Bifurcation Analysis and the Role of Normal Form Symmetries on the Harmonic Forced Inline Oscillation of the Cylinder Wake

2016 ◽  
Author(s):  
N. Nabatian ◽  
N. W. Mureithi
Keyword(s):  
Normal Form ◽  
Numerical Computation ◽  
Vortex Shedding ◽  
Transverse Velocity ◽  
Lift Coefficient ◽  
Cylinder Wake ◽  
Order Model ◽  
Periodic State ◽  
Shedding Frequency ◽  
Low Order

Vortex shedding over a cylinder is strongly affected by the cylinder oscillation. The dynamics of the cylinder wake subjected to harmonic forced excitation in the inline direction at Re = 200 is investigated in this work. Two dominant modes of the transverse velocity field are considered to model and predict the nonlinear interaction of 2D vortex shedding. The normal form symmetries have the main role in the pattern formation. The interaction of two steady modes in the presence of O(2) × S1 symmetry is described by equivariant theory. Considering the symmetries, the amplitude equations are developed with the frequency saturation information included by the addition of complex coefficients. The reduced model is expanded up to 7th order, in order to include the spatio-temporal effects. The coefficients of the model are obtained from 2D simulations of the cylinder wake flow. The physical significance of the inline amplitude oscillation on the wake dynamics is captured by the variation of the two linear coefficients of the low order model. Similarly to the numerical results, as the amplitude of oscillation increases, two limit cycles undergo the symmetry-breaking bifurcation leading to a quasi-periodic state. The existence of the second frequency in addition to the natural shedding frequency is manifested as the small amplitude oscillation in the quasi-periodic state. For a forcing amplitude A/D = 0.5, the quasi-periodic state undergoes a torus doubling bifurcation. The dominant frequency of the bifurcated S mode matches the lift coefficient shedding frequency at A/D = 0.5 obtained from the numerical computation. The lift coefficient signal is not absolutely periodic due to the presence of the other peaks in addition to the dominant one at St = 0.1 representing the quasi-periodic flow pattern. The modulated travelling waves bifurcated from the low order model have mode S as the basic v-velocity mode which verifies the symmetric torus-doubled transverse velocity pattern observed in CFD simulation. Thus the proposed low order model can predict, with reasonable accuracy, the bifurcation sequence of the forced cylinder wake dynamic transitions observed in the numerical computation results.

POD ANALYSIS OF THREE-DIMENSIONAL HARMONICALLY FORCED WAKE FLOW OF A CIRCULAR CYLINDER

2015 ◽  
Vol 39 (4) ◽  
pp. 789-803 ◽  
Author(s):  
Negar Nabatian ◽  
Xiaofei Xu ◽  
Njuki Mureithi
Keyword(s):  
Circular Cylinder ◽  
Vortex Shedding ◽  
Transverse Velocity ◽  
Lift Coefficient ◽  
Oscillation Amplitude ◽  
Three Dimensional ◽  
Wake Flow ◽  
Cylinder Wake ◽  
Pod Analysis ◽  
Vortex Shedding Modes

A 3D numerical simulation of a circular cylinder wake is presented in this paper. The cylinder is harmonically forced in the stream-wise direction. The objective of the present work is to investigate the effect of the oscillation amplitude on the secondary transition of the wake. The frequency of the lift force is then linked to the form of the vortex shedding mode. The relation between these vortex shedding modes using POD analysis of the transverse velocity and the unsteady lift coefficient of 3D simulation is in good agreement with the 2D model. Results show that the 3D spanwise effect, which can change the wake structure, is suppressed at Re = 200 by streamwise oscillation of the cylinder. Thus the 2D analysis can effectively model the temporal instability of the wake flow.

