
Jian Deng- PhD
- Professor at Zhejiang University
Jian Deng
- PhD
- Professor at Zhejiang University
Looking for potential PhD students, and Postdoc researchers.
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117
Publications
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Introduction
DENG Jian received his B.S. degree in Engineering Mechanics and Ph.D. in Fluid Mechanics in 2002 and 2007 respectively, both from Zhejiang University, P.R. China.
Current institution
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Education
September 2002 - March 2007
September 1998 - July 2002
Publications
Publications (117)
We investigate the buckling dynamics of an elastic filament impacted axially by a falling liquid droplet, and identify the buckling modes through a combination of experimental and theoretical analyses. A phase diagram is constructed on a plane defined by two primary parameters: the falling height and the filament length. Two transition boundaries a...
Symmetry breaking (SB) of fluid-structure interaction problems play an important role in our understanding of animals' locomotive and sensing behaviors. In this letter, we study the SB of flexible filaments clamped at one end and placed in a spanwise periodic array in Stokes flow. The equilibrium state of the filament along the streamwise direction...
This study explores the implementation of an online control strategy based on dynamic mode decomposition in the context of flow control. The investigation is conducted mainly with a fixed Reynolds number of Re = 100, focusing on the flow past a circular cylinder constrained between two walls to mitigate vortex shedding. The control approach involve...
The propulsive efficiency of flying and swimming animals propelled by oscillatory appendages typically peaks within a narrow Strouhal number range of 0.20 < St < 0.40. Motivated by the ubiquitous presence of stratification in natural environments, we numerically investigate the optimal Strouhal numbers St m for an oscillating foil in density strati...
We propose a hybrid numerical model for the collective motion of fish groups, which integrates an agent-based model with a CFD-based model. In the agent-based model, the fish group is represented by self-propelled particles (SPPs), incorporating social forces with local interactive rules. The CFD-based model treats the fish body with an undulated f...
A hybrid numerical model for the collective motion of fish groups
Simultaneously detecting hidden solid boundaries and reconstructing flow fields from sparse observations poses a significant inverse challenge in fluid mechanics. This study presents a physics-informed neural network (PINN) framework designed to infer the presence, shape, and motion of static or moving solid boundaries within a flow field. By integ...
Freestream nuclei, also referred to as water quality, are known to significantly affect cavitation inception. However, their effects on fully developed cavitation and the corresponding noise characteristics remain inadequately understood. In this study, a multiscale hydroacoustic model based on the Euler–Lagrangian framework is used to investigate...
This study develops a hybrid multiscale Euler–Lagrange model to investigate the unsteady characteristics of cloud cavitation around a hydrofoil under different water qualities. A homogeneous mixture model is implemented for macroscopic cavity simulations, tracking the vapor–liquid interfaces. In the Lagrangian framework, the dynamics and motion of...
A non-cavitating tip vortex generated by an elliptical hydrofoil is investigated utilizing tomographic particle image velocimetry (TPIV). Focus is placed on its three-dimensional evolution over a relatively large streamwise region, as well as the transport process of turbulent kinetic energy (TKE). Based on the variations in vortex structure and re...
Icing is ubiquitous in nature and engineering applications, and imposes threats to road and air transportations, wind energy infrastructures, etc. However, current active de‐icing solutions, especially the most popular one, i.e., heating, suffer from high energy consumption whilst passive methods are often ineffective at high‐speed, long‐term, or l...
The present work focuses on a specific bouncing behaviour as a spherical particle settling through a density interface in the absence of a neutral buoyant position. This behaviour was initially discovered by Abaid & McLaughlin (2004) in salinity-induced stratification. Both experimental and numerical investigations are conducted to understand this...
The statistical characteristics of bubble plumes generated by inclined liquid jets remain inadequately understood, particularly regarding their spatial evolution. This study employs an optical fiber probe for sequential multi-point measurements to investigate the evolution of bubble plumes produced by shallow-angle inclined jets impacting both quie...
Understanding whether a droplet adheres to or detaches from a flexible filament upon axial impact is of significant interest, particularly in the context of raindrop impact on natural surfaces. This process involves dynamic buckling followed by mode coarsening, dissipating the initial droplet kinetic energy and converting the remaining into elastic...
