- Email: [email protected]
Steady forces on a cylinder with oblique vortex shedding
Journal of Fluids and Structures 44 (2014) 310–315
Contents lists available at ScienceDirect
Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Brief Communication
Steady forces on a cylinder with oblique vortex shedding Sanjay Mittal n, Sidharth GS Department of Aerospace Engineering, Indian Institute of Technology Kanpur, UP 208016, India
a r t i c l e in f o
abstract
Article history: Received 9 March 2013 Accepted 2 November 2013 Available online 6 December 2013
We present a curious situation of a fluid-flow wherein the body experiences non-fluctuating fluid-flow force despite being associated with an unsteady flow comprising of sustained vortex shedding. The flow past a circular cylinder at Re¼100 is investigated. It is shown that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by a cylinder placed in uniform flow. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. It is found that, both, the parallel and oblique shedding are linearly unstable eigenmodes of the Re¼ 100 steady flow past a cylinder. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Oblique shedding Linear stability analysis Global analysis Finite element method Steady forces
1. Introduction The flow past a circular cylinder has received considerable attention in the last few decades (see, for example, the reviews by Williamson, 1996; Bearman, 1984; Berger and Wille, 1972; Zdravkovich, 1997). It involves rich physics and is associated with many interesting phenomena over a range of Reynolds number. The Reynolds number is defined as Re ¼ U 1 D=ν. Here, U 1 is the free-stream speed, D is the diameter of the cylinder and ν is the coefficient of kinematic viscosity. Beyond Re 6:28 the flow past a cylinder undergoes separation (Sen et al., 2009). This leads to the formation of two symmetric, counter-rotating vortices in the wake which increases in size with an increase in Re. The steady flow past a circular cylinder becomes unstable beyond Re 47 via a Hopf bifurcation (Kumar and Mittal, 2006a, 2006b). After an initial linear growth, it achieves a state of limit cycle due to the non-linear processes (Verma and Mittal, 2011). The alternate shedding of vortices results in fluctuations in pressure and the von Karman vortex street (Williamson, 1996). In the classical configuration of the vortex street, for a nominally two-dimensional cylinder, the axes of the shed vortices are parallel to the axis of the cylinder. This is referred to as parallel shedding. In this situation, the body experiences unsteady force. Usually, the unsteady force in the transverse direction, with respect to the free-stream, is much larger compared to that in the in-line direction. This unsteady force may lead to noise/vibrations of the body. Verma and Mittal (2011) reported the another mode of wake instability for Re 4 110, approximately. Beyond Re 180, spanwise undulations appear in the wake marking the onset of three-dimensional instabilities (Williamson, 1996). It has been observed in several laboratory experiments that the vortices can also be shed at an oblique angle to the axis of the cylinder (Berger and Wille, 1972; Tritton, 1971; Gerich and Eckelmann, 1982; Williamson, 1989). This is referred to as oblique shedding. The obliqueness of the vortices has been attributed to the conditions at the ends of the cylinder. The slant in the shed vortices may also be generated by the spanwise variation in the geometry of the cylinder as in the case of tapered or stepped cylinders (Visscher et al., 2011; Valles et al., 2002; Satish et al., 2013). It was demonstrated by Williamson
n
Corresponding author. Tel.: þ91 512 2597906; fax: þ91 512 2597561. E-mail address: [email protected] (S. Mittal).
0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.11.009
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
311
(1989) that even for a cylinder of large aspect ratio (ratio of the spanwise length to diameter) the end-conditions influence the entire span of the cylinder. The oblique vortices, from both the ends of the cylinder, form a chevron pattern which is symmetric about the mid-span. It is possible to manipulate the end conditions to promote parallel shedding. This is achieved, for example, by inward angling of the leading edge of the end-plates (Williamson, 1989), or by placing control cylinders at the ends (Hammache and Gharib, 1991). The dependence of the oblique angle of the vortices on the conditions at the end-walls has also been investigated computationally (Mittal, 2001; Behara and Mittal, 2010). It has been shown that the oblique shedding angle varies linearly with the thickness of the boundary layer on the end wall. Allowing the velocity to slip on the end walls promotes parallel shedding. It was shown by Williamson (1989) that the vortex shedding frequency for the oblique and parallel shedding is related via the oblique angle of the vortices. Thus, a large number of vortex shedding states, each with a different oblique angle, are possible. This idea was utilized to explain the significant scatter in the measurement of Strouhal number from various experiments. The existence of different oblique shedding modes was also used to explain the discontinuity in the St Re number curve at Re¼64 (Williamson, 1989). Theoretical models, based on transverse stability theory (Albarède and Monkewitz, 1992; Triantafyllou, 1992; Leweke et al., 1997), have been further proposed to explain this phenomenon and the associated spanwise cells. Stability analysis (Noack and Eckelmann, 1994; Konig et al., 1993) has shown that three-dimensional unstable perturbations in the steady wake of cylinder are associated with discrete shedding modes. The focus, so far, in all the work related to oblique vortex shedding, has been toward understanding the onset of the slant in the wake, its modeling, its implications on the scatter in the data, and on techniques to induce parallel shedding. There has been no effort, that we are aware of, to understand the implications of the oblique vortex shedding on the fluid force acting on the body. It is also not clear from the earlier studies if the oblique and parallel shedding are related in any fundamental way. The primary objective of the present work is to investigate the effect of oblique shedding on the fluid force acting on the body and contrast it to the situation with parallel shedding. The other objective is to determine if any fundamental relationship between the two modes of vortex shedding exists, and utilize it to explain the difference in the time variation of fluid force in the two cases. We first show, via Direct Numerical Simulation (DNS) of the Re¼100 flow past a cylinder, that the oblique vortex shedding is associated with spanwise periodicity. The fluid force, on a segment of the cylinder comprising of an integral number of spanwise wavelengths of the oblique waves, does not vary with time. The spanwise variation of the unsteady pressure field, during one cycle of vortex shedding, is studied to explain the distinction in the time-variation of force coefficients for oblique and parallel shedding. By carrying out a global linear stability analysis of the two-dimensional steady flow past a cylinder, we show that there are several unstable oblique modes with differing spanwise periodicity. We further show that the parallel mode is a special case of the oblique mode whose wavelength of periodicity along the span is infinite. 