Lock-On Vortex Shedding Patterns and Bifurcation Analysis of the Forced Streamwise Oscillation of the Cylinder Wake

2015 ◽  
Vol 25 (09) ◽  
pp. 1530022
Author(s):  
N. Nabatian ◽  
N. W. Mureithi
Keyword(s):  
Frequency Ratio ◽  
Transverse Velocity ◽  
Lift Coefficient ◽  
Oscillation Amplitude ◽  
Vortex Formation ◽  
Excitation Frequency ◽  
Immersed Boundary ◽  
Cylinder Wake ◽  
Single Vortex ◽  
Frequency Ratios

The two-dimensional numerical simulation of the flow over a cylinder forced to oscillate in the streamwise direction for Re = 200 is performed in CFX ANSYS. The controlled-vibration comprises of prescribed inline vibration from displacement amplitude-to-cylinder diameter A/D = 0.05 up to 0.5 with the excitation frequency ratios of 1, 1.5 and 2 including the harmonic and superharmonic excitation regions. The immersed boundary method is used to model the cylinder oscillation. Modal decomposition of the transverse velocity field via the proper orthogonal decomposition (POD) method is applied to uncover the interaction of symmetric and antisymmetric modes of the near wake. A model using the first two POD modes is developed based on symmetry group equivariance. The model predicts the mode interactions and bifurcated solution branches for all cases, and is shown to be in good agreement with numerical as well as previous experimental results. Lock-on is determined for a range of values of the oscillation amplitudes and frequency ratios. It is shown that the lock-on range widens with increasing nondimensional oscillation amplitude. The asymmetric 2S, P + S and symmetric pattern S with symbol S for a single vortex and P for a vortex pair shed per cycle, as well as a regime in which vortex formation is not synchronized with cylinder motion are observed in the cylinder wake depending on the combination of oscillation amplitude and frequency ratio. The frequency ratio variation from 1 to 2 leads to the switching from asymmetric to symmetric modes. The symmetric flow pattern corresponds to a near zero lift coefficient on the cylinder.

A Numerical Investigation of the Effects of Flow Pulsations on the Vortex Shedding, Drag and Lift Forces Over a Cylinder

2011 ◽  
Author(s):  
Eric D’herde ◽  
Laila Guessous
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Vortex Shedding ◽  
Free Stream ◽  
Lift Coefficient ◽  
Stream Velocity ◽  
Vortex Shedding Frequency ◽  
The Mean ◽  
Shedding Frequency ◽  
Drag And Lift

Flow over a cylinder is a fundamental fluid mechanics problem that involves a simple geometry, yet increasingly complex flow patterns as the Reynolds number is increased, most notably the development of a Karman vortex with a natural vortex shedding frequency when the Reynolds number exceeds a value of about 40. The goal of this ongoing study is to numerically investigate the effect of an incoming free-stream velocity pulsation with a mean Reynolds number of 100 on the drag and lift forces over and vorticity dynamics behind a circular cylinder. This paper reports on initial results involving unsteady, laminar and incompressible flows over a circular cylinder. Sinusoidal free-stream pulsations with amplitudes Av varying between 25% and 75% of the mean free-stream velocity and frequencies varying between 0.25 and 5 times the natural shedding frequency fs were considered. Of particular interest to us is the interaction between the pulsating frequency and natural vortex shedding frequency and the resulting effects on drag. Interestingly, at frequencies close to the natural frequency, and to twice the natural frequency, a sudden drop in the mean value of the drag coefficient is observed. The first drop in the drag coefficient, i.e. near f = fs, is also accompanied by a change in the flow and vortex shedding patterns observed behind the cylinder. This change in vortex shedding pattern manifests itself as a departure from symmetrical shedding, and in a non-zero mean lift coefficient value. The second drop, i.e. near f = 2 fs, has similar characteristics, except that the mean lift coefficient remains at zero.

