Numerical Study of Laminar Flow and Vortex-Induced Vibration on Cylinder Subjects to Free and Forced Oscillation at Low Reynolds Numbers
Abstract
:1. Introduction
2. Model Equations
2.1. Fluid Flow Model
2.2. Structure Model
2.3. Boundary Conditions
- (i)
- At the inlet (inflow), uniform flow conditions were assumed, that is,
- (ii)
- At the outlet (outflow), the flow variables had zero diffusion, which means that the boundary conditions at the outlet were extrapolated from the domain to result in the upstream flow conditions. The velocity and pressure were updated according to the extrapolation at the outflow. This is based on the assumption of a fully developed flow experiencing no change in the area at the outlet boundary [28].
- (iii)
- At the upper and lower walls, where the viscous effects were negligible, boundary slip conditions were assumed.
- (iv)
- On the cylindrical structure wall, the no-slip condition was applied, according to which all the velocity components on the cylinder surface were zero due to the fluid viscous effects.
- (v)
- Moreover, the outlet Neumann conditions were set to zero for both velocity and pressure.
2.4. Dimensionless Parameters
3. Computational Procedure
4. Test Cases
4.1. VIV of Circular Cylinder
4.2. Computational Domain Schematics
- (i)
- In the first scenario, four springs with dampening effects were included. As a result, all springs applied four forces caused by spring effects, causing the damper to slow down. It is also important to remember that all springs were of the same size and that the cylinder could not spin on its axis. Nearly symmetrical stresses in the upper and lower regions of the cylinder were created by the fluid flow as it passed over them. Technically, the point displacement file displays this scenario, which is for the 2-DoF structure.
- (ii)
- In the second scenario, more restrictions were added to the above, which forced the cylinder to travel only along the y-axis; this was termed the 1-DoF case.
- (iii)
- The third scenario had two springs and two dampers. As the cylinder moved along both the in-line and transverse paths, this also applied to the 2-DoF system. The density of the cylinder was set to 9.97 kg for each of the three cases, whereas the stiffness and damping values were 2 N · s/m and 4 N/m, respectively.
4.3. Mesh Generation
5. Results and Discussion
5.1. Freely Oscillating Cylindrical Structure
5.1.1. Discussion of Simulation Predictions of VIV for 1- and 2-DoF
5.1.2. Drag and Lift Factors in the First Scenario
5.1.3. Drag and Lift Factors in the Second Scenario
- (i)
- Force = stiffness × displacement. This indicates that the force caused by the spring varies linearly with the stiffness value.
- (ii)
- The movement of the damper is directly proportional to the velocity of its motion: Force = damping × velocity. This indicates that the damper generates greater force as a result of faster movement.
- The chosen three cases were assumed to be under-damped. This implies that the damper was not sufficient to halt oscillation; hence, all three had the same oscillation structure with the highest amplitude.
- In addition, all the considered cases had similar drag factors until 30 s, provided that the movement paths were nearly identical during this initial period. The decrease in the drag factor must be equally associated with the generation of the first vortex and its release, with the next vortex successively originating and then being released.
- The decreased damping contained less strength to react to the velocity force, resulting in the following:
- 3.1.
- The larger drag factor from 30 s to nearly 63 s was caused by faster cylinder movement for similar forces acting on it, with the movement of the cylinder producing huge vortexes and distorting the fluid.
- 3.2.
- The smaller drag factor after 65 s was associated with a greater capability of the cylinder to weave and bob as eddies were created near the wake of the cylinder.
- The decreased-stiffness scenario regularly contained a very small drag factor from the initial 30 s period, provided that it needed to be shifted for a longer distance for approximately the same forces to be exerted by the fluid on the cylinder surface.
- In the lift case, the predictions were in line with those noticed when comparing the 1-DoF and 2-DoF cases having four springs, provided that the additional capability to shift the cylinder led to a small lift value, for the case where eddies could be created and discharged with small dragging efforts.
- In the last analogy, it could be observed that a decrement in stiffness created smaller amplitudes of the drag and lift after the initial 30 s period.
5.1.4. Drag and Lift Factors in the Third Scenario
- The decreased damping led to quicker motion, which clearly indicates a rapid response interval for the cylinder to shift along the vortexes at a relatively higher rate compared with the remaining two types. Therefore, the greatest amplitudes in the first 30 s were greater/smaller compared with the benchmark 2-DoF situation.