2. Results A cylinder of aspect ratio AR ¼120, resides in a hexahedral domain. The x-axis is along the free stream flow while the z-axis is aligned with the axis of the cylinder. The cylinder occupies the entire span of the domain. Experiments carried out on high aspect ratio cylinders at Re¼100, in the past, show that the vortices in the wake form a chevron pattern and posses symmetry along the centerline (Williamson, 1989). Taking advantage of the symmetry, only one half of the span is simulated and symmetry conditions are applied at the mid-span. It should be noted that the imposition of the symmetry boundary conditions, in the numerical set-up, does not interfere with the development of obliqueness in the vortices shed in the wake of the cylinder. The upstream and downstream boundaries are located at a distance of 50D and 100D, respectively from the center of the cylinder, while the height of the domain is 100D. Uniform flow is prescribed at the upstream face of the domain. At the outflow boundary the stress vector is set to zero. To promote oblique shedding, no-slip condition on the velocity is specified on one of the end walls (z¼ 0) for x=D Z 5:0. Symmetry conditions are prescribed on this boundary for x=D o5:0. No-slip condition is also specified on the cylinder. Computations are carried out for the Re ¼100 flow with a mesh consisting of 14 million nodes and 13.7 million 8-noded hexahedral elements, approximately. A stabilized finite element formulation is utilized to solve the incompressible flow equations in primitive variables (Mittal, 2001). Equal-in-order interpolation functions are used for velocity and pressure. The simulation is carried out up to a non-dimensionalized time of 2500. t¼ 0 corresponds to the onset of vortex shedding. Fully developed oblique shedding is attained at t 1200 corresponding to 96 cycles of vortex shedding. Computations are also carried out for the case of parallel shedding. This is achieved by assigning symmetry conditions on both the end walls. The force on the cylinder, due to the fluid flow, is computed by integrating the stress, r, on its surface. The sectional force coefficients, per unit span, are defined as Z Z 1 1 C l ðzÞ ¼ 1 2 r n^ y dΓ; C d ðzÞ ¼ 1 2 r n^ x dΓ: ð1Þ 2 ρU 1 D Γ 2D 2 ρU 1 D Γ 2D Here, ρ is the density of fluid, n^ is the unit normal vector and Γ 2D is the circle lying on the cylinder at a spanwise location z. The lift and drag coefficients, for a finite span of the cylinder (b ¼ z2 z1 ), can be computed by integrating the sectional coefficients along span: Z Z 1 z2 1 z2 C l ðzÞ dz; C D ¼ C ðzÞ dz: ð2Þ CL ¼ b z1 b z1 d
312
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
Fig. 1 shows instantaneous pictures of the fully developed unsteady flows with parallel and oblique shedding. The flow with oblique shedding has a two cell structure: central and end. The bulk of the span is occupied by the central cell and consists of oblique vortices. The end cell forms in the region close to the no-slip wall. Compared to the central cell, it is associated with a lower shedding frequency. The Strouhal number, corresponding to the vortex shedding frequency, for the central cell is 0.1590; it is 0.1637 for the parallel shedding. These values are in excellent agreement with the experimental values of 0.160 and 0.164 reported for oblique and parallel shedding, respectively, by Williamson (1989) at Re¼100 for cylinders with similar aspect ratios. The cosine rule (Williamson, 1989) relating the vortex shedding frequency of the parallel and oblique modes (Stparallel ¼ Stoblique cos θ) is satisfied to a reasonable approximation. The oblique shedding angle for the central cell is θ 151. The periodic structure along the span, in the central cell of the oblique shedding, is quite apparent from Fig. 1. In the initial stages of the simulation for oblique shedding, the vortices in the central cell region are shed parallel to the cylinder. Oblique shedding begins from the end cell region and propagates outward in the form of an oblique front (Yang et al., 1993; Williamson, 1989). The difference in the vortex shedding frequency in the two cells leads to the appearance of vortex dislocations. More details, including those that demonstrate the adequacy of the resolution, can be found in our earlier work (Behara and Mittal, 2010). Fig. 2 shows the time histories of the force coefficients for the oblique and parallel shedding for the fully developed unsteady state. The time is non-dimensionalized with the free stream speed and the diameter of the cylinder. It can be observed that the amplitude of oscillations is significantly smaller for the oblique shedding, compared to the values for parallel shedding. To investigate this further, we analyze the time- and spanwise variation of sectional force coefficients as defined by Eq. (1). Fig. 3 shows the time history of spanwise distribution of sectional force coefficients for the fully developed unsteady flow. We note the very periodic temporal variation of Cl and Cd, including that in the end-cell. The signature of the periodic generation of vortex dislocations can be clearly observed in this figure. In the region of central cell, we identify two complete spanwise wavelengths of the oblique waves (20:0 r z=D r 54:8). This region is marked in broken lines in Fig. 3. The time histories of the force coefficients integrated across the entire span as well as along the segment with
Fig. 1. Re¼ 100, AR ¼ 120 DNS of flow past a cylinder: isosurfaces of the cross-stream component of velocity (v ¼ 7 0:1) for the fully developed unsteady flow. Symmetry boundary condition is applied at z=D ¼ 60 and only half the span is considered. The boundary condition at z=D ¼ 0 corresponds to (a) noslip wall and (b) slip wall.
Fig. 2. Re¼ 100, AR ¼120 DNS of flow past a cylinder: time histories of lift and drag coefficients for oblique (red color) and parallel shedding (black color). Also shown (in blue color) is the time variation of the contribution from the region 20:0 r z=D r 54:8 corresponding to two spanwise wavelengths of the oblique waves. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
313
Fig. 3. Re¼100, AR¼ 120 DNS of flow past a cylinder: time- and spanwise variation of the sectional force coefficients ((a) Cl, and (b) Cd). The region 20:0 r z=D r 54:8, marked by broken lines, corresponds to two spanwise wavelengths of the oblique waves.