The underwater launch of an axisymmetric body involves complex cavity-structure interactions. Studying the evolution of cavity pressure around an axisymmetric body is crucial for researching its motion stability. In this work, we propose a deep neural network model for cavity pressure distribution recovery, called CPDR-net. This model can reconstru...
Time-dependent flow fields are typically generated by a computational fluid dynamics method, which is an extremely time-consuming process. However, the latent relationship between the flow fields is governed by the Navier–Stokes equations and can be described by an operator. We therefore train a deep operator network (DeepONet) to learn the tempora...
Reynolds-averaged Navier-Stokes (RANS) simulations have found widespread use in engineering applications, yet their accuracy is compromised, especially in complex flows, due to imprecise closure term estimations. Machine learning advancements have opened new avenues for turbulence modeling by extracting features from high-fidelity data to correct R...
We investigate numerically the propulsion characteristics of an oscillating foil undergoing coupled heave and pitch motion in a linearly density-stratified flow. A parameter space defined by the internal Froude number ( $1 \le Fr \le 10$ ) and the maximum angle of attack ( $5^\circ \le {\alpha _0} \le 30^\circ$ ) is considered in our study. The res...
Physics-informed neural networks (PINNs) employed in fluid mechanics deal primarily with stationary boundaries. This hinders the capability to address a wide range of flow problems involving moving bodies. To this end, we propose a novel extension, which enables PINNs to solve incompressible flows with time-dependent moving boundaries. More specifi...
We study numerically a spherical particle settling through a density transition layer at moderate Reynolds numbers Reu=69∼259 for the upper fluid. We investigate how the transition layer thickness affects the particle's bouncing behavior as it crosses the interface. The previous intuitive understanding was that the bounce occurs when the relative t...
The coordinated movement of multiple swimmers is a crucial component of fish schools. Fish swimming in different formations, such as tandem, side-by-side, diamond, and phalanx, can achieve significant energetic advantages. However, the energetic benefits of nonstraight swimming behaviors, such as the collective motion of a milling pattern, are not...
The interactions between a vortex ring and a gas bubble released at the axis are studied numerically, which shed light on understanding more complicated bubble–turbulence coupling. We fix the Reynolds number at $Re_{\tau }=7500$ and consider various Weber numbers in the range of $We=130\unicode{x2013}870$ . We find that the translating speed of the...
Collective motion of animal groups such as fish schools and bird flocks in three-dimensional (3D) space are modeled by considering a topological (Voronoi) neighborhood. The tridimensionality of the group is quantified. Apart from the patterns of swarming, schooling, and milling, we identify a 3D bait ball around the phase transition boundary. More...
We introduce a short-range repulsive force model to tackle near-contact interactions when the collision occurs between bubbles. In contrast to the previous numerical method adopting the adaptive mesh refinement technique, such a mesoscale model can be applied to a relatively coarse mesh, which can prevent the nonphysical coalescence between bubbles...
It is of interest to investigate how a swimming animal performs in a density-stratified fluid. This paper studies a simplified swimmer, a pitching NACA0015 airfoil, considering its locomotion in both homogeneous, or unstratified, and stratified fluid flows. A direct comparison is made between these two conditions through two-dimensional numerical s...
We study experimentally the reduction of mean force and suppression of force fluctuation on a circular cylinder by attaching flexible membranes to the surface of the
cylinder. The experiments were conducted in a wind tunnel. The configuration of a
single membrane with a length of two cylinder diameters, i.e. L/D = 2, is found to
be optimal for mean...
Using two-dimensional numerical simulations, we investigate a circular cylinder in a stratified flow with a pycnocline of a smooth density profile. We are particularly concerned about the difference between stratified flows and their homogeneous, or unstratified, counterparts in drag coefficients. It is well known that the characteristics of a stra...
We investigate numerically the dynamics of freely rising gas bubbles driven by buoyancy at low Reynolds numbers, focusing on two distinct characteristics: the clustering morphology and the probability density functions (p.d.f.) of the velocity fluctuations for bubbles. A modified Volume-of-fluid (VOF) method is implemented in our direct numerical s...