- The decreased stiffness enabled the cylinder to shift away and create tiny vortexes with lower intensity compared with the previous types.
- The decreased damping permitted a lower drag factor compared with the actual state;
- The lowest drag factor was produced with reduced stiffness;
- The two variants had almost similar frequency as observed in the reference 2-DoF situation.
5.1.5. Drag and Lift Factors in Reduced-Damping Scenario
- For uniform periodic oscillation in the two cases, the prediction of 1-DoF revealed a stronger drag factor compared with the respective reference cases.
- With regard to the lift, the predictions were nearly the same as the previously obtained results.
5.1.6. Drag and Lift Factors in Scenario with Reduced Spring Stiffness and One and Two Degrees of Freedom
5.1.7. Lift and Drag Factors for 2-DoF System with Four Springs and Two Springs Plus Two Dampers
- The dissimilar nature of the damper/spring structure could be noticed in the respective plot from the 25–30 s mark. This behavior could have been due to the non-alignment of dampers and springs, causing the cylinder to shift faster, based on the position of the cylinder.
- The greatest amplitude of the drag factor was small in case 3 compared with the predictions in other cases.
- The reduced damping led to a shorter reaction time and was systematically identified, provided it took approximately 40 s less to attain the periodic working area.
5.2. Forced Oscillating Cylindrical Structure
- Figure 12 demonstrates that the highest values of the drag coefficient align with the lowest and highest points of the lift, while the lowest values of the drag coefficient corresponded to the locations where the lift was zero.
- Furthermore, Figure 13 demonstrates a clear correlation between the peaks of the lift and increases in the movement of the cylinder. This indicates that the highest dislocation values corresponded to the highest values of the drag factor. The zero-lift locations corresponded to the instances when the cylinder crossed the central axis during forced oscillation.
6. Conclusions
- Damping and stiffness impacted the behavior of the cylinder along with the flow, and there were important noticeable variations in the response time and starting speed in which the process responded to the flow; this was also true in the case in which the constant periodic oscillation was stabilized.
- For the 2-DoF scenario, during the initial phase, a decrease in damping led to a significantly high drag coefficient, whereas a decrease in stiffness decreased the maximum drag coefficient.
- The behavior of the cylinder was significantly altered by the linear nature of the springs and dampers. In particular, the cylinder response period and the drag factor were significantly affected by the stiffness and damping factors.
- If the mechanism is intended to include more drag, then the dampers should be kept at a minimum or withdrawn for stiff springs. If the mechanism requires minimum drag, moderate damping magnitudes are required with fairly relaxed springs.
- At the point of zero lift, when the forces forming the vortex were in equilibrium, which occurred at approximately , the wake in close proximity to the cylinder was nearly parallel to the fluid flow, which moved from left to right.
- The drag and lift coefficients reach their greatest values when (a) a domain of high-velocity flow passes over top of the cylinder while it is in the upper position or (b) when a high-velocity flow is directed towards the cylinder while it is in the lower position, which was noted at around 30 s in this study.
- The observed peaks in the drag and lift coefficients align with the expected behavior of wings and airfoils under identical operating conditions, specifically when comparing the maximum values for wings/airfoils.
