In-line Oscillation Research Articles

Publisher's Note: “Suppression of vortex shedding in the wake of a circular cylinder through high-frequency in-line oscillation” [Phys. Fluids 35, 073605 (2023)

Publisher's Note: “Suppression of vortex shedding in the wake of a circular cylinder through high-frequency in-line oscillation” [Phys. Fluids <b>35</b>, 073605 (2023)

Vortex-induced vibration of an elastically mounted cylinder subject to planar inflow shear

We have numerically investigated the effect of planar inflow shear on the vortex-induced vibration of a three-dimensional circular cylinder mounted on elastic supports in streamwise and cross-stream directions. At a Reynolds number Re=300, the reduced velocity (U∗) is varied in the range U∗=2−12 for a fixed mass ratio m∗=2. The effect of inflow shear is brought about by varying the shear parameter βy=0,0.05,0.1. We have observed four distinct branches in the structural response curve: Initial Branch (IB), High Amplitude Branch-1 (HAB1), High Amplitude Branch-2 (HAB2), and Desynchronization Branch (DS). The response of the cylinder is locked in with the near wake during IB, HAB1, and HAB2, where the inline oscillation amplitude of the cylinder triggers for increasing βy. The cylinder oscillates with a tilt against the high-velocity side at larger βy for HAB1 and HAB2 responses. During IB response at βy=0, the near wake shows the formation of vortex filaments clustered through the spanwise extent. However, in the HAB1 and HAB2 regimes, the propagation of secondary instability yields random occurrences of vortex splits in the wake. Similar consequences of vortex splits are found at βy=0.1 for IB, HAB1, and HAB2 regimes, where we observe fragmented vortex filaments wrapping around the vortex rings. For βy=0, we have found mode B-type wake instability in the IB response at a plane x=2, where the vortex structures form a staggered array with a spatial periodicity λ≈D. At z=0 plane, the vortex shedding mode is “2S” for the IB response at βy=0.05, while the “P+S” mode is at βy=0.1. The vortex strands experience stretching in the DS branch, where no specific shedding mode is discerned. The cross-stream evolution of time-averaged streamwise velocity shows a data collapse with an asymmetric localization above the cylinder for increasing βy. The streamwise and spanwise velocity fluctuations at βy=0.1 are insignificant compared to the cross-stream. However, for the IB and DS regimes, the growth of the rms velocities reveals spanwise invariance. In contrast, we found small-scale fluctuations of cross-stream velocities about the mean in the HAB1 regime. The temporal growth of the sectional drag coefficients (CDs) along the span shows a negative value in the IB regime for βy=0,0.5. However, we observe the quasiperiodic behavior of CDs along the span for the HAB1 regime at βy=0. The spanwise growth of CDs for the DS regimes shows intermittent transitions via the positive, zero, and negative values for different βy. For both IB and DS regimes at βy=0, the time evolution of sectional lift coefficients (CLs) is asymmetric along the span and represents a helical pattern. However, for increasing βy, the growth of CLs for HAB1 and HAB2 regimes is intermittent. An increase in βy triggers the magnitudes of CLs in the DS branch, where the spanwise evolution shows symmetric periodic bands of sectional lift coefficients.

Suppression of vortex shedding in the wake of a circular cylinder through high-frequency in-line oscillation