We study numerically the effects of free surface on a flapping-foil based ocean current energy extractor. A two-dimensional NACA0015 foil, subjected to a coupled pitching and heaving motions, at a fixed Reynolds number of 900, and varied Froude numbers between 0 and 2, is considered. We fix the non-dimensional frequency at f∗ = 0.16 and pitching am...
Fish have evolved diverse and robust locomotive strategies to swim efficiently in complex fluid environments. However, we know little, if anything, about how these strategies can be achieved. Although most studies suggest that fish rely on the lateral line system to sense local flow and optimise body undulation, recent work has shown that fish are...
For a gas bubble rising in quiescent liquid, most previous drag force models correlate the drag coefficient Cd to dimensionless parameters such as the Reynolds number Re, the Eötvös number Eo, and the Morton number Mo. However, it is still an open question if the current models can be applied to non-buoyancy-driven bubbles in a wide range of parame...
In this paper, the effect of strip spacing on the aerodynamic performance of a high-speed pantograph with a double-strip is numerically studied using a shear stress transport turbulence model. Seven different strip spacings varied from 100 to 700 mm are considered. The errors in pantograph resistance between numerical simulation and wind tunnel tes...
While the collective movements of fish schools evading predators in nature are complex, they can be fundamentally represented by simplified mathematical models. Here we develop a numerical model, which considers self-propelled particles subject to phenomenological behavioural rules and the hydrodynamic interactions between individuals. We introduce...
We investigate the flow around a circular cylinder vibrating with a prescribed magnitude and frequency at Reynolds number 230. The Navier-stokes equations and the adjoint equations are solved to search for the three-dimensional (3D) nonlinear optimal initial perturbation, which is amplified by the attached shear via the Orr mechanism and then activ...
After spraying, the cooling medium retains potential cooling capability and, consequently, can form a falling film to further cool the downward area. In this paper, the flow patterns and the heat transfer coefficients are correlated respectively in the spray zone and the falling film zone on a vertical semi-cylindrical heating surface. The incident...
In this paper, an artificial neural network (ANN) trained through a deep reinforcement learning (DRL) agent is used to perform flow control. The target is to look for the wake stabilization mechanism in an active way. The flow past a 2-D cylinder with a Reynolds number 240 is addressed with and without a control strategy. The control strategy is ba...
Interface-resolved numerical simulations of particle-laden turbulent flows in a vertical channel filled with Bingham fluids - Volume 883 - Chenlin Zhu, Zhaosheng Yu, Xueming Shao, Jian Deng
A direction adaptive approach for the reduction of drag and the suppression of lift fluctuation in flow passing a circular cylinder is developed. Flexible filaments are attached to the surface of the cylinder, and different configurations, including the number, lengths and angles of attachment of the filaments, as well as their tension and bending...
Flying fish have been observed jumping out of warm ocean waters worldwide. Before take-off, the flying fish are seen to taxi on the water surface by rapidly beating their semi-submerged tail fins, which process may help them airborne with enough speed to glide over a long distance. To understand the underlying physical mechanisms, here, we study a...
In this paper, vortex induced vibration of a circular cylinder with an attached filament is numerically studied by a Penalty Immersed boundary method (PIBM). In the current work, we study the effects of a single filament on the flow wake of a fixed cylinder, as well as its dynamic response when it is free to move transversely. Two key parameters, t...
Flying fish is a family of unique aerial–aquatic animals, which can both swim in the water and glide over the sea surface. Most previous studies on their aerodynamic characteristics were based on field observations or measurements of their morphometric parameters. In the present study, we consider three different flying fish models, of which the pr...
To control the shedding of cavitation, an obstacle is placed on the surface of a flat hydrofoil. Both experimental and numerical studies are carried out. Images of cavitation evolution are recorded by a high-speed camera. 3-D simulations are performed to investigate the cavitating flows around the hydrofoil. The results show that the re-entrant jet...
This paper studies numerically the collapse of a cluster of cavitation bubbles (as a primitive model for a bubble cloud) near a solid wall. The homogeneous two-phase mixture model is used, with the liquid-vapor interface resolved by volume of fluid method. The liquid is treated as compressible, allowing the propagation of pressure waves at the spee...
We perform two-dimensional simulations of unsteady flow separation around a NACA0015 airfoil. We consider five different angles of attack, 10°, 12.5°, 15°, 17.5°, and 20°, with the Reynolds number varying from 100 to 1300. The central aim is to study the wake dynamics under the low-Reynolds-number condition, when the flow is most likely experiencin...