- The current analysis was limited to , and the FSI and VIV analyses were confined to a flow over a cylinder. This work could be extended to include higher Reynolds numbers and objects with other shapes. Because of research limitations, turbulence modeling was not used in this study.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Anagnostopoulos, P. Numerical investigation of response and wake characteristics of a vortex-excite cylinder in a uniform stream. J. Fluids Struct. 1994, 8, 367–390. [Google Scholar] [CrossRef]
- Anagnostopoulos, P. Numerical study of flow past a cylinder excited transversely to the incident stream. Part I: Look-in zone, hydrodynamic forces and wake geometry. J. Fluids Struct. 2000, 14, 819–851. [Google Scholar] [CrossRef]
- Govardhan, R.; Williamson, C. Modes of vortex formation and frequency response of a freely vibrating cylinder. J. Fluid Mech. 2000, 420, 85–130. [Google Scholar] [CrossRef]
- Williamson, C.H.K.; Govardhan, R. Vortex-induced vibrations. Annu. Rev. Fluid Mech. 2004, 36, 413–455. [Google Scholar] [CrossRef]
- Williamson, C.; Jauvtis, N. A high-amplitude 2T mode of vortex-induced vibration for a light body in motion. Eur. J. Mech. B/Fluids 2004, 23, 107–114. [Google Scholar] [CrossRef]
- Bourguet, R.; Karniadakis, G.; Triantafyllou, M. Vortex-induced vibrations of a long flexible cylinder in shear flow. J. Fluid Mech. 2011, 677, 342–382. [Google Scholar] [CrossRef]
- Mittal, S. Excitation of shear layer instability in flow past a cylinder at low Reynolds number. Int. J. Numer. Methods Fluids 2005, 49, 1147–1167. [Google Scholar] [CrossRef]
- Jung, J.H.; Jeong, K.L.; Gill, J.H.; Jung, D. Large eddy simulation of free motion of marine riser using OpenFOAM. J. Ocean. Eng. Technol. 2019, 33, 387–393. [Google Scholar] [CrossRef]
- Leong, C.; Wei, T. Two-degree-of-freedom vortex-induced vibration of a pivoted cylinder below critical mass ratio. Proc. R. Soc. A Math. Phys. Eng. Sci. 2008, 464, 2907–2927. [Google Scholar] [CrossRef]
- Srinil, N.; Zanganeh, H.; Day, A. Two-degree-of-freedom VIV of circular cylinder with variable natural frequency ratio: Experimental and numerical investigations. Ocean. Eng. 2013, 73, 179–194. [Google Scholar]
- Xie, F.; Deng, J.; Xiao, Q.; Zheng, Y. A numerical simulation of VIV on a flexible circular cylinder. Fluid Dyn. Res. 2012, 44, 045508. [Google Scholar] [CrossRef]
- Xiao, Z.J.; Yang, S.H.; Yu, C.; Zhang, Z.; Sun, L.; Lai, J.; Huang, Y.P. Effects of cavitation on vortex-induced vibration of a flexible circular cylinder simulated by fluid-structure interaction method. J. Hydrodyn. 2022, 34, 499–509. [Google Scholar] [CrossRef]
- Deng, D.; Zhao, W.; Wan, D. Numerical study of vortex-induced vibration of a flexible cylinder with large aspect ratios in oscillatory flows. Ocean. Eng. 2021, 238, 109730. [Google Scholar] [CrossRef]
- Raghavan, K.; Bernitsas, M. Experimental investigation of Reynolds number effect on vortex induced vibration of rigid circular cylinder on elastic supports. Ocean. Eng. 2011, 38, 719–731. [Google Scholar] [CrossRef]
- Koubogiannis, D.G.; Benetatos, M.V.N. Numerical Assessment of Flow Energy Harvesting Potential in a Micro-Channel. Fluids 2023, 8, 222. [Google Scholar] [CrossRef]
- Xiao, D.; Hao, Z.; Zhou, T.; Zhu, H. Experimental Studies on Vortex-Induced Vibration of a Piggyback Pipeline. Fluids 2024, 9, 39. [Google Scholar] [CrossRef]
- Gonçalves, R.A.; Teixeira, P.R.D.F.; Didier, E.L. Numerical simulations of low Reynolds number flows past elastically mounted cylinder. Eng. Térmica (Therm. Eng.) 2012, 11, 61–67. [Google Scholar] [CrossRef]
- Li, M.; Bernitsas, C.; Jing, G.; Hai, S. Hydrokinetic Power Conversion Using Vortex-Induced Oscillation with Cubic Restoring Force. Energies 2020, 13, 3283. [Google Scholar] [CrossRef]
- Zhang, D.; Feng, L.; Yang, H.; Li, T.; Sun, H. Vortex-Induced Vibration Characteristics of a PTC Cylinder with a Free Surface Effect. Energies 2020, 13, 907. [Google Scholar] [CrossRef]
