In the absence of synchronization, the nature of the flow is found to be quasi-periodic (rotation z) or chaotic (rotations y and xyz). It is found that the rotational tendency of the trace is the largest and smallest for rotations around x and y, respectively.
Uniform flow past bluff structures
Bluff structures are designed to carry loads (bridges or buildings), contain flow (nuclear reactors or riser tubes) or provide heat transfer surface (heat exchanger tubes or electronic cooling systems). Flow past a bluff structure (or wake flow) is characterized by a dominant coefficient of pressure resistance than the frictional resistance.
Vortex-induced vibration
Vortex-induced vibration of a circular cylinder
Depending on the magnitude (high or low) of the mass damping parameter, different types of responses were recorded. For a high mass damping parameter, experiments conducted by Feng [6] in air at m∗ = 100 observed two branches of the response, namely “initial” and “bottom”.
Forced vibration of a circular cylinder
Similarity between free and forced vibration
The wake modes revealed by Williamson & Roshko [12] in 1988 for the forced vibration of a circular cylinder were later confirmed by the free vibration study of Govardhan & Williamson [17] in 2000. Observation of the same phenomena and modes suggests a strong correlation between these two study methods for a circular cylinder.
Numerical simulations
Immersed Boundary Method
A forcing function was incorporated into the Navier-Stokes equation to connect the submerged boundary and the flow. Recently, Peter & De [45] have highlighted the advantages of using GCIBM in the non-inertial reference frame.
Need of the present work
Thesis statement
Can forced transverse oscillation of a sphere in uniform flow predict the VIV modes as in the case of a circular cylinder.
Organization of the thesis
Governing equations
Using the scales from Table 2.1, the normalized form of the governing equations in the non-inertial reference frame is obtained as (for convenience, ∗ is omitted).
Numerical details
Time integration
An explicit, multi-time-level technique (AB) for the non-linear term and CN for all the linear terms yield.
Finite volume method
Since the mesh is stationary, the integration of the temporal acceleration over the CV reduces to . If the surface vector Se(Se1, Se2, Se3) is written in terms of the components of.
GCIBM modified
When the mass flow converges, as in the explicit procedure, a finite velocity field is obtained. The use of a non-inertial frame avoids these problems, since the body remains stationary and only the boundary that is adapted to the boundary conditions moves.
Linear solver
Using the above factors, the original system (Aφ=b) becomes. performing symbolic multiplication of the diagonals leads to the beginning. To complete the factorization, N must hold the extra diagonals (those closed in Eq. Stone [56] proposed that convergence can be improved if N, instead of just the extra diagonals, has all the diagonals that appear in LU. If the equation of above is compared with Eq. 2.32) the nonzero diagonals of N are obtained as.
Solution algorithm
Parallel implementation
- Computational details
- Body points on the sphere surface
- Computation of the fluid forces
- Domain size and Grid spacing
- Domain independence study
- Mesh independence study
- Steady axisymmetric flow, Re = 100
- Steady planar symmetric flow, Re = 250
- Unsteady planar symmetric flow, Re = 300
- Convergence behavior of different linear solvers
- Performance of the parallelized method
Moreover, domain decomposition causes non-uniform distribution of the ghost cells in the processors. These views clearly show the extent of the computational domain and non-uniform grid distribution. The performance of the parallel simulations is assessed using speed-up (S = Ts/Tp) and efficiency (η = S/p), where Ts and Tp are respectively the sequential and parallel (with p processors) CPU times.
Transverse oscillation of a circular cylinder
Computational details
A square domain on side1.2 with grid size 161×161 was used to provide a uniform finer mesh engulfing the cylinder, generating grid resolution ∆x= ∆y= 0.007in and around the cylinder. We defined a frequency ratio (fR=fe/fo) where is the vortex shedding frequency from the stationary cylinder. If Sto = foD/U is the Strouhal number associated with the vortex shedding from a stationary cylinder, then the forced Strouhal number (Ste =feD/U) can be expressed asSte =fRSto; in this study Sto was found to be 0.201.
Results
If Sto = foD/U is the Strouhal number associated with vortex shedding from a stationary cylinder, then the forced Strouhal number (Ste =feD/U) can be expressed asSte =fRSto; in this studySto turned out to be 0.201. The complex waveforms reflect [10] the existence of two frequencies: 1) the natural vortex shedding frequency (fo) of a stationary circular cylinder and 2) the forced frequency (fe). The purely sinusoidal signals are a consequence of the wake blocking condition (0.8≤fR ≤1.0), where the ratio of the forced frequency to the vortex shedding frequency is unity. Outside the locking regime, the dominant vortex shedding frequency (fs) varies almost linearly with fR (see Fig. 3.13).