Low Order Model Dynamics of the Forced Cylinder Wake

2009 ◽  
Author(s):  
N. W. Mureithi ◽  
K. Huynh ◽  
A. Pham
Keyword(s):  
Mode Locking ◽  
Model Parameters ◽  
Cylinder Wake ◽  
Flow State ◽  
Order Model ◽  
2D Simulation ◽  
Equivariant Bifurcation Theory ◽  
Preliminary Experimental Data ◽  
Pod Analysis ◽  
Low Order

The periodically forced cylinder wake exhibits complex but highly symmetrical patterns. In recent work, the authors have exploited symmetry-group equivariant bifurcation theory to derive low order equations describing, approximately, the dominant nonlinear dynamics of wake mode interactions. The models have been shown to qualitatively predict the observed bifurcations suggesting that the Karman wake remains, dynamically, a fairly simple system at least when viewed in 2D. Preliminary experimental data are presented supporting the feasibility of using 2D simulation results for the derivation of the low order model parameters. A POD analysis of the wake PIV velocity field yields flow modes closely similarly to those obtained via 2D CFD computations for Re in the 1000 range. The paper presents new results of simulations for Re = 200. For this low Reynolds number, the forced Karman wake exhibits rich dynamics dominated by quasi-periodicity, mode locking, torus doubling and chaos. The low Re torus breakdown may be explained by the Afraimovich-Shilnikov theorem. Interestingly, in a previous analysis for the higher Re number, Re = 1000, transition to a period-doubled flow state was found to occur via a route akin to the Takens-Bogdanov bifurcation scenario.

Response of a circular cylinder wake to superharmonic excitation

2001 ◽  
Vol 442 ◽  
pp. 67-88 ◽  
Author(s):  
SEUNG-JIN BAEK ◽  
SANG BONG LEE ◽  
HYUNG JIN SUNG
Keyword(s):  
Numerical Simulation ◽  
Phase Diagram ◽  
Circular Cylinder ◽  
Lift Coefficient ◽  
Vortex Formation ◽  
Cylinder Wake ◽  
Wake Flows ◽  
Frequency Responses ◽  
Formation Mode ◽  
Shedding Frequency

A systematic numerical analysis is performed for superharmonic excitations in a wake where a circular cylinder is rotationally oscillated in time. Emphasis is placed on identifying the secondary and tertiary lock-on in the forced wakes. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation has been conducted to portray the unsteady dynamics of wake flows behind a circular cylinder. The Reynolds number based on the diameter is Re = 106, and the forcing magnitude is 0.10 [les ] Ωmax [les ] 0.40. The tertiary lock-on is observed, where the shedding frequency (St0) is one third of the forcing frequency (Sf), i.e. the 1/3 subharmonic lock-on. The phase shift of CL with respect to the forcing frequency is observed. It is similar to that of the primary lock-on. However, in the secondary superharmonic excitation, modulated oscillations are observed, i.e. the lock-on does not exist. As Ωmax increases, St0 is gradually shifted from the natural shedding frequency (St*0) to lower values. The magnitudes and phases of Sf and St0 are analysed by the phase diagram. The vorticity contours are employed to examine the vortex formation mode against the forcing conditions.

Quasi-periodicity in the wake of a rotationally oscillating cylinder

2000 ◽  
Vol 408 ◽  
pp. 275-300 ◽  
Author(s):  
SEUNG-JIN BAEK ◽  
HYUNG JIN SUNG
Keyword(s):  
Lift Coefficient ◽  
Phase Changes ◽  
Circle Map ◽  
Van Der Pol ◽  
Wake Flows ◽  
Frequency Selection ◽  
Frequency Responses ◽  
Periodic State ◽  
Main Emphasis ◽  
Shedding Frequency

A systematic numerical analysis is performed for the quasi-periodicity in the wake where a circular cylinder is rotationally oscillated in time. The main emphasis is placed on the identification of frequency selection subjected to the controlled perturbations in the vicinity of lock-on. The frequency responses are scrutinized by measuring the lift coefficient (CL). A direct numerical simulation is made to portray the unsteady dynamics of wake flows at Re = 110. It is found that, after the shedding frequency is bifurcated at the boundary of lock-on, one frequency follows the forcing frequency and the other gradually converges to the natural shedding frequency. The asymptotic convergence phenomena are observed by solving the Van der Pol equation and the circle map. A new frequency selection formula is proposed. The quasi-periodic states are interpreted in terms of the forcing frequency, shedding frequency and modulated frequencies by employing the torus concept and the CL(t) diagram. In the quasi-periodic state, the variation of magnitudes and relevant phase changes of CL with forcing phase are examined.