This paper presents a new flow control approach to suppress the vortex shedding in the wake of a circular cylinder through high-frequency oscillation. The circular cylinder is forced to oscillate in the streamwise direction at high-frequency and low amplitude, corresponding to a high Stokes number (β = 100–1000) and low Keulegan–Carpenter number (KC = 0.001–4). Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations of an oscillating circular cylinder in steady current have been carried out in the parameter space of KC, Rec, and β. Our numerical results show that when the flow remains in the two-dimensional vortex shedding regime, the cylinder wake sequentially experiences transitions from the vortex shedding regime to the suppression of the vortex shedding regime and finally to the symmetry breaking regime, with increasing KC. Corresponding wake characteristics and variations of hydrodynamic forces over the three wake regimes are explored. Three quantities that represent shear-layer characteristics, including the length of separating shear layers, the circulation of shear layers and wake recirculation length, reach maxima at the onset of suppression. The physical mechanisms for the suppression of vortex shedding and occurrence of symmetry breaking are also explained. Once the flow becomes 3-D, vortex shedding from the cylinder cannot be suppressed, primarily because the outer shear layers induced by the steady approaching flow are enhanced in 3-D flows. The cylinder oscillation over the frequency range investigated in the present study delays wake transition to 3-D. The cylinder oscillation alters the 3-D vortical structure and its spanwise wavelength significantly.

Vortex shedding analysis of flows past forced-oscillation cylinder with dynamic mode decomposition

Direct numerical simulations are performed for flow past circular cylinders by the lattice Boltzmann method coupled with immersed moving boundary method. By analyzing the flows past a single cylinder at a wide range of cross-flow or in-line oscillation amplitude (0.25≤A/D≤1.5) and frequency (0.5≤fe/f0≤1.5), the results find that the vortex shedding modes inside and outside “lock-in” interval are of significant difference. The vortex shedding mode in the “unlock-in” state is 2S, but C(2S) and P + S shedding modes can be found in the lock-in state. Dynamic mode decomposition is used to analyze characteristic flow features, which shows that mode 1 is the main factor reflecting the flow field structure and mode 2 represents the vortex shedding mode in this work. The vortex shedding modes of flows past a tandem and side-by-side cross-flow double oscillating cylinders are systemically investigated. For tandem double oscillation cylinders, the results of modal decomposition suggest that the shear layer of upstream oscillating cylinder is separated behind the downstream cylinder at a space rate of L/D≤2, but separated behind the upstream cylinder at L/D≥3. Mode 2 at L/D=4 differs from other vortex shedding modes due to the strong inhibition effect by the downstream cylinder on the vortex formation of upstream cylinder. For side-by-side double oscillation cylinders, the wake of two cylinders is a single vortex street at H/D=1, a bistable flow at H/D=2 or 3, a coupled vortex street at H/D=4, and close to a single cylinder at H/D&amp;gt;4. The results of modal decomposition are disordered at H/D=2 due to the interaction between two cylinders and effect of gap flow.

Benchmarking of VIV numerical analysis with prototype response for fatigue assessment of inverse catenary coldwater pipelines

High density polyethylene pipe has been used to draw coldwater for low temperature thermal desalination plants since the year 2006 at Lakshadweep Islands, India. The pipeline endured in-line oscillations due to shear current in one of the desalination plants, and this gave an opportunity to observe the response parameters of the prototype pipeline. The recorded oscillation parameters are utilized as a benchmark to verify the results of the vortex induced vibration (VIV) analysis in frequency domain and time domain, and the results show good concurrence. The in-line fatigue damage assessment is carried out for a range of current profiles showing the need for mitigation of pipeline VIV. The helical strakes on a partial length of the pipeline is modeled numerically showing significant suppression of VIV response as well as substantial increase in fatigue life of the pipeline. The benchmarking of the VIV response analysis with prototype measurements enhances the reliability of using numerical VIV fatigue analysis and mitigation measures.