Uniform flow past a rotating circular cylinder
Computational details
Results
The influence of the forced transverse oscillation of the sphere on the resulting trace is studied in detail. This chapter reports the results thus obtained, and based on the observations, the unstable wake modes are revealed in light of the VIV sphere. The sphere oscillates with a range of force frequencies at a fixed amplitude and Reynolds number.
Introduction
Vibrations of the sphere in the vertical direction (z) were found to be less than 5% of those in the transverse direction (y). In the first region, the vortex ejection pattern was identical to that of a stationary sphere without transverse oscillation. The vortex separation and wake structure were similar to those observed by G-W in the second region.
Problem set-up
Computational details
Time evolution of force coefficients
- Wake synchronization regime and Mode III
- Energy transfer and decomposition of transverse force
- Synchronized Modes I and II
- Characterization of VIV Modes
- VIV Modes and time traces
A 100% increase in mean drag coefficient (< CD >) was observed by Williamson & Govardhan [48] from that of stationary sphere values. The time signals atfR = 1.3 and 0.8 for one oscillation cycle of the sphere are shown in Fig. 4.6(c) shows different periodicity of CtotalandCvortextfrom the sphere oscillation atfR = 0.2 suggesting the loss of wake organization achieved in the synchronous regime, a feature of mode III.
Vortical motion
Periodic vortex shedding
On the other hand, outside this regime, no specific shedding pattern is observed in Figure Zdravkovich [11] for a circular cylinder in the synchronization regime. He noted that the oscillating cylinder also imposes the timing of the vortex shedding, along with the frequency of the wake. .
Wake structures and planar symmetry
4.7(c), 4.8(c)), the vortex rings from each side merge and move away downstream as vortex rings. The vortex lobes at fR = 1.3 are shorter due to the shorter vortex formation length discussed in section 4.4.1. Vortex rings are not seen because the frequency is so low that coupling of the loops hardly occurs.
Instantaneous vorticity field
In the same field of view (FOV) or observation window, more structures are seen at fR = 1.3. He reported that in the low frequency limit of the "lock-in" regime, the vortex path widens as the longitudinal spacing between the vortex filaments increases. Although the flow is periodic, but its period is not the same as that of the sphere oscillation (fe/fs 6= 1 discussed in section 4.3.1), therefore, the structures in frames (i) and (viii) are not repeated. .
Summary
- Rotation about stream-wise axis
- Rotation about transverse axes
- Rotation about an inclined axis
- Need of the present work
The effect of simultaneous transverse oscillation and rotation of the sphere on the resulting wake is studied in detail. Moreover, the velocity on the surface of the sphere is imposed according to the angular velocity. They found that counterclockwise rotation of the sphere induces a positive vorticity in the direction of the flow, making the flow periodic and spiral.
Computational details
Rotation of the sphere is taken into account by additionally imposing velocity on the surface of the sphere. If (xb, yb, zb) represent Cartesian coordinates of a point on the surface, then are the imposed velocities.
Time signals
ZeroCH signal for rotation about z demonstrates the preservation of plane symmetry observed for purely oscillating sphere (previous chapter). Largest and smallest ranges of β for rotation aboutxandy, respectively, reveal extreme rotation in the wake. For rotation about y, the amplitude of the higher harmonics is more indicative of growth of nonlinearity, resulting in jagged lobes in Fig.
The phase plots for β show departures from this ideal range implying that the oscillations of the sphere prevent full rotation of the wave. The lack of wave synchronization atfR = 0.8 implies that the imposing character of the transverse oscillation is overwhelmed by the rotational motion. The larger closed area in the β −ys curve for x-rotation at fR = 0.8 compared to fR= 1.3 implies higher wake rotation.
Global flow parameters
The increase < CD >at fR = 0.8 can be explained from the point of view of rotational motion. At a lower rocking inertia, fR = 0.8, the flow is driven more by rotation than by rocking which causes an increase in
Vortical motions
Figure 5.12(a)-(c) shows the time evolution of the force coefficients, the planar streamwise vorticity contours and the three-dimensional wake structure in eight cases (marked with black dots in the signals) during a sphere oscillation cycle. This narrowing of the wake is expected based on the observations of Giacobello et al. This observation suggests the deflection of the wave towards positivity (advanced side of the sphere).
