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Stabilisation of the Absolute Instability of a Flow Past a Cylinder via Spanwise Forcing at Re = 180
Available online at www.sciencedirect.com
ScienceDirect Procedia IUTAM 14 (2015) 115 – 121
IUTAM ABCM Symposium on Laminar Turbulent Transition
Stabilisation of the absolute instability of a flow past a cylinder via spanwise forcing at Re = 180. G. Roccoa,∗, S. J. Sherwin1 a Department
of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ
Abstract Vortex shedding in the wake of bluff bodies is often an undesired phenomenon which generates unsteady loads, vibrations and fluctuations of the aerodynamic forces. Consequently, the attenuation or suppression of the self-sustained oscillations associated to the vortex shedding is a fundamental problem in a wide range of engineering applications. Three-dimensional control techniques to control the vortex shedding are characterised by a variation of the control input along the spanwise direction and offer a promising methodology due to their versatility and high potential efficiency 1 . In the present paper, the control of vortex dynamics of the wake of a flow past a cylinder at Reynolds number Re = 180 is performed by means of spanwise distributed forcing 1 2 ; starting from a fully developed shedding, a sufficiently high spanwise forcing is introduced on the surface of the cylinder, close to the separation regions, to stabilise the near-wake in a time-independent state, similarly to the effect of a sinusoidal stagnation surface 3,4 . The effects of the forcing on the drag reduction and the dynamics of the vorticity have been investigated using a spanwise gaussian forcing, which generates a significant redistribution of the spanwise vorticity into streamwise and vertical components was observed, leading to a minor susceptibility of the three-dimensional shear layers to roll-up into the vortex street 4? . An insight into the main physical mechanisms underlying the suppression is provided by the hydrodynamic stability theory. Three different regimes were found for different forcing amplitude and the computation of the leading modes helps to shed light on some mechanisms responsible for the suppression of the Von-Karman shedding. c 2015 2014Published The Authors. Published by Elsevier B.V. © by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: stabilization; vortex shedding; hydrodynamic instability; forcing;
1. Introduction An efficient control of the vortex shedding is a fundamental problem in several engineering applications due to the drag, vibrations and noise it generates. Methods aimed at suppressing the vortex shedding can be classified in two main categories. Following Choi et al. (2008) 5 , we will consider two dimensional and three-dimensional controls. In two dimensional controls the input has a constant profile along the spanwise direction, while three-dimensional controls introduce a modification in one direction, e.g. z. Example of two-dimensional controls are the end plates 8 , base bleed 3 , splitter plates 9 and secondary small cylinders 10 , while segmented trailing edge 11 , wavy trailing edge 12 ∗
Corresponding author. Tel.: +44-(0)20 20 7594 5100. E-mail address: [email protected]
2210-9838 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.030
116
G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
and spanwise-periodic blowing and suction 1 are examples of 3D techniques. Three-dimensional control techniques might be more efficient than two-dimensional techniques, but the underlying mechanisms leading to the reduction of the drag coefficient and the suppression of the vortex shedding are still poorly understood, despite noteworthy progresses have been made by means of linear stability analysis. Hwang et al. (2013) 2 found that the application of spanwise waviness to a flow over a bluff body is able to stabilise the near-wake instabilities. Del Guercio et. al (2013) 13 applied spanwise forcing to a parallel wake model and more recently to a flow past a cylinder 14 . Rocco (2013) 6 applied different spanwise forcing functions to a flow past a cylinder at Re = 60 and used linear stability to compute the three-dimensional eigenmodes, confirming the complete stabilisation of the wake in a global stability analysis framework, and identifying a significant shift of the wavemaker region towards the forcing regions. In the present paper, we focus on the dynamics of a flow past a circular cylinder at Re = 180. Differently from simulations at lower Reynolds numbers 7 , three different regimes characterise the suppression of the shedding. Specifically, an increase on the intensity of the spanwise forcing leads to the appearance of Λ-structures emerging from the nearbase structures. This phenomenon arise further questions about the relevant mechanisms behind the stabilisation of the wakes past bluff-bodies and their connection with the ones already detected at lower Reynolds numbers. Linear stability analysis is used to address these questions: a slight decrease in the magnitude of the growth rate was detected when the forcing amplitude is above a threshold value and a significant change of the eigenmode topology was found. 2. Methodology Let us consider the Navier-Stokes equations for a Newtonian fluid and incompressible flow: ∂u 1 2 + u · ∇u = −∇p + ∇ u. ∂t Re
(1)
∇ · u = 0.