This paper investigates numerically the collapses of both a single cavitation bubble and a cluster consisting of 8 bubbles, concerning mainly on the conversions between different forms of energy. Direct numerical simulation (DNS) with volume of fluid (VOF) method is applied, considering the detailed resolution of the vapor-liquid interfaces. First,...
We consider the self-induced motions of three-dimensional oblate spheroids of density $\unicode[STIX]{x1D70C}_{s}$ with varying aspect ratios $AR=b/c\leqslant 1$ , where $b$ and $c$ are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that $y_{s}(t)=A\sin (2\unicode[STIX]{x0...
This paper investigates the three-dimensional instabilities of the flow past a periodically heaving airfoil. By comparison with a pitching foil [Deng et al., Phys. Rev. E 92, 063013 (2015)], here we present distinctive characteristics for the heaving foil, particulary regarding its Floquet modes. By increasing the frequency (Sr), or equivalently de...
In order to study the unsteady aerodynamics effects in railway tunnels, the 3D Reynolds average Navier–Stokes equations of a viscous compressible fluid are solved, and the two-equation k-ε model is used in the simulation of turbulence, while the dynamic grid technique is employed for moving bodies. We focus on obtaining the changing tendencies of t...
Two tandem filaments in a uniform flow are simulated by using a penalty immersed boundary method. First, we investigate two identical filaments, duplicating the 'inverted drafting'. We then fix the structural parameters of the leader while vary that of the follower in kb2 and ρs2. It is found as the follower is stiffer or lighter than the leader, t...
We perform a Floquet stability analysis in the wake of a NACA0015 airfoil, with four angles of attack, α=20°, 17.5°, 15°, and 12.5°, considered. The central aim is to predict the secondary instabilities at the fixed moderate angles of attack, which are sufficiently large for massive separation from the airfoil, while at the same time allowing the f...
We investigate experimentally the nonlinear structures that develop from interacting vortex rings induced by a sinusoidally oscillating ellipsoidal disk in fluid at rest. We vary the scaled amplitude or Keulegan-Carpenter number 0.3<NKC=2πA/c<1.5, where A is the oscillation amplitude and c is the major diameter of the disk, and the scaled frequency...
We propose a natural model to probe in a controlled fashion the instability of interacting vortex rings shed from the edge of an oblate spheroid disk of major diameter c, undergoing oscillations of frequency f0 and amplitude A. We perform a Floquet stability analysis to determine the characteristics of the instability modes, which depend strongly o...
We study numerically the propulsive wakes produced by a flapping foil. Both pure pitching and pure heaving motions are considered, respectively, at a fixed Reynolds number of Re = 1700. As the major innovation of this paper, we find an interesting coincidence that the efficiency maximum agrees well with the 2D-3D transition boundary, by plotting th...
We used a computational fluid dynamics (CFD) model to study the inspiratory airflow profiles of patients with anterior nasal cavity stenosis who underwent curative surgery, by comparing pre- and postoperative airflow characteristics. Twenty patients with severe anterior nasal cavity stenosis, including one case of bilateral stenosis, underwent comp...
The modified immersed boundary method is introduced and applied to study the propulsive mechanism of a tandem flapping wings system. The effects of tandem wings distance and phase lag between the two flapping wings are investigated. Thrust force of the upstream wing is nearly constant and close to the magnitude of single flapping wing system. Thrus...
Using two-dimensional direct numerical simulations, we investigate the flow in a fluid of kinematic viscosity $\nu$ and density $\rho$ around elliptical foils of density $\rho_s$ with major axis $c$ and minor axis $b$ for three different aspect ratios: $AR=b/c=1$ (a circle); $AR=0.5$; and $AR=0.1$. The vertical location of these foils $y_s(t)=A \si...
In this paper, we investigate the three-dimensional transition of a propulsive wake, which is generated by a pitching foil, mimicking the caudal fin of a fish. We note that a fish fin is usually of very short aspect ratio, here we assume an infinite span, aiming to reveal the fundamental physical mechanisms. For the base flow, three sequential wake...