Forced oscillation experiments for each vibration mode of closely spaced parallel pipes in steady flow

In the industrially widespread use of multiple bundled pipes, the effects of mutual interference, in addition to the vibrations generated in single columns, must be considered, and estimating their behavior is critical to the design. In this study, based on previous research that identified the vibration modes occurring in two bundled pipes placed in a steady flow, such as (1) translational motion in the crossflow direction, (2) approaching and moving away, and (3) torsional motion, we developed four types of oscillators that can emulate the oscillation patterns (mode (2) uses different mechanisms for in-line and crossflow direction oscillation.), conducted forced oscillation experiments, and measured fluid forces. The mechanism of generation of the behavior patterns, measured in the towing and vibration measurement experiments of the elastic body model, was examined from the viewpoint of fluid force. A database of added mass and linear damping coefficients with parameters of distance between the center of the column and flow direction, in addition to oscillation patterns, was compiled for the development of a future two-tube behavior estimation program.A comparison of the measured fluid force of two columns with that of a single column, obtained by forced oscillation experiments for three vibration modes, showed that two closely spaced columns may vibrate over a wider frequency band and have larger amplitudes than a single column because of the expanded region of negative linear damping coefficients. Based on the results of the fluid force measurements for each vibration mode, rotational motion is more likely to occur in the tandem configuration, whereas approaching and moving away motion is more likely to occur in the side-by-side configuration because of the interchanged positive and negative added mass coefficients and strength of the negative linear damping coefficient.

Numerical investigation of noise suppression and amplification in forced oscillations of single and tandem cylinders in high Reynolds number turbulent flows

The noise generated by a three-dimensional turbulent wake behind an oscillating single or tandem cylinder(s) is studied using detached eddy simulation and the Ffowcs Williams-Hawkings (FW-H) method. This hybrid methodology is validated using some experimental data of a turbulent flow past a single or tandem stationary cylinder(s) and some numerical data (obtained from a direct numerical simulation) of a laminar flow past tandem oscillating cylinders. For an oscillating cylinder at a fixed Reynolds number Re=120,000, the sound energy is suppressed compared to that of a stationary cylinder provided the value of oscillation frequency approaches 57% of the original vortex-shedding frequency of the stationary cylinder. This noise reduction arises from the structural oscillations suppressing the noise generated by the vortex shedding from the stationary cylinder—the additional source of noise stemming from the structural oscillations is too small to compensate for the mechanism responsible for the noise suppression. The noise generated by the cylinder motion includes contributions from the loading and thickness noise, whereas the noise generated by vortex shedding from the stationary cylinder only includes the loading noise. The flow past tandem stationary cylinders exhibits a larger number of spectral peaks in the sound pressure power spectrum and the peaks occur at a lower frequency compared to the flow past a single cylinder. The drag force fluctuations result in the propagation of the sound pressure in the in-line direction, whereas the lift force fluctuations result in the propagation of the sound pressure in the transverse direction. Five different scenarios of stationary and/or oscillating tandem cylinders are studied. The second and third invariants of the velocity gradient tensor are used to visualize the distinct vortex structures in the wake for the five different scenarios of tandem cylinders. A single cylinder oscillating in the transverse direction exhibits a larger sound pressure level (SPL) than the streamwise oscillation of the same cylinder. In contrast, the streamwise (in-line) oscillation of one or the other cylinder in a tandem array results in a larger SPL than the transverse oscillation of one or the other cylinder. Finally, the SPL is more sensitive to the oscillations of the rear (downstream) cylinder than the front (upstream) cylinder.

The space effect on WIV interference between a fixed and oscillating diamond cylinder at a low Reynolds number of 100

To clarify the space effect on wake-induced vibration (WIV) of twin tandem diamond cylinders, the fixed upstream cylinder and the downstream cylinder freely oscillating in both in-line and transverse directions have been numerically investigated. The nondimensional center-to-center spacing ratio L* is varied in the range of 2–6, the Reynolds number Re is 100, the reduced velocity Ur is 3–20, and the mass ratio m* is 3. According to the wake modes and dynamic responses, the mechanism for the wake excitation of twin diamond cylinders with different spacing is further discussed. At a smaller spacing (L* = 2 and 3), the reattachment and co-shedding regimes appear with the increasing Ur, but when it comes to a larger spacing (L* = 4–6), only the co-shedding one occurs. The downstream cylinder shares a similar lock-in region at each spacing, while the corresponding flow patterns are not the same. The vortex shedding mode of the downstream cylinder within the lock-in region is 2S at L* = 2, while it is 2P at L* = 3–6. Additionally, the downstream diamond cylinder displays significant in-line and transverse oscillations at each spacing ratio, with the largest amplitude appearing at L* = 3.