(2)
where u is the velocity normalised by the undisturbed value u∞ , t is the normalised time, p is the static pressure and Re is the Reynolds number Re = u∞ D/ν. These equations are discretised using a Fourier-spectral/hp element method 15,17 . To study the stability of the base flow, the flow is decomposed in the sum of a base flow and perturbations, u = U + u and p = P + p . Substituting into the Navier-Stokes equations and neglecting the second order terms, we obtain the linearised Navier-Stokes equations: 1 2 ∂u + U · ∇u + u · ∇U = −∇p + ∇ u. ∂t Re
(3)
∇ · u = 0.
(4)
These equations can be re-written as: ∂u = A(U)u . ∂t
(5)
ˆ exp(λt), where λ ∈ C. The stability of the system We assume the perturbations to be in the form u (x, t) = u(x) is then associated to the evaluation of the eigenvalue of operator A and the system is unstable if the the leading eigenvalue is outside the unit circle. In the present paper the solution of the eigenproblem is performed using an Arnoldi method based on Barkley et al. (2008) 18 . 3. Discretisation In this paper we consider a flow past a circular cylinder at Re = 180. In figure (3) the geometry and the mesh of the problem are reported. The upstream boundary Li is 50D from the centre of the cylinder, while the side boundary, Lc , 45D from the centre of the body. Finally the downstream boundary LO is 50D such that eventual instabilities have sufficient space to evolve. Dirichlet boundary conditions were imposed for the velocity at the upstream and side
G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
117
Fig. 1. Geometry of the problem and relative mesh.
boundaries, while Neumann boundary conditions were applied on the outflow. A homogeneous boundary condition was applied on the outflow, while high order boundary conditions 16 were applied on all the others. The domain is discretised using 700 quadrilateral elements with a eight-order polynomial expansions, while a second-order time integration scheme was chosen. The spanwise dimension Lz was chosen to be equal to 5D. The number of Fourier modes in the spanwise direction was set to nF = 32, which are sufficient to capture the dynamics of the flow physics for the considered case. 4. Suppression of the vortex shedding via distributed forcing In this paper, we use distributed spanwise forcing 1,6 to suppress the vortex shedding and compare the behaviour of the resulting flow with the results obtained at lower Reynolds number 6 . Starting from a fully developed vortexshedding, at Re = 180, a three-dimensional distributed forcing a Gaussian distributed forcing is applied on the surface of the cylinder. Two forcing functions are applied on both the top and bottom surfaces of the cylinder, close to the separation points: in the streamwise and vertical directions, a gaussian forcing, φ xy , with standard deviation ζ xy = 0.1 was applied, while in the spanwise direction a gaussian forcing with ζz = 0.02 was chosen: (θ + θc )2 φ xy (θ)|top/bottom = exp − . (6) 2ζ xy2 ⎡ ⎤ ⎢⎢⎢ z − Lz /2 2 ⎥⎥⎥ ⎢ ⎥⎥⎦ . φz (z) = exp ⎢⎣− (7) 2ζz2 φ(x, y, z) = A · φ xy (x, y) · φz (z).
(8)
Numerical experiments showed that ζ = 0.02, is small enough to suppress the oscillation of the vortex shedding. We study the changes in the wake topology when the forcing amplitude A is progressively increased. For very small A < 0.2, the distributed forcing is not able to suppress the von-Karman street and unsteady coherent structures can be detected along the wake (regime I). These structures show a high degree of three-dimensionality and do not disappear or become weaker if longer time are considered (regime I). When A 0.2 a complete stabilisation of the wake was observed, which shows a symmetrical profile, with both vertical and horizontal connections. The drag coefficient is subject to a reduction of about 17% and the Strouhal frequency is zero. However, this behaviour is maintained only if the forcing amplitude is A < 0.25 (regime II). Higher amplitudes were found to generate a smaller drag reduction (about 15%) and large unsteady hairpin vortices were seen to emerge from the near-base structure (regime III), with a wake topology resembling the one of a sphere at low Reynolds number 19 . This behaviour is reported in figure 4; as we can see, the absolute instability is progressively weakened when the forcing amplitude is increased. When A 0.2, the flow has been stabilised, consistently with the complete suppression of vortex shedding. However, when A > 0.25 a new type of unsteadiness was detected.