As an extension of the previous study on three-dimensional transition of the wake behind a pitching foil [J. Deng \& C.P. Caulfield, Phys. Rev. E 91, 043017 (2015)], this investigation draws a comprehensive map on the pitching frequency-amplitude phase space. First, by fixing the Reynolds number at $Re=1700$ and varying the pitching frequency and a...
Spool valves play an important role in fluid power system. Cavitation phenomena happen frequently inside the spool valves, which cause structure damages, noise and lower down hydrodynamic performance. A numerical tools incorporating the cavitation model, are developed to predict the flow structure and cavitation pattern in the spool valve. Two majo...
The inertia plays a significant role in the response of a system undergoing flow-induced vibrations, which has been extensively investigated by previous researchers. However, the inertial effects of an energy harvester employing the mechanism of flow-induced vibrations have attracted little attention. This paper concentrates on a semi-passive energ...
We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: Bénard-von Kármán (BvK) vortex streets; reverse BvK vortex streets; and...
Numerical simulations are used to investigate the effect of variation of the aspect ratio and the structure of pitching motions on the energy extraction efficiency and wake topology of flapping foils. The central aim is to predict the energy extraction performance and efficiency of a flapping-foil-based energy harvesting system (EHS) in realistic w...
A novel ocean flow energy harvesting device was examined by using the computational fluid dynamic method. The device consists of an oscillating foil mounted on a damper (representing a power generator) and a rotational spring. The basic principle of the fully passive energy harvester relies upon fluid-induced vibrations. The energy harvesting conce...
For the analysis of high-speed train pantograph-catenary current collection quality considering the impact of pantograph average steady-state and unsteady aerodynamic forces, based on the pantograph model with unsteady aerodynamics and the assumption of uniform flow, the unsteady drag and lift forces of pantograph components were obtained by comput...
In this paper, numerical simulations of a flexible circular cylinder subjected to a vortex-induced vibration (VIV) are conducted. The Reynolds number for simulations is fixed at 1000. The finite volume method is applied for modeling fluid flow with the moving meshes feature. The dynamic response of a flexible cylinder fixed at both ends is modeled...
In order to study the unsteady aerodynamics effects in railway tunnels, the 3D Reynolds average Navier-Stokes equations of a viscous compressible fluid are solved, and the two-equation k-ɛ model is used in the simulation of turbulence, while the dynamic grid technique is employed for moving bodies. We focus on obtaining the changing tendencies of t...
The vortex-induced vibration of a flexible circular cylinder is investigated at a constant Reynolds number of 1 000. The finite-volume method on moving meshes is applied for the fluid flow, and the Euler-Bernoulli beam theory is used to model the dynamic response of a flexible cylinder. The relationship between the reduced velocity and the amplitud...
Setup computational fluid dynamics (CFD) model of the nasal cavity in patients with Crouzon syndrome analyze inspiratory airflow hydrokinetics of its nasal cavity. After changing the morphosis structure of the nasal cavity by operation, compare the preoperative and postoperative alteration of the airflow field of the nasal cavity and evaluate the e...
The hydrodynamic performances of a fishlike undulating foil in the wake of a D-section cylinder are numerically investigated by using a modified immersed boundary method. The results regarding the effects of various controlling parameters, including the distance between the foil and the D-cylinder, the frequency and the phase angle of foil’s undula...
The IBM/ICM method is developed to simulate the interactions between moving bodies and free surfaces. This method is a combination of Immersed Boundary Method (IBM) and Interface Capture Method (ICM), inheriting the IBM's capability of using non-body conformal grids to represent the effect of a moving body in the flow by only adding a pseudo body-f...
Numerical simulations of cavitating flows on 2D NACA0015 hydrofoils with and without obstacle are performed. Cavitation model is based on a transfer equation for a void mass fraction and an improved RNG k-ɛ model is adopted to study the turbulent cavitating flows around the foils. Different arrangements and geometric parameters of the obstacles are...
An immersed-boundary method is used to investigate the flapping wings with different aspect ratios ranging from 1 to 5. The numerical results on wake structures and the performance of the propulsion are given. Unlike the case of the two-dimensional flapping foil, the wing-tip vortices appear for the flow past a three-dimensional flapping wing, whic...