Simultaneous streamwise and cross-stream oscillations of a diamond oscillator at low Reynolds numbers

A two-dimensional stabilized space-time finite-element-based in-house solver is used to explore flow around an elastically mounted rigid diamond oscillator undergoing undamped vortex-induced vibrations in transverse and stream-wise directions simultaneously. The computations are carried over a reduced velocity (U*) range of 1–12 keeping the Reynolds number (Re) and the mass ratio (m*) fixed at 100 and 10, respectively. In the current investigation, a five-branch response behavior with the presence of an “upper branch” is identified, which is absent for transverse-only oscillations. A shift in normalized time-averaged phase difference (ϕ/π) from 0 to 1 is related to the transition from the upper to the lower branch. The non-dimensional oscillation frequencies in both streamwise and cross-stream directions (Fx and Fy, respectively) collapse on the upper branch, matching the oscillator's non-dimensional natural frequency (FN). This leads to periodic single-looped “raindrop-shaped” cylinder trajectories. On this branch, the vortex-shedding mode is asymmetric, and the mean lift is non-zero (Clavg≠0). For all other response branches, Clavg≈0 and the vortex-shedding modes are symmetric. The presence of multiple frequencies in the in-line oscillations makes the initial branch quasi-periodic, whereas all other response branches are periodic. The addition of in-line motion modifies the fluid loading on the oscillator significantly.

Investigation of the influence of cylinder span length on vortex shedding from an in-line oscillating circular cylinder

In order to investigate the dependability of the circular cylinder span length of a lock-in phenomenon, closed circuit water channel experimental apparatus was used, and the visualization experiment was performed. In the visualization experiment, the aspect of the vortex shedding from the circular cylinder which is carrying out the in-line forced oscillation was observed by varying the water level of a test section. In the experiment, adjustment was carried out so that flow velocity might constitute about 0.06 m/s every water level. Main experiment parameters are a circular cylinder diameter (6 mm, 10 mm and 16 mm), amplitude ratio (2a/d=0.5 and 1.0),oscillation frequency ratio (range of f/fK is from 0 to 6), and circular cylinder span length (200 mm, 150 mm and 100 mm). As a result, although changes of the configuration of the lock-in corresponding to the variation of circular cylinder oscillating frequency were the same, it was found that the appearance ranges differ. Distribution of the lock-in range was classified into three groups by the aspect ratio. In the closed circuit water channel experiment, sunken span length was required 150 mm or more, and the aspect ratio L/d obtained the indicator in which the circular cylinder diameter which becomes 15 or more was selected.

An enhanced hybrid piezoelectric–electromagnetic energy harvester using dual-mass system for vortex-induced vibrations

This article proposes a novel hybrid piezoelectric–electromagnetic vortex-induced vibration energy harvester from flow of water inside of a pipe. The piezoelectric energy harvester was modeled with a macro-fiber composite P2-type while the electromechanical transduction was modeled by an elastic magnet coupled to the bluff body movement. A dual-mass configuration was proposed to increase the energy harvesting efficiency. Theoretical models and the submerged natural frequencies of the hybrid energy harvesters were outlined. Computational fluid dynamics and finite element analysis with ANSYS were used to visualize the response in synchronization and output the voltage extracted from the harvesting mechanisms. The addition of a secondary system improves the amount of harvestable energy and outputs more energy than just a single system. This demonstrates the superiority of a dual-mass hybrid system. A tuned secondary beam was used for L-body configurations to make use of inline oscillations, and the secondary piezoelectric output improved for all configurations. Secondary beam tuning also improved the performance of the harvester by any amount between 21% and 52% when compared against a single-mass hybrid energy harvester. A comparative study showed that the L-vertical and vertical bluff-body-tuned was the best performing hybrid-PE energy harvester based on total voltage output.