Fig. 4. Re¼ 100 global linear stability analysis of the two-dimensional steady flow past a cylinder: real part of the global modes corresponding to the most unstable eigenvalues for β ¼ 0:0 (left) and 0.4 (right). The upper row shows the isosurfaces of cross-stream component of velocity (v ¼ 7 0:001) while the lower frames show the variation in the x y plane at z ¼0.
two oblique cells are shown in Fig. 2. It is observed that the force coefficients corresponding to the region of central cell with integral number of oblique waves are virtually constant with time. The mean drag coefficient from this region is 1.325 while the rms value of the unsteady fluctuation is Oð10 4 Þ. Next, a global linear stability analysis of the Re ¼100 two-dimensional steady flow is carried out. The perturbations are assumed to be periodic in the spanwise direction and of the form u′ðx; y; z; tÞ ¼ ðu^ r ðx; yÞ þiu^ i ðx; yÞÞeiβz eλt . Here, β is the spanwise wavenumber of the perturbation; β ¼ 0:0 corresponds to parallel shedding, while non-zero values of β represent the oblique modes of shedding. u^ r and u^ i represent the real and imaginary parts of the eigenmode. The real part of the eigenvalue, λ, represents the growth rate while the imaginary part is related to the vortex shedding frequency. The analysis is carried out for flows with 60 rRe r200. For each Re, the growth rate is maximum for β ¼ 0 and decreases with an increase in β. The critical β, for which growth rate is zero, corresponds to neutral stability. For Re ¼ 60 the critical β is 0.5, approximately. It increases with an increase in Re. The oblique angle of the vortices (inclination to the axis of the cylinder) increases with an increase in β. This confirms, as has also been presented earlier (Williamson, 1989), that the oblique shedding is as intrinsic to the flow as is the parallel shedding. The mode for β ¼ 0:0 has the largest growth rate. Therefore, the parallel shedding mode is the preferred mode. The results from the linear stability analysis have been confirmed via computations on meshes with increased spatial resolution. Fig. 4 shows the most unstable eigenmode for β ¼ 0:0 and 0.4. The inclination of the vortices, to the cylinder, is clearly seen for β ¼ 0:4. The two modes look very similar in the x y plane. To further investigate the oblique shedding mode, we carry out DNS with periodic boundary conditions on the end walls. The initial condition for the simulation is the real part of the eigenmode, corresponding to the eigenvalue with the largest real part for β ¼ 0:4 from the global linear stability analysis, superposed with the steady-state solution. The unstable mode for β ¼ 0:4 is associated with an oblique angle of shedding that is close to the range of angles observed in experiments. At the same time, the corresponding growth rate is large enough to warrant its rapid growth. The extent of the domain
314
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
along the z-axis is one spanwise wavelength of the global mode ( ¼ 5πD). Computations are also carried out for the parallel mode of shedding (β ¼ 0). The time evolution of the force coefficients, for the two simulations, is shown in Fig. 5. The initial growth is similar to that predicted by the linear stability theory. The time-variation of CD (Fig. 5) clearly shows that, compared to the oblique mode, the parallel mode grows faster. The main difference between the two time histories is the absence of unsteady oscillations in the force coefficients for the fully developed state of the oblique shedding. The rms value of the lift is Oð10 3 Þ and decays to zero, asymptotically with time. We note that the mechanism by which oblique shedding suppresses unsteadiness in the forces is quite different from that due to the use of helical strakes. The spanwise phase variation of forces, in oblique shedding, stems from the spatio-temporal nature of the instability for the flow past a nominally two-dimensional smooth cylinder. However, phase-incoherence induced by strakes is a phenomenon forced by the geometric variation along the span of the cylinder. At Re ¼100, the three-dimensionality in the flow past a straked cylinder is restricted to the near wake of the cylinder. The spanwise nature of the moderate to far wake is essentially identical to parallel shedding. The periodic vortices shed from the strakes induce spanwise velocities of opposite signs. In contrast, the spanwise velocity remains constant over the central cell region in oblique shedding. The reduction in spanwise correlation in the wake is an important mechanism of VIV mitigation by helical strakes at higher Reynolds numbers (Zhou et al., 2011). Moreover, the unstable spanwise modes for the straked cylinder, that are responsible for suppression of the amplitude of shedding, arise beyond Re¼120 (Gomez et al., 2013).
Fig. 5. Re ¼100, AR ¼ 5π DNS of flow past a cylinder with periodic boundary conditions at both end walls: time evolution of force coefficients for simulations initiated with the real part of the most unstable eigenmode for β ¼ 0:0 and 0.4, obtained from the global linear stability analysis of the steady base flow. Also shown are the isosurfaces of cross-stream component of velocity (v ¼ 7 0:1) for the fully developed unsteady flow at t¼ 354.
Fig. 6. St¼100, AR ¼ 5π, β ¼ 0:4 DNS of flow past a cylinder with periodic boundary conditions at both end walls: perturbation of the pressure coefficient, with respect to the time-averaged distribution, on the surface of the cylinder at various time instants during one cycle of the fully developed unsteady flow. The cylinder has been flattened out. θ ¼ 0 represents the front stagnation point while θ ¼ 7 π corresponds to the base point of the cylinder.