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G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
Fig. 2. Visualisation of the different regimes.
5. Stability analysis and stabilisation of the flow In the previous section we described the different regimes occurring when the amplitude of the forcing is increased, and we detected a new unsteady behaviour when A > 0.25. We investigate whether higher forcing amplitudes destabilise the wake and we try to detect the most relevant mechanisms in regime III. We try to answer these questions using stability analysis. Direct stability analysis was performed in all the three regimes and the profile of the growth rate as a function of the amplitude is reported in figure 5. To perform the stability analysis in regime I and III, a steady base flow was used, which was computed by inserting a symmetry condition along the centreplane and then time-marching the Navier-Stokes equations. The behaviour for regimes I and II is very similar to the profiles detected at lower Reynolds numbers 6,7 (i.e. Re = 60): a monotonic decrease which is associated with a progressive weakening of the near-wake absolute instability. When A 0.2 the base flow was found to be stable. The main different with previous results at Re = 60 is the slower reduction of the growth rate with the forcing amplitude, which is probably related to the fact that we are operating at a Reynolds number closer to the on-set of three-dimensional instabilities. The main differences with the above mentioned case are in regime III: increasing the forcing amplitude, the decay rate is subject to a slight increase despite the flow is still stable. Hence, we can conclude that in the range of forcing amplitude we considered ( 0 < A < 3), the flow is stable above a threshold value Acr , but when A > 0.25 the distur-
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Fig. 3. Growth rate as a function of the forcing amplitude.
Fig. 4. Profile of the eigenmode. The top right and left panels represent regime I, bottom left represents regime II and bottom right regime III
bances decay more slowly than in regime II. The general profiles of the growth rate as function of the forcing is not a monotonic function but shows a minimum. Regarding the eigenmode, in regimes I and II, they have a profile similar to the profiles found at Re = 60. The corresponding profiles of the base flows are reported in figures 4. When no forcing is applied the eigenmode shows a row of extended vortices with alternating sign. Increasing the amplitude, the three-dimensionality of the flow becomes more pronounced and the modes tend to be located in the near-wake region, while it spatially decays downstream from the cylinder. This mechanism continues until the flow becomes stable, then a different trend was detected. In regime III (bottom right panel in figure 5), the mode extends throughout the domain, even if it is more intense in the near-wake region. This indicates that the far wake plays an important role on the stability of this configuration. We will address this behaviour considering the equation of transport of vorticity perturbations, which can be obtained considering the curl of the linearised Navier-Stokes equations: 1 2 2 ∂ω + U · ∇ω = ω · ∇U + Ω · ∇u − u · ∇Ω + ∇ω ∂t Re
(9)
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Fig. 5. Profile of the vorticity perturbation magnitude |ω | for a) regime II (A=0.22) and b) regime III (A=0.35)
where U and Ω represent the base flow velocity and its vorticity respectively. The tilting mechanisms by the base flow shear (first term on the right hand side) are the dominant ones, hence we will focus on this term to understand the key factors for the stabilisation. Following Hwang et al. (2013) 2 , we can rewrite equation 9 as: Dωz ∼ ωz E xz Dt
(10)
Dωz ∼ ωx E xz . Dt
(11)