- Narváez, G.F.; Schettini, E.B.; Silvestrini, J.H. Numerical simulation of flow-induced vibration of two cylinders elastically mounted in tandem by immersed moving boundary method. Appl. Math. Model. 2020, 77, 1331–1347. [Google Scholar] [CrossRef]
- Wu, Y.; Lien, F.-S.; Yee, E.; Chen, G. Numerical Investigation of Flow-Induced Vibration for Cylinder-Plate Assembly at low Reynolds Number. Fluids 2023, 8, 118. [Google Scholar] [CrossRef]
- Sciortino, G.; Lombardi, V.; Prestininzi, P. Modelling of Cantilever-Based Flow Energy Harvesters Featuring C-Shaped Vibration Inducers: The Role of the Fluid/Beam Interaction. Appl. Sci. 2023, 13, 416. [Google Scholar] [CrossRef]
- Michna, J.; Rogowski, K. Numerical study of the effect of the Reynolds number and the turbulence intensity on the performance of the NACA 0018 Airfoil at the low Reynolds number regime. Processes 2022, 10, 1004. [Google Scholar] [CrossRef]
- Placzek, A.; Sigrist, J.; Hamdouni, A. Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: Forced and free oscillations. Comput. Fluids 2009, 38, 80–100. [Google Scholar] [CrossRef]
- Bearman, P.W. Understanding and predicting vortex-induced vibrations. J. Fluid Mech. 2009, 634, 1–4. [Google Scholar] [CrossRef]
- Bearman, P.W. Circular cylinder wakes and vortex-induced vibrations. J. Fluids Struct. 2011, 27, 648–658. [Google Scholar] [CrossRef]
- Schafer, F.; Uffinger, T.; Becker, S.; Grabinger, J.; Kaltenbacher, M. Fluid-structure interaction and computational aeroacoustics of the flow past a thin flexible structure. J. Acoust. Soc. Am. 2008, 123, 3570–3575. [Google Scholar] [CrossRef]
- Patnana, V.K.; Bharti, R.P.; Chhabra, R.P. Two-dimensional unsteady flow of power-law fluids over a cylinder. Chem. Eng. Sci. 2009, 64, 2978–2999. [Google Scholar] [CrossRef]
- Hirsch, C. Numerical Computation of Internal and External Flows: The Fundamentals of Computational Fluid Dynamics; Butterworth-Heinemann: Oxford, UK, 2007. [Google Scholar]
- Muzaferija, S. Adaptive Finite Volume Method for Flow Prediction Using Unstructured Meshes and Multigrid Approach. Ph.D. Thesis, Imperial College, University of London, London, UK, 1994. [Google Scholar]
- OpenFOAM Foundation. OpenFOAM User Guide; Version 2.2.2; OpenFOAM Foundation: London, UK, 2013. [Google Scholar]
- Maric, T.; Hopken, J.; Mooney, K. The OpenFOAM Technology Primer; Sourceflux UG (haftungsbeschrankt): Duisburg, Germany, 2014. [Google Scholar]
- Yakush, S.E.; Sivakov, N.S.; Melikhov, O.I.; Melikhov, V.I. Numerical Modeling of Water Jet Plunging in Molten Heavy Metal Pool. Mathematics 2024, 12, 12. [Google Scholar] [CrossRef]
- Ke, L.; Wang, Q. Numerical Simulation of the Stability of Low Viscosity Ratio Viscoelastic Lid-Driven Cavity Flow Based on the Log-Conformation Representation (LCR) Algorithm. Mathematics 2024, 12, 430. [Google Scholar] [CrossRef]
- Aghajanpour, A.; Khatibi, S. Numerical Study of Velocity and Mixture Fraction Fields in a Turbulent Non-Reacting Propane Jet Flow Issuing into Parallel Co-Flowing Air in Isothermal Condition through OpenFOAM. AppliedMath 2023, 3, 468–496. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Al Manthari, M.S.; Azeez, C.; Sankar, M.; Pushpa, B.V. Numerical Study of Laminar Flow and Vortex-Induced Vibration on Cylinder Subjects to Free and Forced Oscillation at Low Reynolds Numbers. Fluids 2024, 9, 175. https://doi.org/10.3390/fluids9080175
Al Manthari MS, Azeez C, Sankar M, Pushpa BV. Numerical Study of Laminar Flow and Vortex-Induced Vibration on Cylinder Subjects to Free and Forced Oscillation at Low Reynolds Numbers. Fluids. 2024; 9(8):175. https://doi.org/10.3390/fluids9080175
Chicago/Turabian StyleAl Manthari, M. S., Carlton Azeez, M. Sankar, and B. V. Pushpa. 2024. "Numerical Study of Laminar Flow and Vortex-Induced Vibration on Cylinder Subjects to Free and Forced Oscillation at Low Reynolds Numbers" Fluids 9, no. 8: 175. https://doi.org/10.3390/fluids9080175