Experimental investigation on vortex-induced vibration and solid-structure impact of a near-bottom horizontal flexible pipeline in oblique shear flow

Vortex-induced vibrations and solid-structure impacts of a horizontal flexible pipeline in the vicinity of a bottom wall boundary are experimentally investigated in a water flume. A neutrally-buoyant submerged pipe, with a length-to-diameter ratio of about 87 and fixed end supports, is subject to a shear flow with an oblique angle of 30°. The initial gap-to-diameter ratio between the wall and the pipe is equal to 0.2. A non-intrusive measurement with high-speed cameras is employed to simultaneously capture the space–time varying in-line and cross-flow vibrations in the normal flow reduced velocity range of 3–15 with a maximum Reynolds number of about 2780. Experimental results highlight new pipe equilibrium profiles and modal response branches being significantly broadened due to the wall proximity. Two solid-pipe impact features are discussed, including single-segment and two-segment alternating impacts depending on the dominant oscillation mode of the flexible pipe. Both the response amplitude and frequency increase with the reduced velocity, resulting in a more frequent impact with a greater impact force. A temporal evolution from a single point contact to a segment contact is observed for the two-segment alternating impact. Nevertheless, the lengths of the two contact segments are different as a result of the asymmetric response profile. In all cases, the pipe transversely impacts the bottom wall introducing a perturbation transmission along the pipe, which leads to the multi-mode, multi-frequency and asynchronous in-line and cross-flow oscillations with a 1:1 and 2:1 resonance. Due to a momentum energy consumption, the pipe experiences a slow post-impact bouncing phenomenon, contributing to a longer temporal period of the fluid–pipe interaction with a wall impact.

An Improved Hybrid Cartesian/Immersed Boundary Method for Flow Simulation with Moving Boundaries

To simulate flow problems with moving fluid-solid interfaces, a boundary treatment technique based on a hybrid Cartesian/immersed boundary method is presented. In the vicinity of the immersed boundary, a forcing term is added into the governing flow equations to modify the discretization scheme for the viscous term during its spatial discretization. Addition of such a term enforces desired boundary condition on the immersed boundary. A one-dimensional linear interpolation scheme is used to obtained required information for the modified discretization scheme when dealing with a moving boundary. The present method is essentially second-order accurate and can obtain results that are in good agreement with benchmark references when simulating the flow induced by the in-line oscillation of a circular cylinder.

On the Huse-Muren model for wake interaction with vertical offshore cylinders

There are many offshore applications that incorporate vertical cylindrical-shaped structural elements such as marine risers, TLP tendons and near shore pier designs. A better understanding of complex wake flows on the drag dominated force predictions is very much of interest for offshore design problems. This study investigates the Huse-Muren analytical wake model, whose mathematical formulation was founded on the earlier definitive studies of Prandtl and Schlichting. Developing a deeper understanding of their key findings that were based upon central assumptions and key approximations is central to this research. Analysis of their model leads to the development of dimensionless expressions for the drag coefficient correction ratio. Both the DNV-RP recommended drag force coefficients and some of Sarpkaya's U-tube measurements are used to investigate whether their analytical model could be used to bound the experimental data. The Keulegan-Carpenter number, A0/D ratios, Sarpkaya's Beta parameter and Reynold's number are used to interpret the range of model applicability. For the current cross flow case, variables identified in Huse and Muren's formulation lead to a dimensionless expression involving the period of in-line oscillation and the velocity of the cross-current resulting in a parameter similar to the Keulegan-Carpenter number. The ratio of this parameter to the Keulegan-Carpenter number allows one to address the drag coefficient correction ratio where a cylinder is simultaneously subjected to in-line oscillatory flow and a steady cross current coming from the transverse direction. A numerical simulation is performed to investigate the behavior of the correction ratio in this flow condition, and the results are presented and discussed.