where E xz is the deformation rate tensor, which is related to the spanwise shear of the base flow. Hence, if ω is zero in a specific part of the domain, the above-mentioned mechanism is inhibited. This is what happens at moderate amplitude of forcing: the mode was seen to be located in a small streamwise region close to the cylinder, and so the vorticity of the perturbations. (figure 5). This is not the case for higher forcing amplitudes which show a profile extending in a wide portion of the domain, and the subsequent appearance of alternating Λ-vortices. Specifically, the perturbations in the far-wake are probably responsible for a local destabilisation, with subsequent convective instabilities. In particular, a transient growth analysis could provide additional insight into the dynamics and the development of hairpin vortices, which remains one of the main points of investigation. 6. Conclusion In this paper we investigated the instability arising in a flow past circular cylinder subject to distributed spanwise forcing at Re = 180. Results were compared to previous studies at lower Reynolds number (Re = 60), to understand the different mechanisms involved in the suppression of vortex shedding. The main difference with simulations at Re = 60 6,7 are the necessity of larger forcing amplitudes to suppress the von-Karman street and the presence of unsteady hairpin vortices if the forcing amplitude is increase above a critical value (A 0.25 in the present case). Results from global stability analysis point out that these structures do not destabilise the flow, but a decrease of
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121
the decay rate of disturbances was found. A first explanation of such mechanisms comes from the structure of the eigenmode, which has a prominent magnitude throughout the domain. A significant profile of the streamwise vorticity perturbation can be associated to the gradual spatial and temporal evolution of the Λ-structures 2 . This phenomenon is probably related to the presence of convective instabilities, hence a transient growth analysis would help to explain the dynamics of the flow in regime III. References 1. Kim J., Choi H., Distributed forcing of flow over a circular cylinder Physics of Fluids 2005; 17 (033103). 2. Hwang Y., Kim J., Choi H., Stabilization of absolute instability in spanwise wavy two-dimensional wakes J. Fluid Mech. 2013; 727: 346-378. 3. Bearman P. W., Owen J.C., Special brief note: reduction of bluff-body drag and suppression of vortex shedding by introduction of wavy separation lines Journal of Fluids and Structures 1988; 12: 123-130. 4. Darekar R. M., Sherwin S.J., Flow past a square-section cylinder with a wavy stagnation face J. Fluid Mechanics 2001; 426: 263-295. 5. Choi H., Woo-Pyung J., Kim J., Control of flow over a bluff body Ann. Rev. Fluid Mech. 2008; 40: 113-139. 6. Rocco G., Advanced Instability Methods using Spectral/hp discretisations and their applications to complex geometries PhD thesis 2014; Imperial College London. 7. Rocco G., Sherwin S.J., The role of the spanwise forcing on vortex shedding suppression in a flow past a cylinder Instability and Control of Massively Separated Flows. 2014. 8. Nishioka M,m Sato H., Measurements of the velocity distributions in the wake of a circular cylinder at low Reynolds numbers J. Fluid Mech. 1974; 765: 97-112. 9. On the development of turbulent wakes from vortex streets NACA Tech. Rep.. 10. Strykowski P.J., Sreenivasan K.R., On the formation and suppression of vortex shedding at low Reynolds number J. Fluid Mech.; 218, 71-107. 11. Tanner M., A method of reducing the base drag of wings with blunt trailing edges Aeronaut. Q. 1972; 23: 15-23. 12. Tombazis N., Bearman P.W. A study of three-dimensional aspect of vortex shedding from a bluff body with mild geometric disturbance J. Fluid Mech.; 330: 80-112. 13. Del Guercio G., Cossu C., Pujals G., Stabilizing effects of optimally amplified streaks J. Fluid Mech. 2013; 739: 37-56. 14. Del Guercio G., Cossu C., Pujals G., Optimal streaks in the circular cylinder wake and suppression of the global instability J. Fluid Mech. 2014; 752: 572-588. 15. Karniadakis G.E., Spectral element-Fourier methods for incompressible turbulent flows C.M.A.M.E. 1990; 90: 362. 16. Karniadakis G.E., Israeli M., Orszag S.A., High-order splitting methods for incompressible Navier-Stokes equations J. Comput. Phys.; 97: 414. 17. Karniadakis G.E., Spetral/hp element methods for computational fluid dynamics, 2nd edition, Oxford University Press. 18. Barkley D., Blackburn H.M., Sherwin S.J., Direct optimal growth analysis for timesteppers Int J. Numer. Meth. Fluids 2008; 57: 1434-1458. 19. Johnson T.A., Patel V.C. Flow past a sphere up to Reynolds number 300 J. Fluid Mech. 1999; 378:19-70.
