬂow visualization tools. A strong interaction between the gust and shed vortices was observed at the lower gust frequency producing highly unsteady wide wake structure and frequency locking. Consequently a strong merged frequency peak in the lift spectrum was observed. At higher gust frequencies the two frequencies remained distinct. The gust peak became weaker and the vortex shedding peak became comparatively stronger with increase in gust frequency as observed in the frequency spectra. At x = 0.375 Hz and above, the vortex shedding frequency asymptotically approached that of uniform ﬂow for both Re = 100 and

200. The time history of stagnation and upper-lower separation point angles revealed interesting differences with uniform ﬂow including enhanced amplitudes of variation and asymmetry in the separation points about horizontal axis of symmetry. References  D. Brika, A. Laneville, Vortex induced oscillations of two ﬂexible circular cylinders coupled mechanically, J. Wind Eng. Ind. Aerodyn. 69 (1997) 293–302.  D. Brika, A. Laneville, Wake interference between two circular cylinders, J. Wind Eng. Ind. Aerodyn. 72 (1997) 61–70.  B.S. Carmo, S.J. Sherwin, P.W. Bearman, R.H.J. Willden, Flow-induced vibration of a circular cylinder subjected to wake interference at low Reynolds number, J. Fluids Struct. 27 (2011) 503–522.  V.V. Golubev, B.D. Dreyer, T.M. Hollenshade, High – accuracy viscous analysis of unsteady ﬂexible airfoil response to impinging gust, in: 15th AIAA/CEAS Aeroacoustics Conference, Miami, Florida, 11–13 May, 2009.  V. Golubev, M. Visbal, Unsteady viscous analysis of low–Re gust-airfoil interaction, in: V European Conference on Computational Fluid Dynamics, Lisban, Portugal, 14–17 June, 2010.  Z. Gu, On interference between two circular cylinders at supercritical Reynolds number, J. Wind Eng. Ind. Aerodyn. 62 (1996) 175–190.  Z. Gu, T. Sun, On interference between two circular cylinders in staggered arrangement at high subcritical Reynolds numbers, J. Wind Eng. Ind. Aerodyn. 80 (1999) 287–309.  A.B. Harichandan, A. Roy, Numerical investigation of ﬂow past single and tandem cylindrical bodies in the vicinity of a plane wall, J. Fluids Struct. 33 (2012) 19–43.  A.B. Harichandan, A. Roy, Numerical investigation of low Reynolds number ﬂow past two and three circular cylinders

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Further reading  Z. Gu, T. Sun, Classiﬁcations of ﬂow pattern on three circular cylinders in equilateral-triangular arrangements, J. Wind Eng. Ind. Aerodyn. 89 (2001) 553–568.