The objective of this article is a numerical analysis of the temperature distributions around the notches of a spool valve, because of the viscous heating effect caused by high-speed flow of the large-viscosity hydraulic oil. Two valve spools fabricated with different shapes of notches are investigated, of which spool A is an earlier one, being rep...
In this paper, the steady high speed oil flow in the spool valve and the local temperature distributions around the notches are simulated using a finite volume based CFD code. The temperature increase is due to the viscous heating effect, which is generally significant in highly-viscous fluid flow. Moreover, the deformation of the spool is simulate...
The flow field of the foil with both the heave and pitch motion was simulated by using immersed-boundary method. The affection of the maximum pitch angle, flapping frequency and the phase difference of pitch and heave motion were analyzed systemically. It is shown that the reversed Kaman vortex street was generated, and then the foil get the positi...
The turbulent vortex induced vibrations of a circular cylinder were simulated. The flow was predicted to be turbulent, and was solved by the ALE domain decomposition method, in which the Poisson pressure equation formation was adopted. The standard k-ε model and RNG k-ε model were employed for turbulence. In this study, the Reynolds number was from...
The temperature distributions around the notches in spool valve were simulated using a finite-volume based CFD code. The temperature
increase is due to the viscous heating effect, which is generally significant in high speed oil flow. Two valve spools with
different types of notches are investigated, of which spool A is an earlier one, and spool B...
The turbulent flow around a circular cylinder with vortex-induced vibration (VIV) was simulated by plane standard k-ε and RNG k-ε models aiming at the problems of turbulent vortex induced vibration in engineering. The domain decomposition method (DDM) combined with an arbitrary Lagrangian-Eulerian(ALE) method was applied. The equations were discret...
Parallelized solver based on unstructured grid system was developed for supersonic combustion problem. The solver which is aimed to supersonic combustion was validated with numerical experiments of several Scramjet models, one is the two-dimensional Scramjet model of German Aerospace Center (DLR), the other model come from the University of Queensl...
The flow behind a three-dimensional rotationally oscillating circular cylinder was studied by a numerical method. The computations were performed at a Reynolds number of 260, which is at a level that the flow wake has developed into a three-dimensional state called Mode-B. The purpose of this paper is to examine the influence of various rotational...
In this study, the hydrodynamic interactions between two tandem foils undergoing fishlike swimming motion are investigated numerically by solving the Navier-Stokes equations with the immersed-boundary method. The two foils represent two tandem propellers attached on a concept ship. The thrusts and efficiencies at three typical Strouhal numbers, i.e...
The flow past a stationary circular cylinder and a downstream elastic circular cylinder in cruciform arrangement is investigated at a constant Reynolds number of 150. The virtual boundary method is employed in this study. After the validation of the numerical method, two cases are simulated. In Case 1, both cylinders are stationary. A critical spac...
In this paper, a new immersed-boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite-difference approach on a staggered mesh together with a fractional-step...
Two-dimensional numerical simulations are performed to study the propulsive performance of fish-like swimming foils using the immersed-boundary method. A single fish as well as two fishes in tandem arrangement are studied. First, the effect of the phase speed on the propulsive performance of a single fish is analyzed. The wake structures and pressu...
In this paper, the computational fluid dynamics was applied to fish-like swimming, and the propulsion mechanism of this motion was focused. Although previous researchers have suggested that a diamond shape of fish school is helpful for drag reduction and efficiency enhancement, and individuals can benefit from such a school, experimental data or nu...
The spatial evolutions of vortices and transition to three dimensionality in the wake of two tandem circular cylinders are numerically studied. The virtual body method developed from virtual boundary method is applied to model the no-slip boundary condition of the cylinders. Two different aspects of this problem are considered. Firstly, the spacing...
The virtual body method is applied to simulate the flow passing around a single oscillating circular cylinder and two circular cylinders in cruciform arrangement with the downstream one forced oscillating. The synchronization and phase 'jump' phenomena are observed in a single oscillating circular and confirm the validity of this numerical method f...
The flow around two tandem circular cylinders was studied by a three-dimensional numerical simulation of the Navier-stokes equations at Re=220. The improved virtual boundary method was applied to model the no-slip boundary condition of the cylinders. The results show that as the spacing ratio L/D≥4, the three dimensionality occurs in the wake. When...