Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

Vortex shedding from a square cylinder in presence of a moving wall

A numerical study on the flow past a square cylinder placed parallel to a wall, which is moving at the speed of the far field has been made. Flow has been investigated in the laminar Reynolds number (based on the cylinder length) range. We have studied the flow field for different values of the cylinder to wall separation length. The governing unsteady Navier–Stokes equations are discretized through the finite volume method on a staggered grid system. A SIMPLE type of algorithm has been used to compute the discretized equations iteratively. A shear layer of negative vortex generates along the surface of the wall, which influences the vortex shedding behind the cylinder. The flow‐field is distinct from the flow in presence of a stationary wall. An alternate vortex shedding occurs for all values of gap height in the unsteady regime of the flow. The strong positive vortex pushes the negative vortex upwards in the wake. The gap flow in the undersurface of the cylinder is strong and the velocity profile overshoots. The cylinder experiences a downward force for certain values of the Reynolds number and gap height. The drag and lift are higher at lower values of the Reynolds number. Copyright © 2005 John Wiley & Sons, Ltd.     

串列双圆柱绕流问题的数值模拟

本文运用有限体积方法,对绕串列放置的双圆柱的二维不可压缩流动进行了数值计算。为研究两圆柱不同间距对圆柱相互作用和尾流特征的影响,选取间距比Ｌ／Ｄ（Ｌ为两圆柱中心间的距离,Ｄ为圆柱直径）在１．５～５．０之间每隔０．５共八个有代表性的间距进行了计算模拟。计算均在Ｒｅ＝２００条件下进行。计算结果表明：对该绕流问题,流动特征在很大程度上取决于间距的大小。且间距存在一临界值,间距比从小于临界值变化到大于临界     

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De ≈ 1.3, for an extensibility parameter of the model of L² = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L² = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De ≈ 4.0–4.5.     

棱柱绕流流动的数值模拟

利用数值方法对长宽比为1／3, 1和3的棱柱绕流在雷诺数为100的非稳态流动特性进行了分析和研究。采用有限体积法对棱柱绕流的二维流动N-S方程进行离散求解,分析和研究了非稳态的棱柱绕流流场,升力系数,阻力系数和涡动特性,数值模拟的结果与相关文献的数据比较吻合。通过上述研究能够为了解棱柱绕流的非稳态流动特性提供有力的帮助。而对棱柱三维流动的模拟分析和对雷诺数的变化对棱柱流动特性的影响进行研究,将为掌握棱柱绕流的工程特性打下基础。     

A quantitative study of the flow around an impulsively started circular cylinder

The detailed flow structure behind an impulsively started circular cylinder has been investigated experimentally. The Reynolds number based on the steady state velocity and the diameter of the cylinder was 500 to 3,000. This work is unique in that unsteady spatial velocities were measured simultaneously by a quantitative visualization technique — Laser Induced Photochemical Anemometry (LIPA). The surface vorticity at g/q = π/2 and vorticity distribution behind the cylinder in the Lagrangian coordinates (i.e. coordinates fixed on the cylinder) were calculated from the measured velocities. The surface vorticity shows in the early stage of flow development a close agreement with the previous results obtained by analytical and numerical approaches. The large-field velocity and vorticity information provides an insight into the formation process of the vortices downstream of the cylinder. In addition to the quantitative information, the results of visualized flow pattern obtained by LIPA technique are also presented. A preliminary version of this paper was presented at the Twelfth Symposium on Turbulence, University of Missouri-Rolla, Sept. 24–26, 1990     

Flow past a sphere in density-stratified fluid

Stratified flow past a three-dimensional obstacle such as a sphere has been a long-lasting subject of geophysical, environmental and engineering fluid dynamics. In order to investigate the effect of the stratification on the near wake, in particular, the unsteady vortex formation behind a sphere, numerical simulations of stratified flows past a sphere are conducted. The time-dependent Navier–Stokes equations are solved using a three-dimensional finite element method and a modified explicit time integration scheme. Laminar flow regime is considered, and linear stratification of density is assumed under Boussinesq approximation. The effects of stratification is implemented by density transport without diffusion. The computed results include the characteristics of the near wake as well as the effects of stratification on the separation angle. Under increased stratification, the separation on the sphere is suppressed and the wake structure behind the sphere becomes planar, resembling that behind a vertical cylinder. With further increase in stratification, the wake becomes unsteady, and consists of planar vortex shedding similar to von Karman vortex streets.     

Influences of suction and blowing on vortex shedding behind a square cylinder in a channel

The structure of confined wakes behind a square cylinder in a channel subject to a locally uniform suction or blowing at both the channel walls is presented. A pressure based finite-difference technique has been used to solve the unsteady Navier-Stokes equations. It is observed that the amplitude of the lift coefficient decreases with increase in the blowing velocity. Coefficients of drag also decrease for the application of uniform blowing and for a suitable value of the blowing parameter, the flow becomes steady and symmetric. The amplitude of the lift coefficients increases up to a certain limit of suction velocity and after that it suddenly decreases and flow becomes steady. Coefficients of drag also gives the same feature. Effects of the suction and blowing on the vortex-shedding region are analyzed in detail and presented graphically.     