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
315
The Strouhal number for the parallel and oblique vortex shedding for the fully developed unsteady flow, estimated by recording signals from probes placed in the near wake, in this work is 0.1637 and 0.1562, respectively. The oblique shedding angle, for β ¼ 0:4, varies from 121 immediately behind the cylinder to 201 in the far wake. Such a streamwise variation of the oblique angle has also been observed in laboratory experiments (Koopman, 1967). The empirical cosine rule (Williamson, 1989) relating the oblique shedding angle to vortex shedding frequency for parallel and oblique shedding (Stparallel ¼ Stoblique cos θ) is satisfied to a good approximation beyond the mid-wake (x Z5D). Fig. 6 shows the disturbance in the pressure coefficient, with respect to the time-averaged pressure, on the surface of the cylinder at several time instants during one cycle of the fully developed vortex shedding for β ¼ 0:4. The perturbations are periodic along the span and appear to travel outward. Unlike parallel shedding, the spanwise spatial periodicity of the perturbations results in zero lift and steady drag at each time instant in the limit cycle. This is true for all cases of oblique shedding, no matter how small the inclination angle of the vortices is, provided integral number of spanwise wavelengths are included. Of course, the wavelength increases with a decrease in obliqueness. The physical reason for the vanishing unsteadiness is the phase variation over the cylinder that spans 1801. On the contrary, the unsteady lift and drag are nonzero when the vortices are shed exactly parallel to the cylinder. In that sense, the parallel shedding is a singular case of oblique shedding. 3. Conclusions The flow past a cylinder at Re ¼100 has been investigated. It is shown via linear stability analysis that both the parallel and oblique vortex shedding are linearly unstable eigenmodes of the two dimensional steady base flow. This reaffirms the belief that, similar to the parallel shedding, the obliqueness of the shed vortices is intrinsic to the flow. In fact, there exist a vast number of unstable eigenmodes for the Re ¼ 100 flow past a cylinder, each corresponding to a certain angle of oblique vortices. The parallel vortex shedding is one of the many possible states and is associated with the largest growth rate. The spatio-temporal structure of the vortex shedding instability has been demonstrated. The spatio-temporal periodicity along the cylinder surface leads to steady forces on the cylinder in the presence of oblique shedding. In contrast, in the case of a strictly parallel shedding, the unsteadiness in the forces is quite large. In that sense, the parallel shedding is a singular case from the point of view of unsteady forces. From the point of view of flow-control, this work suggests that, it might be possible to design a device that promotes oblique shedding, as opposed to parallel shedding of vortices, and reduces/ eliminates the associated noise/vibrations. References Albarède, Pierre, Monkewitz, Peter A., 1992. A model for the formation of oblique shedding and chevron patterns in cylinder wakes. Physics of Fluids A 4, 744. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195. Behara, Suresh, Mittal, Sanjay, 2010. Flow past a circular cylinder at low Reynolds number: oblique vortex shedding. Physics of Fluids 22, 054102. Berger, E., Wille, R., 1972. Periodic flow phenomena. Annual Review of Fluid Mechanics 4, 313. Gerich, D., Eckelmann, H., 1982. Influence of end plates and free ends on the shedding frequency of circular cylinders. Journal of Fluid Mechanics 122, 109. Gomez, F., Quesada, J.H., Gomez, R., Theofilis, V., Carmo, B., Meneghini, J., 2013. Stability analysis of the flow around a cylinder fitted with helical strakes. In: 43rd AIAA Fluid Dynamics Conference. Hammache, M., Gharib, M., 1991. An experimental study of the parallel and oblique vortex shedding from circular cylinders. Journal of Fluid Mechanics 232, 567. Konig, M., Noack, B.R., Eckelmann, H., 1993. Discrete shedding modes in the von Karman vortex Street. Physics of Fluids A 5, 1846. Koopman, G.H., 1967. On the wind induced vibrations of circular cylinders. Thesis, Catholic University of America, School of Engineering and Architecture. Kumar, B., Mittal, S., 2006a. Effect of blockage on critical parameters for flow past a circular cylinder at low Reynolds number. International Journal for Numerical Methods in Fluids 50, 987. Kumar, B., Mittal, S., 2006b. Prediction of the critical Reynolds number for flow past a circular cylinder. Computer Methods in Applied Mechanics and Engineering 195, 6046. Leweke, T., Provansal, M., Miller, G.D., Williamson, C.H.K., 1997. Cell formation in cylinder wakes at low Reynolds numbers. Physical Review Letters 78, 1259. Mittal, S., 2001. Computation of three-dimensional flows past circular cylinder of low aspect ratio. Physics of Fluids 13, 177. Noack, B.R., Eckelmann, H., 1994. A global stability analysis of steady and periodic cylinder wake. Journal of Fluid Mechanics 270, 89. Satish, Paluri, Patwardhan, Saurabh S., Ramesh, O.N., 2013. Effect of steady rotation on low Reynolds number vortex shedding behind a circular cylinder. Journal of Fluids and Structures 41, 175. Sen, S., Mittal, S., Biswas, G., 2009. Steady separated flow past a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 620, 89. Triantafyllou, G.S., 1992. Three-dimensional flow patterns in two-dimensional wakes. ASME Journal of Fluids Engineering 114, 356. Tritton, D.J., 1971. A note on vortex streets behind circular cylinders at low Reynolds numbers. Journal of Fluid Mechanics 45, 203. Valles, B., Andersson, H.I., Jenssen, C.B., 2002. Oblique vortex shedding behind tapered cylinders. Journal of Fluids and Structures 16, 453. Verma, A., Mittal, S., 2011. A new unstable mode in the wake of a circular cylinder. Physics of Fluids 23, 121701. Visscher, Jan, Pettersen, Bjornar, Andersson, Helge I., 2011. Experimental study on the wake behind tapered circular cylinders. Journal of Fluids and Structures 27, 1228. Williamson, C.H.K., 1989. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 206, 579. Williamson, C.H.K., 1996. Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics 28, 477. Yang, P.M., Mansy, H., Williams, D.R., 1993. Oblique and parallel wave interaction in the near wake of a circular cylinder. Physics of Fluids A 5, 1657. Zdravkovich, M.M., 1997. Flow Around Circular Cylinders, vol. 1. Oxford University Press. Zhou, T., Mohd. Razali, S.F., Hao, Z., Cheng, L., 2011. On the study of vortex-induced vibration of a cylinder with strakes. Journal of Fluids and Structures 27, 903.