ScienceDirect Procedia IUTAM 14 (2015) 115 – 121
IUTAM ABCM Symposium on Laminar Turbulent Transition
Stabilisation of the absolute instability of a flow past a cylinder via spanwise forcing at Re = 180. G. Roccoa,∗, S. J. Sherwin1 a Department
of Aeronautics, Imperial College London, South Kensington Campus, SW7 2AZ
Abstract Vortex shedding in the wake of bluff bodies is often an undesired phenomenon which generates unsteady loads, vibrations and fluctuations of the aerodynamic forces. Consequently, the attenuation or suppression of the self-sustained oscillations associated to the vortex shedding is a fundamental problem in a wide range of engineering applications. Three-dimensional control techniques to control the vortex shedding are characterised by a variation of the control input along the spanwise direction and offer a promising methodology due to their versatility and high potential efficiency 1 . In the present paper, the control of vortex dynamics of the wake of a flow past a cylinder at Reynolds number Re = 180 is performed by means of spanwise distributed forcing 1 2 ; starting from a fully developed shedding, a sufficiently high spanwise forcing is introduced on the surface of the cylinder, close to the separation regions, to stabilise the near-wake in a time-independent state, similarly to the effect of a sinusoidal stagnation surface 3,4 . The effects of the forcing on the drag reduction and the dynamics of the vorticity have been investigated using a spanwise gaussian forcing, which generates a significant redistribution of the spanwise vorticity into streamwise and vertical components was observed, leading to a minor susceptibility of the three-dimensional shear layers to roll-up into the vortex street 4? . An insight into the main physical mechanisms underlying the suppression is provided by the hydrodynamic stability theory. Three different regimes were found for different forcing amplitude and the computation of the leading modes helps to shed light on some mechanisms responsible for the suppression of the Von-Karman shedding. c 2015 2014Published The Authors. Published by Elsevier B.V. © by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering). Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) Keywords: stabilization; vortex shedding; hydrodynamic instability; forcing;
1. Introduction An efficient control of the vortex shedding is a fundamental problem in several engineering applications due to the drag, vibrations and noise it generates. Methods aimed at suppressing the vortex shedding can be classified in two main categories. Following Choi et al. (2008) 5 , we will consider two dimensional and three-dimensional controls. In two dimensional controls the input has a constant profile along the spanwise direction, while three-dimensional controls introduce a modification in one direction, e.g. z. Example of two-dimensional controls are the end plates 8 , base bleed 3 , splitter plates 9 and secondary small cylinders 10 , while segmented trailing edge 11 , wavy trailing edge 12 ∗
Corresponding author. Tel.: +44-(0)20 20 7594 5100. E-mail address: [email protected]
2210-9838 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Selection and peer-review under responsibility of ABCM (Brazilian Society of Mechanical Sciences and Engineering) doi:10.1016/j.piutam.2015.03.030
116
G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
and spanwise-periodic blowing and suction 1 are examples of 3D techniques. Three-dimensional control techniques might be more efficient than two-dimensional techniques, but the underlying mechanisms leading to the reduction of the drag coefficient and the suppression of the vortex shedding are still poorly understood, despite noteworthy progresses have been made by means of linear stability analysis. Hwang et al. (2013) 2 found that the application of spanwise waviness to a flow over a bluff body is able to stabilise the near-wake instabilities. Del Guercio et. al (2013) 13 applied spanwise forcing to a parallel wake model and more recently to a flow past a cylinder 14 . Rocco (2013) 6 applied different spanwise forcing functions to a flow past a cylinder at Re = 60 and used linear stability to compute the three-dimensional eigenmodes, confirming the complete stabilisation of the wake in a global stability analysis framework, and identifying a significant shift of the wavemaker region towards the forcing regions. In the present paper, we focus on the dynamics of a flow past a circular cylinder at Re = 180. Differently from simulations at lower Reynolds numbers 7 , three different regimes characterise the suppression of the shedding. Specifically, an increase on the intensity of the spanwise forcing leads to the appearance of Λ-structures emerging from the nearbase structures. This phenomenon arise further questions about the relevant mechanisms behind the stabilisation of the wakes past bluff-bodies and their connection with the ones already detected at lower Reynolds numbers. Linear stability analysis is used to address these questions: a slight decrease in the magnitude of the growth rate was detected when the forcing amplitude is above a threshold value and a significant change of the eigenmode topology was found. 2. Methodology Let us consider the Navier-Stokes equations for a Newtonian fluid and incompressible flow: ∂u 1 2 + u · ∇u = −∇p + ∇ u. ∂t Re
(1)
∇ · u = 0.