Steady forces on a cylinder with oblique vortex shedding

We present a curious situation of a fluid-flow wherein the body experiences non-fluctuating fluid-flow force despite being associated with an unsteady flow comprising of sustained vortex shedding. The flow past a circular cylinder at Re=100 is investigated. It is shown that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by a cylinder placed in uniform flow. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. It is found that, both, the parallel and oblique shedding are linearly unstable eigenmodes of the Re=100 steady flow past a cylinder.     

Near-wake flow structure of elliptic cylinders close to a free surface: effect of cylinder aspect ratio

The flow fields behind elliptic cylinders adjacent to a free surface were investigated experimentally in a circulating water channel. A range of cylinder aspect ratios (AR=2, 3, 4) were considered, while the cross-sectional area of the elliptical cylinder was kept constant. The main objective of this study was to investigate the effect of cylinder aspect ratio and a free surface on the flow structure in the near-wake behind elliptic cylinders. For each elliptic cylinder, the flow structure was analyzed for various values of the submergence depth of the cylinder beneath the free surface. The flow fields were measured using a single-frame double-exposure PIV (Particle Image Velocimetry) system. For each experimental condition, 350 instantaneous velocity fields were obtained and ensemble-averaged to obtain the mean velocity field and spatial distribution of the mean vorticity statistics. The results show that near-wake can be classified into three typical flow patterns: formation of a Coanda flow, generation of substantial jet-like flow, and attachment of this jet flow to the free surface. The general flow structure observed behind the elliptic cylinders resembles the structure previously reported for a circular cylinder submerged near a free surface. However, the wake width and the angle of downward deflection of the shear layer developed from the lower surface of the elliptic cylinder differ from those observed for a circular cylinder. These trends are enhanced as cylinder aspect ratio is increased. In addition, the free surface distortion is also discussed in the paper.     

Vortex Induced Vibrations of a Pair of Cylinders at Reynolds Number 1000

Vortex induced vibrations of two equal-sized cylinders in tandem and staggered arrangement placed in uniform incompressible flow is studied. A stabilized finite element formulation is utilized to solve the governing equations. The Reynolds number for these 2D simulations is 1000. The cylinders are separated by 5.5 times the cylinder diameter in the streamwise direction. For the staggered arrangement, the cross-flow spacing between the two cylinders is 0.7 times the cylinder diameter. In this arrangement, the downstream cylinder lies in the wake of the upstream one and therefore experiences an unsteady inflow. The wake looses its temporal periodicity, beyond a few diameters downstream of the front cylinder. The upstream cylinder responds as an isolated single cylinder while the downstream one undergoes disorganized motion. Soft-lock-in is observed in almost all the cases.     

On the flows produced by sudden application of a constant pressure gradient or by impulsive motion of a boundary

Some properties of the time-dependent Navier-Stokes equations are discussed for flows impulsively started from rest by sudden application of a constant pressure gradient or by the impulsive motion of a boundary. Five illustrative examples are given. They are: unsteady flow in a circular cylinder moving parallel to its length, starting flow in a circular pipe, unsteady flow in a rotating cylinder, starting flow in a rectangular channel moving parallel to its length and unsteady flow in a channel of rectangular cross-section. It is found that the expressions of the quantities such as velocity, flux and skin friction are in series forms which may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa. It is shown that if their expressions can be found for one of large values of the time or small values of the time, these expressions can be used for the other.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

Vibrations of a cylinder in a uniform flow in the presence of a no-slip side-wall

A circular cylinder placed in a uniform flow, and that spans the entire length between two side walls, may experience either parallel or oblique vortex shedding depending on the end conditions. It was shown by Mittal and Sidharth (2014) that the spatio-temporal periodicity of the oblique vortex shedding results in constant-in-time force experienced by the cylinder. On the contrary, parallel vortex shedding leads to fluid force that fluctuates with time. The free vibrations of a circular cylinder, in the presence of a wall, are investigated. For comparison, computations with end walls, where a slip condition on velocity is specified, are also carried out. The Reynolds number, based on the diameter of the cylinder and free-stream speed of the flow, is Re=100. The initial condition for the free vibrations is the fully developed unsteady flow past a stationary cylinder with oblique shedding. It is found that as the amplitude of vibration of the cylinder builds up, the vortices shed from the cylinder align with its axis leading to parallel shedding. The response of the cylinder is associated with two branches: initial and lower. On the lower branch, the response of the cylinder is virtually identical from two- and three-dimensional computations. The flow as well as the response is different on the initial branch and outside the synchronization regime. Forced vibrations confirm the phenomena.     