Contents lists available at ScienceDirect
Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Brief Communication
Steady forces on a cylinder with oblique vortex shedding Sanjay Mittal n, Sidharth GS Department of Aerospace Engineering, Indian Institute of Technology Kanpur, UP 208016, India
a r t i c l e in f o
abstract
Article history: Received 9 March 2013 Accepted 2 November 2013 Available online 6 December 2013
We present a curious situation of a fluid-flow wherein the body experiences non-fluctuating fluid-flow force despite being associated with an unsteady flow comprising of sustained vortex shedding. The flow past a circular cylinder at Re¼100 is investigated. It is shown that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by a cylinder placed in uniform flow. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. It is found that, both, the parallel and oblique shedding are linearly unstable eigenmodes of the Re¼ 100 steady flow past a cylinder. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Oblique shedding Linear stability analysis Global analysis Finite element method Steady forces
1. Introduction The flow past a circular cylinder has received considerable attention in the last few decades (see, for example, the reviews by Williamson, 1996; Bearman, 1984; Berger and Wille, 1972; Zdravkovich, 1997). It involves rich physics and is associated with many interesting phenomena over a range of Reynolds number. The Reynolds number is defined as Re ¼ U 1 D=ν. Here, U 1 is the free-stream speed, D is the diameter of the cylinder and ν is the coefficient of kinematic viscosity. Beyond Re 6:28 the flow past a cylinder undergoes separation (Sen et al., 2009). This leads to the formation of two symmetric, counter-rotating vortices in the wake which increases in size with an increase in Re. The steady flow past a circular cylinder becomes unstable beyond Re 47 via a Hopf bifurcation (Kumar and Mittal, 2006a, 2006b). After an initial linear growth, it achieves a state of limit cycle due to the non-linear processes (Verma and Mittal, 2011). The alternate shedding of vortices results in fluctuations in pressure and the von Karman vortex street (Williamson, 1996). In the classical configuration of the vortex street, for a nominally two-dimensional cylinder, the axes of the shed vortices are parallel to the axis of the cylinder. This is referred to as parallel shedding. In this situation, the body experiences unsteady force. Usually, the unsteady force in the transverse direction, with respect to the free-stream, is much larger compared to that in the in-line direction. This unsteady force may lead to noise/vibrations of the body. Verma and Mittal (2011) reported the another mode of wake instability for Re 4 110, approximately. Beyond Re 180, spanwise undulations appear in the wake marking the onset of three-dimensional instabilities (Williamson, 1996). It has been observed in several laboratory experiments that the vortices can also be shed at an oblique angle to the axis of the cylinder (Berger and Wille, 1972; Tritton, 1971; Gerich and Eckelmann, 1982; Williamson, 1989). This is referred to as oblique shedding. The obliqueness of the vortices has been attributed to the conditions at the ends of the cylinder. The slant in the shed vortices may also be generated by the spanwise variation in the geometry of the cylinder as in the case of tapered or stepped cylinders (Visscher et al., 2011; Valles et al., 2002; Satish et al., 2013). It was demonstrated by Williamson
n
Corresponding author. Tel.: þ91 512 2597906; fax: þ91 512 2597561. E-mail address: [email protected] (S. Mittal).
0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2013.11.009
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
311
(1989) that even for a cylinder of large aspect ratio (ratio of the spanwise length to diameter) the end-conditions influence the entire span of the cylinder. The oblique vortices, from both the ends of the cylinder, form a chevron pattern which is symmetric about the mid-span. It is possible to manipulate the end conditions to promote parallel shedding. This is achieved, for example, by inward angling of the leading edge of the end-plates (Williamson, 1989), or by placing control cylinders at the ends (Hammache and Gharib, 1991). The dependence of the oblique angle of the vortices on the conditions at the end-walls has also been investigated computationally (Mittal, 2001; Behara and Mittal, 2010). It has been shown that the oblique shedding angle varies linearly with the thickness of the boundary layer on the end wall. Allowing the velocity to slip on the end walls promotes parallel shedding. It was shown by Williamson (1989) that the vortex shedding frequency for the oblique and parallel shedding is related via the oblique angle of the vortices. Thus, a large number of vortex shedding states, each with a different oblique angle, are possible. This idea was utilized to explain the significant scatter in the measurement of Strouhal number from various experiments. The existence of different oblique shedding modes was also used to explain the discontinuity in the St Re number curve at Re¼64 (Williamson, 1989). Theoretical models, based on transverse stability theory (Albarède and Monkewitz, 1992; Triantafyllou, 1992; Leweke et al., 1997), have been further proposed to explain this phenomenon and the associated spanwise cells. Stability analysis (Noack and Eckelmann, 1994; Konig et al., 1993) has shown that three-dimensional unstable perturbations in the steady wake of cylinder are associated with discrete shedding modes. The focus, so far, in all the work related to oblique vortex shedding, has been toward understanding the onset of the slant in the wake, its modeling, its implications on the scatter in the data, and on techniques to induce parallel shedding. There has been no effort, that we are aware of, to understand the implications of the oblique vortex shedding on the fluid force acting on the body. It is also not clear from the earlier studies if the oblique and parallel shedding are related in any fundamental way. The primary objective of the present work is to investigate the effect of oblique shedding on the fluid force acting on the body and contrast it to the situation with parallel shedding. The other objective is to determine if any fundamental relationship between the two modes of vortex shedding exists, and utilize it to explain the difference in the time variation of fluid force in the two cases. We first show, via Direct Numerical Simulation (DNS) of the Re¼100 flow past a cylinder, that the oblique vortex shedding is associated with spanwise periodicity. The fluid force, on a segment of the cylinder comprising of an integral number of spanwise wavelengths of the oblique waves, does not vary with time. The spanwise variation of the unsteady pressure field, during one cycle of vortex shedding, is studied to explain the distinction in the time-variation of force coefficients for oblique and parallel shedding. By carrying out a global linear stability analysis of the two-dimensional steady flow past a cylinder, we show that there are several unstable oblique modes with differing spanwise periodicity. We further show that the parallel mode is a special case of the oblique mode whose wavelength of periodicity along the span is infinite. 