(2)
where u is the velocity normalised by the undisturbed value u∞ , t is the normalised time, p is the static pressure and Re is the Reynolds number Re = u∞ D/ν. These equations are discretised using a Fourier-spectral/hp element method 15,17 . To study the stability of the base flow, the flow is decomposed in the sum of a base flow and perturbations, u = U + u and p = P + p . Substituting into the Navier-Stokes equations and neglecting the second order terms, we obtain the linearised Navier-Stokes equations: 1 2 ∂u + U · ∇u + u · ∇U = −∇p + ∇ u. ∂t Re
(3)
∇ · u = 0.
(4)
These equations can be re-written as: ∂u = A(U)u . ∂t
(5)
ˆ exp(λt), where λ ∈ C. The stability of the system We assume the perturbations to be in the form u (x, t) = u(x) is then associated to the evaluation of the eigenvalue of operator A and the system is unstable if the the leading eigenvalue is outside the unit circle. In the present paper the solution of the eigenproblem is performed using an Arnoldi method based on Barkley et al. (2008) 18 . 3. Discretisation In this paper we consider a flow past a circular cylinder at Re = 180. In figure (3) the geometry and the mesh of the problem are reported. The upstream boundary Li is 50D from the centre of the cylinder, while the side boundary, Lc , 45D from the centre of the body. Finally the downstream boundary LO is 50D such that eventual instabilities have sufficient space to evolve. Dirichlet boundary conditions were imposed for the velocity at the upstream and side
G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
117
Fig. 1. Geometry of the problem and relative mesh.
boundaries, while Neumann boundary conditions were applied on the outflow. A homogeneous boundary condition was applied on the outflow, while high order boundary conditions 16 were applied on all the others. The domain is discretised using 700 quadrilateral elements with a eight-order polynomial expansions, while a second-order time integration scheme was chosen. The spanwise dimension Lz was chosen to be equal to 5D. The number of Fourier modes in the spanwise direction was set to nF = 32, which are sufficient to capture the dynamics of the flow physics for the considered case. 4. Suppression of the vortex shedding via distributed forcing In this paper, we use distributed spanwise forcing 1,6 to suppress the vortex shedding and compare the behaviour of the resulting flow with the results obtained at lower Reynolds number 6 . Starting from a fully developed vortexshedding, at Re = 180, a three-dimensional distributed forcing a Gaussian distributed forcing is applied on the surface of the cylinder. Two forcing functions are applied on both the top and bottom surfaces of the cylinder, close to the separation points: in the streamwise and vertical directions, a gaussian forcing, φ xy , with standard deviation ζ xy = 0.1 was applied, while in the spanwise direction a gaussian forcing with ζz = 0.02 was chosen: (θ + θc )2 φ xy (θ)|top/bottom = exp − . (6) 2ζ xy2 ⎡ ⎤ ⎢⎢⎢ z − Lz /2 2 ⎥⎥⎥ ⎢ ⎥⎥⎦ . φz (z) = exp ⎢⎣− (7) 2ζz2 φ(x, y, z) = A · φ xy (x, y) · φz (z).
(8)
Numerical experiments showed that ζ = 0.02, is small enough to suppress the oscillation of the vortex shedding. We study the changes in the wake topology when the forcing amplitude A is progressively increased. For very small A < 0.2, the distributed forcing is not able to suppress the von-Karman street and unsteady coherent structures can be detected along the wake (regime I). These structures show a high degree of three-dimensionality and do not disappear or become weaker if longer time are considered (regime I). When A 0.2 a complete stabilisation of the wake was observed, which shows a symmetrical profile, with both vertical and horizontal connections. The drag coefficient is subject to a reduction of about 17% and the Strouhal frequency is zero. However, this behaviour is maintained only if the forcing amplitude is A < 0.25 (regime II). Higher amplitudes were found to generate a smaller drag reduction (about 15%) and large unsteady hairpin vortices were seen to emerge from the near-base structure (regime III), with a wake topology resembling the one of a sphere at low Reynolds number 19 . This behaviour is reported in figure 4; as we can see, the absolute instability is progressively weakened when the forcing amplitude is increased. When A 0.2, the flow has been stabilised, consistently with the complete suppression of vortex shedding. However, when A > 0.25 a new type of unsteadiness was detected.