Numerical modelling of the oscillatory flow effect around submarine pipeliness

This work describes the dynamics of an underwater pipeline subjected to an incident flow composed of a uniform flow and a sinusoidal component. The motion equations of the pipeline and the fluid flow around the pipeline are solved simultaneously with a numerical model that considers in-line oscillations (one degree of freedom) and a second numerical model that additionally includes cross flow (two degrees of freedom). The amplitude response and drag forces on the pipe are compared for both models considering parameters such as the difference between the excitation and natural frequencies of pipelines and the relative value between the sinusoidal and uniform components of the incident flow. Important differences in numerical predictions of both models are observed when the excitation frequency is greater than the natural frequency of the system and when the amplitude of the oscillatory component of the incident flow is greater than the amplitude of the uniform flow.

Application of wake oscillators to two-dimensional vortex-induced vibrations of circular cylinders in oscillatory flows

A nonlinear time-domain simulation model for predicting two-dimensional vortex-induced vibration (VIV) of a flexibly mounted circular cylinder in planar and oscillatory flow is presented. This model is based on the utilization of van der Pol wake oscillators, being unconventional since wake oscillators have typically been applied to steady flow VIV predictions. The time-varying relative flow–cylinder velocities and accelerations are accounted for in deriving the coupled hydrodynamic lift, drag and inertia forces leading to the cylinder cross-flow and in-line oscillations. The system fluid–structure interaction equations explicitly contain the time-dependent and hybrid trigonometric terms. Depending on the Keulegan–Carpenter number (KC) incorporating the flow maximum velocity and excitation frequency, the model calibration is performed, entailing a set of empirical coefficients and expressions as a function of KC and mass ratio. Parametric investigations in cases of varying KC, reduced flow velocity, cylinder-to-flow frequency ratio and mass ratio are carried out, capturing some qualitative features of oscillatory flow VIV and exploring the effects of system parameters on response prediction characteristics. The model dependence of hydrodynamic coefficients on the Reynolds number is studied. Discrepancies and limitations versus advantages of the present model with different feasible solution scenarios are illuminated to inform the implementation of wake oscillators as a computationally efficient prediction model for VIV in oscillatory flows.

A robust sharp interface based immersed boundary framework for moving body problems with applications to laminar incompressible flows

This study presents a robust, sharp interface immersed boundary (IBM) framework for moving body problems. The in-house solver makes use of a density-based finite volume framework for solving unsteady, 3D Favre averaged Navier Stokes equation in a generalized curvilinear coordinate system. The immersed boundary formulation is capable of handling arbitrarily complex three-dimensional bodies. The sharp interface approach allows for the exact imposition of boundary conditions at the immersed surface by reconstructing the flow field along its local normal. The implemented reconstruction schemes maintain second-order accuracy. The study focuses on issues of mass conservation and spurious temporal oscillations (pressure as well as force) that the sharp interface IBM approach typically faces when encountering moving body problems. A ghost node-based field extension technique provides an efficient way to improve mass conservation and as a result, reduces not only spurious oscillations but also increases temporal accuracy. Flow past both bluff body (triangular, circular and spherical), as well as streamlined body (airfoil), is presented here as validation studies. The ability of the present formulation to deal with moving body problems in the laminar incompressible flow regime is demonstrated by presenting cases that involve motions such as pitching and in-line oscillation. The predictions are found to be in good agreement with the published results and measurements as well.