2. Results A cylinder of aspect ratio AR ¼120, resides in a hexahedral domain. The x-axis is along the free stream flow while the z-axis is aligned with the axis of the cylinder. The cylinder occupies the entire span of the domain. Experiments carried out on high aspect ratio cylinders at Re¼100, in the past, show that the vortices in the wake form a chevron pattern and posses symmetry along the centerline (Williamson, 1989). Taking advantage of the symmetry, only one half of the span is simulated and symmetry conditions are applied at the mid-span. It should be noted that the imposition of the symmetry boundary conditions, in the numerical set-up, does not interfere with the development of obliqueness in the vortices shed in the wake of the cylinder. The upstream and downstream boundaries are located at a distance of 50D and 100D, respectively from the center of the cylinder, while the height of the domain is 100D. Uniform flow is prescribed at the upstream face of the domain. At the outflow boundary the stress vector is set to zero. To promote oblique shedding, no-slip condition on the velocity is specified on one of the end walls (z¼ 0) for x=D Z 5:0. Symmetry conditions are prescribed on this boundary for x=D o5:0. No-slip condition is also specified on the cylinder. Computations are carried out for the Re ¼100 flow with a mesh consisting of 14 million nodes and 13.7 million 8-noded hexahedral elements, approximately. A stabilized finite element formulation is utilized to solve the incompressible flow equations in primitive variables (Mittal, 2001). Equal-in-order interpolation functions are used for velocity and pressure. The simulation is carried out up to a non-dimensionalized time of 2500. t¼ 0 corresponds to the onset of vortex shedding. Fully developed oblique shedding is attained at t 1200 corresponding to 96 cycles of vortex shedding. Computations are also carried out for the case of parallel shedding. This is achieved by assigning symmetry conditions on both the end walls. The force on the cylinder, due to the fluid flow, is computed by integrating the stress, r, on its surface. The sectional force coefficients, per unit span, are defined as Z Z 1 1 C l ðzÞ ¼ 1 2 r n^ y dΓ; C d ðzÞ ¼ 1 2 r n^ x dΓ: ð1Þ 2 ρU 1 D Γ 2D 2 ρU 1 D Γ 2D Here, ρ is the density of fluid, n^ is the unit normal vector and Γ 2D is the circle lying on the cylinder at a spanwise location z. The lift and drag coefficients, for a finite span of the cylinder (b ¼ z2 z1 ), can be computed by integrating the sectional coefficients along span: Z Z 1 z2 1 z2 C l ðzÞ dz; C D ¼ C ðzÞ dz: ð2Þ CL ¼ b z1 b z1 d
312
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
Fig. 1 shows instantaneous pictures of the fully developed unsteady flows with parallel and oblique shedding. The flow with oblique shedding has a two cell structure: central and end. The bulk of the span is occupied by the central cell and consists of oblique vortices. The end cell forms in the region close to the no-slip wall. Compared to the central cell, it is associated with a lower shedding frequency. The Strouhal number, corresponding to the vortex shedding frequency, for the central cell is 0.1590; it is 0.1637 for the parallel shedding. These values are in excellent agreement with the experimental values of 0.160 and 0.164 reported for oblique and parallel shedding, respectively, by Williamson (1989) at Re¼100 for cylinders with similar aspect ratios. The cosine rule (Williamson, 1989) relating the vortex shedding frequency of the parallel and oblique modes (Stparallel ¼ Stoblique cos θ) is satisfied to a reasonable approximation. The oblique shedding angle for the central cell is θ 151. The periodic structure along the span, in the central cell of the oblique shedding, is quite apparent from Fig. 1. In the initial stages of the simulation for oblique shedding, the vortices in the central cell region are shed parallel to the cylinder. Oblique shedding begins from the end cell region and propagates outward in the form of an oblique front (Yang et al., 1993; Williamson, 1989). The difference in the vortex shedding frequency in the two cells leads to the appearance of vortex dislocations. More details, including those that demonstrate the adequacy of the resolution, can be found in our earlier work (Behara and Mittal, 2010). Fig. 2 shows the time histories of the force coefficients for the oblique and parallel shedding for the fully developed unsteady state. The time is non-dimensionalized with the free stream speed and the diameter of the cylinder. It can be observed that the amplitude of oscillations is significantly smaller for the oblique shedding, compared to the values for parallel shedding. To investigate this further, we analyze the time- and spanwise variation of sectional force coefficients as defined by Eq. (1). Fig. 3 shows the time history of spanwise distribution of sectional force coefficients for the fully developed unsteady flow. We note the very periodic temporal variation of Cl and Cd, including that in the end-cell. The signature of the periodic generation of vortex dislocations can be clearly observed in this figure. In the region of central cell, we identify two complete spanwise wavelengths of the oblique waves (20:0 r z=D r 54:8). This region is marked in broken lines in Fig. 3. The time histories of the force coefficients integrated across the entire span as well as along the segment with
Fig. 1. Re¼ 100, AR ¼ 120 DNS of flow past a cylinder: isosurfaces of the cross-stream component of velocity (v ¼ 7 0:1) for the fully developed unsteady flow. Symmetry boundary condition is applied at z=D ¼ 60 and only half the span is considered. The boundary condition at z=D ¼ 0 corresponds to (a) noslip wall and (b) slip wall.
Fig. 2. Re¼ 100, AR ¼120 DNS of flow past a cylinder: time histories of lift and drag coefficients for oblique (red color) and parallel shedding (black color). Also shown (in blue color) is the time variation of the contribution from the region 20:0 r z=D r 54:8 corresponding to two spanwise wavelengths of the oblique waves. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
313
Fig. 3. Re¼100, AR¼ 120 DNS of flow past a cylinder: time- and spanwise variation of the sectional force coefficients ((a) Cl, and (b) Cd). The region 20:0 r z=D r 54:8, marked by broken lines, corresponds to two spanwise wavelengths of the oblique waves.