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G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
Fig. 2. Visualisation of the different regimes.
5. Stability analysis and stabilisation of the flow In the previous section we described the different regimes occurring when the amplitude of the forcing is increased, and we detected a new unsteady behaviour when A > 0.25. We investigate whether higher forcing amplitudes destabilise the wake and we try to detect the most relevant mechanisms in regime III. We try to answer these questions using stability analysis. Direct stability analysis was performed in all the three regimes and the profile of the growth rate as a function of the amplitude is reported in figure 5. To perform the stability analysis in regime I and III, a steady base flow was used, which was computed by inserting a symmetry condition along the centreplane and then time-marching the Navier-Stokes equations. The behaviour for regimes I and II is very similar to the profiles detected at lower Reynolds numbers 6,7 (i.e. Re = 60): a monotonic decrease which is associated with a progressive weakening of the near-wake absolute instability. When A 0.2 the base flow was found to be stable. The main different with previous results at Re = 60 is the slower reduction of the growth rate with the forcing amplitude, which is probably related to the fact that we are operating at a Reynolds number closer to the on-set of three-dimensional instabilities. The main differences with the above mentioned case are in regime III: increasing the forcing amplitude, the decay rate is subject to a slight increase despite the flow is still stable. Hence, we can conclude that in the range of forcing amplitude we considered ( 0 < A < 3), the flow is stable above a threshold value Acr , but when A > 0.25 the distur-
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Fig. 3. Growth rate as a function of the forcing amplitude.
Fig. 4. Profile of the eigenmode. The top right and left panels represent regime I, bottom left represents regime II and bottom right regime III
bances decay more slowly than in regime II. The general profiles of the growth rate as function of the forcing is not a monotonic function but shows a minimum. Regarding the eigenmode, in regimes I and II, they have a profile similar to the profiles found at Re = 60. The corresponding profiles of the base flows are reported in figures 4. When no forcing is applied the eigenmode shows a row of extended vortices with alternating sign. Increasing the amplitude, the three-dimensionality of the flow becomes more pronounced and the modes tend to be located in the near-wake region, while it spatially decays downstream from the cylinder. This mechanism continues until the flow becomes stable, then a different trend was detected. In regime III (bottom right panel in figure 5), the mode extends throughout the domain, even if it is more intense in the near-wake region. This indicates that the far wake plays an important role on the stability of this configuration. We will address this behaviour considering the equation of transport of vorticity perturbations, which can be obtained considering the curl of the linearised Navier-Stokes equations: 1 2 2 ∂ω + U · ∇ω = ω · ∇U + Ω · ∇u − u · ∇Ω + ∇ω ∂t Re
(9)
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G. Rocco and S.J. Sherwin / Procedia IUTAM 14 (2015) 115 – 121
Fig. 5. Profile of the vorticity perturbation magnitude |ω | for a) regime II (A=0.22) and b) regime III (A=0.35)
where U and Ω represent the base flow velocity and its vorticity respectively. The tilting mechanisms by the base flow shear (first term on the right hand side) are the dominant ones, hence we will focus on this term to understand the key factors for the stabilisation. Following Hwang et al. (2013) 2 , we can rewrite equation 9 as: Dωz ∼ ωz E xz Dt
(10)
Dωz ∼ ωx E xz . Dt
(11)