Instantaneous structure and statistical feature of unsteady flow in a channel obstructed by a square rod

A two-dimensional numerical computation has been made for an unsteady flow in a channel obstructed by an inserted square rod. The results of the computation made for the flow with a parabolic inlet velocity profile at a specific value of channel Reynolds number are analyzed in detail. The obtained results reveal that momentum transfer is enhanced due to the apparent shear stress resulting from the nonzero value of cross-correlation between the streamwise and normal components of fluctuating velocity, , just as in turbulent shear flows, although the studied flow is quite different from turbulent flows in the sense that it is highly periodical and therefore free from randomness. This periodicity leads to a quick recovery of the velocity defect in some region of the wake of the rod. Special attention is paid to the time variation of flow structure. The crisscross motion of the Karman vortex previously found to occur is discussed again, and how it appears is explained in terms of the interaction between the Karman vortex and the disturbed wall shear layer. In the discussion of this relationship, wavering motion of the separation vorticity layers formed on both sides of the rod and the periodic formation of an isolated vortex island from the lifted tip of the wall vorticity layer are analyzed. The vortex island is found to play an important role not only for the occurrence of the crisscross motion of Karman vortex but also for the generation of the nonzero value of .     

Numerical simulation of shear effect on vortex shedding behind a square cylinder

This paper is concerned with the numerical simulation of the flow structure around a square cylinder in a uniform shear flow. The calculations were conducted by solving the unsteady 2D Navier–Stokes equations with a finite difference method. The effect of the shear parameter K of the approaching flow on the vortex-shedding Strouhal number and the force coefficients acting on the square cylinder is investigated in the range K=0·0–0·25 at various Reynolds numbers from 500 to 1500. The computational results are compared with some existing experimental data and previous studies. The effect of shear rate on the Strouhal number and the force acting on the cylinder has a tendency to reduce the oscillation. The Strouhal number, mean drag and amplitude of the fluctuating force tend to decrease as the shear rate increases, but show no significant change at low shear rate. Increasing the Reynolds number decreases the Strouhal number and increases the force acting on the cylinder. At high shear rate the shedding frequencies of the fluctuating drag and lift coefficients are identical. © 1997 John Wiley & Sons, Ltd.     

Unsteady mixed convection heat transfer from tandem square cylinders in cross flow at low Reynolds numbers

A two-dimensional numerical study is carried out to understand the influence of cross buoyancy on the vortex shedding processes behind two equal isothermal square cylinders placed in a tandem arrangement at low Reynolds numbers. The spacing between the cylinders is fixed with five widths of the cylinder dimension. The flow is considered in an unbounded medium, however, fictitious confining boundaries are chosen to make the problem computationally feasible. Numerical calculations are performed by using a finite volume method based on the PISO algorithm in a collocated grid system. The range of Reynolds number is chosen to be 50–150. The flow is unsteady laminar and two-dimensional in this Reynolds number range. The mixed convection effect is studied for Richardson number range of 0–2 and the Prandtl number is chosen constant as 0.71. The effect of superimposed thermal buoyancy on flow and isotherm patterns are presented and discussed. The global flow and heat transfer quantities such as overall drag and lift coefficients, local and surface average Nusselt numbers and Strouhal number are calculated and discussed for various Reynolds and Richardson numbers.     

Two‐dimensional prediction of time dependent,turbulent flow around a square cylinder confined in a channel

This paper presents two‐dimensional and unsteady RANS computations of time dependent, periodic, turbulent flow around a square block. Two turbulence models are used: the Launder–Sharma low‐Reynolds number k–ε model and a non‐linear extension sensitive to the anisotropy of turbulence. The Reynolds number based on the free stream velocity and obstacle side is Re=2.2×10⁴. The present numerical results have been obtained using a finite volume code that solves the governing equations in a vertical plane, located at the lateral mid‐point of the channel. The pressure field is obtained with the SIMPLE algorithm. A bounded version of the third‐order QUICK scheme is used for the convective terms. Comparisons of the numerical results with the experimental data indicate that a preliminary steady solution of the governing equations using the linear k–ε does not lead to correct flow field predictions in the wake region downstream of the square cylinder. Consequently, the time derivatives of dependent variables are included in the transport equations and are discretized using the second‐order Crank–Nicolson scheme. The unsteady computations using the linear and non‐linear k–ε models significantly improve the velocity field predictions. However, the linear k–ε shows a number of predictive deficiencies, even in unsteady flow computations, especially in the prediction of the turbulence field. The introduction of a non‐linear k–ε model brings the two‐dimensional unsteady predictions of the time‐averaged velocity and turbulence fields and also the predicted values of the global parameters such as the Strouhal number and the drag coefficient to close agreement with the data. Copyright © 2009 John Wiley & Sons, Ltd.     

The effect of an unsteady drag force on the structure of a non-equilibrium region behind a shock wave in a gas-particle mixture

For numerical analysis of shock wave propagation in gas-particle mixtures, drag coefficients of a sphere in steady flows are generally used. However, it is shown both experimentally and numerically that a shock loaded solid sphere experiences unsteady drag forces. The paper describes a model of unsteady drag force and its effect on the structure of the non-equilibrium region behind a shock front traveling in a dusty gas. The results are compared with those obtained by using a steady drag coefficient and are discussed. It is demonstrated that the large drag force at the early stage of the interaction between shock-wave induced flow and a solid particle affects the flow structure that is obtained with a steady drag force.