Fig. 4. Re¼ 100 global linear stability analysis of the two-dimensional steady flow past a cylinder: real part of the global modes corresponding to the most unstable eigenvalues for β ¼ 0:0 (left) and 0.4 (right). The upper row shows the isosurfaces of cross-stream component of velocity (v ¼ 7 0:001) while the lower frames show the variation in the x y plane at z ¼0.
two oblique cells are shown in Fig. 2. It is observed that the force coefficients corresponding to the region of central cell with integral number of oblique waves are virtually constant with time. The mean drag coefficient from this region is 1.325 while the rms value of the unsteady fluctuation is Oð10 4 Þ. Next, a global linear stability analysis of the Re ¼100 two-dimensional steady flow is carried out. The perturbations are assumed to be periodic in the spanwise direction and of the form u′ðx; y; z; tÞ ¼ ðu^ r ðx; yÞ þiu^ i ðx; yÞÞeiβz eλt . Here, β is the spanwise wavenumber of the perturbation; β ¼ 0:0 corresponds to parallel shedding, while non-zero values of β represent the oblique modes of shedding. u^ r and u^ i represent the real and imaginary parts of the eigenmode. The real part of the eigenvalue, λ, represents the growth rate while the imaginary part is related to the vortex shedding frequency. The analysis is carried out for flows with 60 rRe r200. For each Re, the growth rate is maximum for β ¼ 0 and decreases with an increase in β. The critical β, for which growth rate is zero, corresponds to neutral stability. For Re ¼ 60 the critical β is 0.5, approximately. It increases with an increase in Re. The oblique angle of the vortices (inclination to the axis of the cylinder) increases with an increase in β. This confirms, as has also been presented earlier (Williamson, 1989), that the oblique shedding is as intrinsic to the flow as is the parallel shedding. The mode for β ¼ 0:0 has the largest growth rate. Therefore, the parallel shedding mode is the preferred mode. The results from the linear stability analysis have been confirmed via computations on meshes with increased spatial resolution. Fig. 4 shows the most unstable eigenmode for β ¼ 0:0 and 0.4. The inclination of the vortices, to the cylinder, is clearly seen for β ¼ 0:4. The two modes look very similar in the x y plane. To further investigate the oblique shedding mode, we carry out DNS with periodic boundary conditions on the end walls. The initial condition for the simulation is the real part of the eigenmode, corresponding to the eigenvalue with the largest real part for β ¼ 0:4 from the global linear stability analysis, superposed with the steady-state solution. The unstable mode for β ¼ 0:4 is associated with an oblique angle of shedding that is close to the range of angles observed in experiments. At the same time, the corresponding growth rate is large enough to warrant its rapid growth. The extent of the domain
314
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
along the z-axis is one spanwise wavelength of the global mode ( ¼ 5πD). Computations are also carried out for the parallel mode of shedding (β ¼ 0). The time evolution of the force coefficients, for the two simulations, is shown in Fig. 5. The initial growth is similar to that predicted by the linear stability theory. The time-variation of CD (Fig. 5) clearly shows that, compared to the oblique mode, the parallel mode grows faster. The main difference between the two time histories is the absence of unsteady oscillations in the force coefficients for the fully developed state of the oblique shedding. The rms value of the lift is Oð10 3 Þ and decays to zero, asymptotically with time. We note that the mechanism by which oblique shedding suppresses unsteadiness in the forces is quite different from that due to the use of helical strakes. The spanwise phase variation of forces, in oblique shedding, stems from the spatio-temporal nature of the instability for the flow past a nominally two-dimensional smooth cylinder. However, phase-incoherence induced by strakes is a phenomenon forced by the geometric variation along the span of the cylinder. At Re ¼100, the three-dimensionality in the flow past a straked cylinder is restricted to the near wake of the cylinder. The spanwise nature of the moderate to far wake is essentially identical to parallel shedding. The periodic vortices shed from the strakes induce spanwise velocities of opposite signs. In contrast, the spanwise velocity remains constant over the central cell region in oblique shedding. The reduction in spanwise correlation in the wake is an important mechanism of VIV mitigation by helical strakes at higher Reynolds numbers (Zhou et al., 2011). Moreover, the unstable spanwise modes for the straked cylinder, that are responsible for suppression of the amplitude of shedding, arise beyond Re¼120 (Gomez et al., 2013).
Fig. 5. Re ¼100, AR ¼ 5π DNS of flow past a cylinder with periodic boundary conditions at both end walls: time evolution of force coefficients for simulations initiated with the real part of the most unstable eigenmode for β ¼ 0:0 and 0.4, obtained from the global linear stability analysis of the steady base flow. Also shown are the isosurfaces of cross-stream component of velocity (v ¼ 7 0:1) for the fully developed unsteady flow at t¼ 354.
Fig. 6. St¼100, AR ¼ 5π, β ¼ 0:4 DNS of flow past a cylinder with periodic boundary conditions at both end walls: perturbation of the pressure coefficient, with respect to the time-averaged distribution, on the surface of the cylinder at various time instants during one cycle of the fully developed unsteady flow. The cylinder has been flattened out. θ ¼ 0 represents the front stagnation point while θ ¼ 7 π corresponds to the base point of the cylinder.