where E xz is the deformation rate tensor, which is related to the spanwise shear of the base flow. Hence, if ω is zero in a specific part of the domain, the above-mentioned mechanism is inhibited. This is what happens at moderate amplitude of forcing: the mode was seen to be located in a small streamwise region close to the cylinder, and so the vorticity of the perturbations. (figure 5). This is not the case for higher forcing amplitudes which show a profile extending in a wide portion of the domain, and the subsequent appearance of alternating Λ-vortices. Specifically, the perturbations in the far-wake are probably responsible for a local destabilisation, with subsequent convective instabilities. In particular, a transient growth analysis could provide additional insight into the dynamics and the development of hairpin vortices, which remains one of the main points of investigation. 6. Conclusion In this paper we investigated the instability arising in a flow past circular cylinder subject to distributed spanwise forcing at Re = 180. Results were compared to previous studies at lower Reynolds number (Re = 60), to understand the different mechanisms involved in the suppression of vortex shedding. The main difference with simulations at Re = 60 6,7 are the necessity of larger forcing amplitudes to suppress the von-Karman street and the presence of unsteady hairpin vortices if the forcing amplitude is increase above a critical value (A 0.25 in the present case). Results from global stability analysis point out that these structures do not destabilise the flow, but a decrease of
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the decay rate of disturbances was found. A first explanation of such mechanisms comes from the structure of the eigenmode, which has a prominent magnitude throughout the domain. A significant profile of the streamwise vorticity perturbation can be associated to the gradual spatial and temporal evolution of the Λ-structures 2 . This phenomenon is probably related to the presence of convective instabilities, hence a transient growth analysis would help to explain the dynamics of the flow in regime III. References 1. Kim J., Choi H., Distributed forcing of flow over a circular cylinder Physics of Fluids 2005; 17 (033103). 2. Hwang Y., Kim J., Choi H., Stabilization of absolute instability in spanwise wavy two-dimensional wakes J. Fluid Mech. 2013; 727: 346-378. 3. Bearman P. W., Owen J.C., Special brief note: reduction of bluff-body drag and suppression of vortex shedding by introduction of wavy separation lines Journal of Fluids and Structures 1988; 12: 123-130. 4. Darekar R. M., Sherwin S.J., Flow past a square-section cylinder with a wavy stagnation face J. Fluid Mechanics 2001; 426: 263-295. 5. Choi H., Woo-Pyung J., Kim J., Control of flow over a bluff body Ann. Rev. Fluid Mech. 2008; 40: 113-139. 6. Rocco G., Advanced Instability Methods using Spectral/hp discretisations and their applications to complex geometries PhD thesis 2014; Imperial College London. 7. Rocco G., Sherwin S.J., The role of the spanwise forcing on vortex shedding suppression in a flow past a cylinder Instability and Control of Massively Separated Flows. 2014. 8. Nishioka M,m Sato H., Measurements of the velocity distributions in the wake of a circular cylinder at low Reynolds numbers J. Fluid Mech. 1974; 765: 97-112. 9. On the development of turbulent wakes from vortex streets NACA Tech. Rep.. 10. Strykowski P.J., Sreenivasan K.R., On the formation and suppression of vortex shedding at low Reynolds number J. Fluid Mech.; 218, 71-107. 11. Tanner M., A method of reducing the base drag of wings with blunt trailing edges Aeronaut. Q. 1972; 23: 15-23. 12. Tombazis N., Bearman P.W. A study of three-dimensional aspect of vortex shedding from a bluff body with mild geometric disturbance J. Fluid Mech.; 330: 80-112. 13. Del Guercio G., Cossu C., Pujals G., Stabilizing effects of optimally amplified streaks J. Fluid Mech. 2013; 739: 37-56. 14. Del Guercio G., Cossu C., Pujals G., Optimal streaks in the circular cylinder wake and suppression of the global instability J. Fluid Mech. 2014; 752: 572-588. 15. Karniadakis G.E., Spectral element-Fourier methods for incompressible turbulent flows C.M.A.M.E. 1990; 90: 362. 16. Karniadakis G.E., Israeli M., Orszag S.A., High-order splitting methods for incompressible Navier-Stokes equations J. Comput. Phys.; 97: 414. 17. Karniadakis G.E., Spetral/hp element methods for computational fluid dynamics, 2nd edition, Oxford University Press. 18. Barkley D., Blackburn H.M., Sherwin S.J., Direct optimal growth analysis for timesteppers Int J. Numer. Meth. Fluids 2008; 57: 1434-1458. 19. Johnson T.A., Patel V.C. Flow past a sphere up to Reynolds number 300 J. Fluid Mech. 1999; 378:19-70.