Vortex dynamics for flow over a circular cylinder in proximity to a wall

2017 ◽  
Vol 812 ◽  
pp. 698-720 ◽  
Author(s):  
Guo-Sheng He ◽  
Jin-Jun Wang ◽  
Chong Pan ◽  
Li-Hao Feng ◽  
Qi Gao ◽  
...  
Keyword(s):  
Circular Cylinder ◽  
Shear Layer ◽  
Flat Plate ◽  
Vortex Shedding ◽  
Vortex Dynamics ◽  
Wake Vortex ◽  
Secondary Vortex ◽  
Cylinder Wake ◽  
Gap Ratio ◽  
Shedding Frequency

The dynamics of vortical structures in flow over a circular cylinder in the vicinity of a flat plate is investigated using particle image velocimetry (PIV). The cylinder is placed above the flat plate with its axis parallel to the wall and normal to the flow direction. The Reynolds number $Re_{D}$ based on the cylinder diameter $D$ is 1072 and the gap $G$ between the cylinder and the flat plate is varied from gap-to-diameter ratio $G/D=0$ to $G/D=3.0$. The flow statistics and vortex dynamics are strongly dependent on the gap ratio $G/D$. Statistics show that as the cylinder comes close to the wall ($G/D\leqslant 2.0$), the cylinder wake becomes more and more asymmetric and a boundary layer separation is induced on the flat plate downstream of the cylinder. The wake vortex shedding frequency increases with decreasing $G/D$ until a critical gap ratio (about $G/D=0.25$) below which the vortex shedding is irregular. The deflection of the gap flow away from the wall and its following interaction with the upper shear layer may be the cause of the higher shedding frequency. The vortex dynamics is investigated based on the phase-averaged flow field and virtual dye visualization in the instantaneous PIV velocity field. It is revealed that when the cylinder is close to the wall ($G/D=2.0$), the cylinder wake vortices can periodically induce secondary spanwise vortices near the wall. As the cylinder approaches the wall ($G/D=1.0$) the secondary vortex can directly interact with the lower wake vortex, and a further approaching of the cylinder ($G/D=0.5$) can result in more complex interactions among the secondary vortex, the lower wake vortex and the upper wake vortex. The breakdown of vortices into filamentary debris during vortex interactions is clearly revealed by the coloured virtual dye visualizations. For $G/D<0.25$, the lower shear layer is strongly inhibited and only the upper shear layer can shed vortices. Investigation of the vortex formation, evolution and interaction in the flow promotes the understanding of the flow physics for different gap ratios.

Hybrid Reduced Model of Rotor

2013 ◽  
Vol 60 (3) ◽  
pp. 319-333
Author(s):  
Rafał Hein ◽  
Cezary Orlikowski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hybrid Model ◽  
Rotor System ◽  
Reduced Model ◽  
Order Model ◽  
Gyroscopic Effect ◽  
Reduced Models ◽  
Rigid Finite Element Method ◽  
Low Order

Abstract In the paper, the authors describe the method of reduction of a model of rotor system. The proposed approach makes it possible to obtain a low order model including e.g. non-proportional damping or the gyroscopic effect. This method is illustrated using an example of a rotor system. First, a model of the system is built without gyroscopic and damping effects by using the rigid finite element method. Next, this model is reduced. Finally, two identical, low order, reduced models in two perpendicular planes are coupled together by means of gyroscopic and damping interaction to form one model of the system. Thus a hybrid model is obtained. The advantage of the presented method is that the number of gyroscopic and damping interactions does not affect the model range

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