S. Mittal, S. GS / Journal of Fluids and Structures 44 (2014) 310–315
315
The Strouhal number for the parallel and oblique vortex shedding for the fully developed unsteady flow, estimated by recording signals from probes placed in the near wake, in this work is 0.1637 and 0.1562, respectively. The oblique shedding angle, for β ¼ 0:4, varies from 121 immediately behind the cylinder to 201 in the far wake. Such a streamwise variation of the oblique angle has also been observed in laboratory experiments (Koopman, 1967). The empirical cosine rule (Williamson, 1989) relating the oblique shedding angle to vortex shedding frequency for parallel and oblique shedding (Stparallel ¼ Stoblique cos θ) is satisfied to a good approximation beyond the mid-wake (x Z5D). Fig. 6 shows the disturbance in the pressure coefficient, with respect to the time-averaged pressure, on the surface of the cylinder at several time instants during one cycle of the fully developed vortex shedding for β ¼ 0:4. The perturbations are periodic along the span and appear to travel outward. Unlike parallel shedding, the spanwise spatial periodicity of the perturbations results in zero lift and steady drag at each time instant in the limit cycle. This is true for all cases of oblique shedding, no matter how small the inclination angle of the vortices is, provided integral number of spanwise wavelengths are included. Of course, the wavelength increases with a decrease in obliqueness. The physical reason for the vanishing unsteadiness is the phase variation over the cylinder that spans 1801. On the contrary, the unsteady lift and drag are nonzero when the vortices are shed exactly parallel to the cylinder. In that sense, the parallel shedding is a singular case of oblique shedding. 3. Conclusions The flow past a cylinder at Re ¼100 has been investigated. It is shown via linear stability analysis that both the parallel and oblique vortex shedding are linearly unstable eigenmodes of the two dimensional steady base flow. This reaffirms the belief that, similar to the parallel shedding, the obliqueness of the shed vortices is intrinsic to the flow. In fact, there exist a vast number of unstable eigenmodes for the Re ¼ 100 flow past a cylinder, each corresponding to a certain angle of oblique vortices. The parallel vortex shedding is one of the many possible states and is associated with the largest growth rate. The spatio-temporal structure of the vortex shedding instability has been demonstrated. The spatio-temporal periodicity along the cylinder surface leads to steady forces on the cylinder in the presence of oblique shedding. In contrast, in the case of a strictly parallel shedding, the unsteadiness in the forces is quite large. In that sense, the parallel shedding is a singular case from the point of view of unsteady forces. From the point of view of flow-control, this work suggests that, it might be possible to design a device that promotes oblique shedding, as opposed to parallel shedding of vortices, and reduces/ eliminates the associated noise/vibrations. References Albarède, Pierre, Monkewitz, Peter A., 1992. A model for the formation of oblique shedding and chevron patterns in cylinder wakes. Physics of Fluids A 4, 744. Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195. Behara, Suresh, Mittal, Sanjay, 2010. Flow past a circular cylinder at low Reynolds number: oblique vortex shedding. Physics of Fluids 22, 054102. Berger, E., Wille, R., 1972. Periodic flow phenomena. Annual Review of Fluid Mechanics 4, 313. Gerich, D., Eckelmann, H., 1982. Influence of end plates and free ends on the shedding frequency of circular cylinders. Journal of Fluid Mechanics 122, 109. Gomez, F., Quesada, J.H., Gomez, R., Theofilis, V., Carmo, B., Meneghini, J., 2013. Stability analysis of the flow around a cylinder fitted with helical strakes. In: 43rd AIAA Fluid Dynamics Conference. Hammache, M., Gharib, M., 1991. An experimental study of the parallel and oblique vortex shedding from circular cylinders. Journal of Fluid Mechanics 232, 567. Konig, M., Noack, B.R., Eckelmann, H., 1993. Discrete shedding modes in the von Karman vortex Street. Physics of Fluids A 5, 1846. Koopman, G.H., 1967. On the wind induced vibrations of circular cylinders. Thesis, Catholic University of America, School of Engineering and Architecture. Kumar, B., Mittal, S., 2006a. Effect of blockage on critical parameters for flow past a circular cylinder at low Reynolds number. International Journal for Numerical Methods in Fluids 50, 987. Kumar, B., Mittal, S., 2006b. Prediction of the critical Reynolds number for flow past a circular cylinder. Computer Methods in Applied Mechanics and Engineering 195, 6046. Leweke, T., Provansal, M., Miller, G.D., Williamson, C.H.K., 1997. Cell formation in cylinder wakes at low Reynolds numbers. Physical Review Letters 78, 1259. Mittal, S., 2001. Computation of three-dimensional flows past circular cylinder of low aspect ratio. Physics of Fluids 13, 177. Noack, B.R., Eckelmann, H., 1994. A global stability analysis of steady and periodic cylinder wake. Journal of Fluid Mechanics 270, 89. Satish, Paluri, Patwardhan, Saurabh S., Ramesh, O.N., 2013. Effect of steady rotation on low Reynolds number vortex shedding behind a circular cylinder. Journal of Fluids and Structures 41, 175. Sen, S., Mittal, S., Biswas, G., 2009. Steady separated flow past a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 620, 89. Triantafyllou, G.S., 1992. Three-dimensional flow patterns in two-dimensional wakes. ASME Journal of Fluids Engineering 114, 356. Tritton, D.J., 1971. A note on vortex streets behind circular cylinders at low Reynolds numbers. Journal of Fluid Mechanics 45, 203. Valles, B., Andersson, H.I., Jenssen, C.B., 2002. Oblique vortex shedding behind tapered cylinders. Journal of Fluids and Structures 16, 453. Verma, A., Mittal, S., 2011. A new unstable mode in the wake of a circular cylinder. Physics of Fluids 23, 121701. Visscher, Jan, Pettersen, Bjornar, Andersson, Helge I., 2011. Experimental study on the wake behind tapered circular cylinders. Journal of Fluids and Structures 27, 1228. Williamson, C.H.K., 1989. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 206, 579. Williamson, C.H.K., 1996. Vortex dynamics in the cylinder wake. Annual Review of Fluid Mechanics 28, 477. Yang, P.M., Mansy, H., Williams, D.R., 1993. Oblique and parallel wave interaction in the near wake of a circular cylinder. Physics of Fluids A 5, 1657. Zdravkovich, M.M., 1997. Flow Around Circular Cylinders, vol. 1. Oxford University Press. Zhou, T., Mohd. Razali, S.F., Hao, Z., Cheng, L., 2011. On the study of vortex-induced vibration of a cylinder with strakes. Journal of Fluids and Structures 27, 903.