Mapping the properties of the vortex-induced vibrations of flexible cylinders in uniform oncoming flow

Flexible structures placed within an oncoming flow exhibit far more complex vortex-induced dynamics than flexibly mounted rigid cylinders, because they involve the distributed interaction between the structural and wake dynamics along the entire span. Hence, mapping the well-understood properties of rigid cylinder vibrations to those of strings and beams has been elusive. We show here with a combination of experiments, conducted at Reynolds number, $Re$ from 250 to 2300, and computational fluid dynamics that such a mapping is possible for flexible structures in uniform flow undergoing combined cross-flow and in-line oscillations, but only when additional concepts are introduced to model the extended coupling of the flow and the structure. The in-line response consists of largely standing waves that define cells, each cell spanning the distance between adjacent nodes, over which stable vortical patterns form, whose features (‘2S’ versus ‘P$+$S’) depend strongly on the true reduced velocity, $V_{r}=U/f_{y}d$, where $U$ is the inflow velocity, $f_{y}$ is the cross-flow vibration frequency and $d$ is the cylinder diameter, and the phase angle between in-line and cross-flow response; while the cross-flow response may contain travelling waves, breaking the symmetry of the problem. The axial distribution of the highly variable effective added masses in the cross-flow and in-line directions, and the local phase angle between in-line and cross-flow motion determine the single frequency of cross-flow response, while the in-line response vibrates at twice the cross-flow frequency. The cross-flow and in-line lift coefficients in phase with velocity depend strongly on the true reduced velocity but also on the local phase angle between in-line and cross-flow motions. Modal shapes can be defined for in-line and cross-flow, based on the resemblance of the response to conventional modes, which can be in the ratio of either ‘$2n/n$’ or ‘$(2n-1)/n$’, where $n$ is the order of the cross-flow response mode. We use an underwater optical tracking system to reconstruct the sectional fluid forces in a flexible structure and show that, once the cross-flow and in-line motion features are known, employing strip theory and the hydrodynamic coefficients obtained from forced rigid cylinder experiments allows us to predict the distributed forces accurately.

Improving the Accuracy of Fictitious Domain Method Using Indicator Function from Volume Intersection

Fictitious domain method (FDM) is a commonly accepted direct numerical simulation technique for moving boundary problems. Indicator function used to distinguish the solid zone and the fluid zone is an essential part concerning the whole prediction accuracy of FDM. In this work, a new indicator function through volume intersection is developed for FDM. In this method, the arbitrarily polyhedral cells across the interface between fluid and solid are located and subdivided into tetrahedrons. The fraction of the solid volume in each cell is accurately computed to achieve high precision of integration calculation in the particle domain, improving the accuracy of the whole method. The quadrature over the solid domain shows that the newly developed indicator function can provide results with high accuracy for variable integration in both stationary and moving boundary problems. Several numerical tests, including flow around a circular cylinder, a single sphere in a creeping shear flow, settlement of a circular particle in a closed container, and in-line oscillation of a circular cylinder, have been performed. The results show good accuracy and feasibility in dealing with the stationary boundary problem as well as the moving boundary problem. This method is accurate and conservative, which can be a feasible tool for studying problems with moving boundaries.

DFIB-SPH study of submerged horizontal cylinder oscillated close to the free surface of a viscous liquid

The hydrodynamics is studied numerically of a horizontal cylinder undergoing forced in-line oscillation beneath the free surface of otherwise quiescent liquid at low Keulegan–Carpenter and Froude numbers. The direct forcing immersed boundary-smoothed particle hydrodynamics (DFIB-SPH) numerical model uniquely combines two well-established techniques: the direct forcing immersed boundary method and SPH. This facilitates accurate evaluation of the potentially violent free surface motions through SPH and the hydrodynamic force on the solid body using a volume integral. A parameter study is conducted covering a range of Keulegan–Carpenter numbers (KC = 3, 7, and 10) and submergence ratios () at fixed Reynolds number (Re = 100) and Froude number (Fr = 0.35). The flow pattern and transverse force coefficient are found to be affected by the proximity of the cylinder to the free surface. Spectral analysis suggests that free surface wave motions are linked to the transverse force acting on the submerged, oscillating cylinder.