The coil appears to shed Ω-shaped loops with a nose in the center of the loops as seen in Fig.5.15(c). Mittal & Kumar [64] for a rotating circular cylinder observed the thinning (or narrowing) of the track at high rotational speed. The highest rotational effects (due to the highest speed and the lowest fR) for this case make the track thinner.
Summary of features in the flow
- State of the wake
- Nature of vortex shedding
- Nature of flow
- Response of the wake to rotation
- Visualization of the wake
The nature of the temporal evolution of the current for all cases is given in the fifth column of Table 5.2. The wake response to the imposed rotation, denoted by the lift angle β, can be either auxiliary or opposite. Changes to the Ghost Cell Immersed Boundary Method (GCIBM) in Chapter 2 improved its capabilities.
Classification of flow-induced vibrations
Movement of a ship
Classification of cells in GCIBM
Relation between inertial and non-inertial frames of reference
A typical control volume (CV)
Diffusion flux computation at the east face
Geometrical features of a typical finite volume
Schematic diagram for evaluating flow variables at G
Planar view of finite volumes and their neighbors
Flowchart for the solution steps
Schematic diagram for uniform flow past a stationary sphere
Cubical control volume for calculating the force coefficients
Comparison of signals for different grid sizes
Planar and three-dimensional views of the grid
Comparison of computed wake structures with reported results for
Convergence behavior of the linear solvers
Performance of the parallel solver with processors
Time evolution of the coefficient of drag and lift for forced trans-
Ratio of the forcing and the vortex shedding frequency as a function
Comparison of computed global flow parameters for forced trans-
Comparison of the average and rms values of the drag and lift co-
Schematic diagram for the forced transverse oscillation of a sphere 56
Time traces for one oscillation cycle
Instantaneous signals, planar vertical vorticity contours and wake
Instantaneous signals, planar vertical vorticity contours and wake
Instantaneous signals, planar vertical vorticity contours and wake
In the present problem, the flow is affected by oscillations as well as the rotational motion of the sphere. 33] together implies a subsequent deflection towards the advanced side of the sphere together with the narrowing of the wave. The appearance of clockwise transverse vorticity (blue) on the leading (−z) side of the sphere is consistent with the literature [28,30–33].
This difference in deflection of the track is due to the choice of the transverse axis (yorz). On the other hand, an intermediate area for rotation aboutz implies a weak assistance to the rotation of the sprue.
Instantaneous planar stream-wise vorticity contours
Schematic diagram of the oscillating and rotating sphere
Instantaneous signals, planar stream-wise vorticity contours and
The Q structure indicates the wake as a chain of vortex loops oscillating in alternating sides of the sphere as seen in the previous chapter. To see how the appearance of these contours determines the location of the chain, consider frame (v). Furthermore, the fifth instance corresponding to frame (v) shows a positive CL, which suggests that the CLs are exerted in an opposite direction to the chain motion.
Instantaneous signals, planar stream-wise vorticity contours and
This leads to a progressive intensification and thinning of the shear layer released from the leading edge. Poon et al.[33] stated that "most vortices are released from the shear layer on the advancing side". A remarkable discovery [28-33] for a transversely rotating sphere is that the wave deflection and the appearance of the vortex in the clockwise direction is always advancing. side, whichever axis (yorz) is chosen.
Instantaneous signals, planar stream-wise vorticity contours and
This case corresponds to the fourdof (one transverse oscillation and three revolutions about the primary axes) of the sphere, which makes it relevant in practical situations. This case is expected to demonstrate a combination of flow features observed for three primary rotations. As a result, the ball experiences the largest negative reaction force along the direction, as evidenced by a maximum magnitude of < CL > (-0.32) in this case.
Instantaneous signals, planar stream-wise vorticity contours and
In addition to oscillation along the path, the wake exhibits counterclockwise rotation around all three primary axes. Furthermore, in the case of a stationary sphere at very high Re, Taneda [79] experimentally and Jindal et al. [80] numerically found these structures with a deviation from the flow axis, causing non-zero lateral forces .
Instantaneous signals, planar stream-wise vorticity contours and
The least fluctuations marked by the smallest rms value imply damping of the unsteady nature in the flow. In other words, the instability in the flow caused mainly by the oscillation of the sphere is suppressed by the simultaneous streamwise rotation. 5.17(c) indicates the wakes which are twisted horseshoe loops with two legs on each arm of the shoe.
Instantaneous signals, planar stream-wise vorticity contours and
Moreover, some of the vortex generated on the rotating cylinder appeared to be wrapped around it. Therefore, incorporating the non-inertial reference frame alleviates these problems with certain limitations. The effects of oscillation of the sphere along x and z axes can also be considered.
