Numerical solution of Stokes flow generated by vortices: part 1: around an elliptical cylinder

In this paper, the Stokes flow outside a two-dimensional elliptical cylinder is considered for a highly viscous, incompressible fluid flow, driven by a pair of vortices placed at different positions. Relating the coefficients of some of the terms in the asymptotic expansion of the stream function to the force components and the torque on the combined system, together with the imposition of integral constraints, enables the boundary element method to provide a closed system of equations. It is found that excellent agreement is obtained between the numerical results for a single vortex and all the previous analytical expressions. Some new results relating to forces and torques on the elliptical cylinder are presented. The analytical closed-form expression for the single vortex is provided.     

A comparative study on effect of plain- and wavy-wall confinement on wake characteristics of flow past circular cylinder

A first attempt is made for identifying the wake characteristics of circular cylinder confined by a wavy wall at laminar flow regime. Numerical study of flow characteristics past circular cylinder with wavy-wall confinement perpendicular to cylinder axis has been carried out in the range of Reynolds number 20–100. The finite volume-based CFD solver Ansys Fluent (Version 15.0) is used for computations. The results are presented in the form of streamline plots, mean drag co-efficient, flow separation angle and recirculation length. Wavy-wall confinement leads to highly significant changes in the cylinder wake such as the evolution of strong x-plane vortices, enhanced fluid mixing, wake suppression near the crest region and vortex stretching near the trough region on the downstream of the cylinder has been observed. Flow separation angle varies significantly along the axis of the cylinder. Increased wall shear stress on rear surface of the cylinder has also been observed. The part of vorticity magnitude as compared to strain rate has been distinguished and identified using vortex identification methods such as Q-criterion and Lambda-2 criterion.     

Rotating Inclined Cylinder and the Effect of the Tilt Angle on Vortices

We study numerically some possible vortex configurations in a rotating cylinder that is tilted with respect to the rotation axis and where different numbers of vortices can be present at given rotation velocity. In a long cylinder at small tilt angles the vortices tend to align along the cylinder axis and not along the rotation axis. We also show that the axial flow along the cylinder axis, caused by the tilt, will result in the Ostermeier-Glaberson instability above some critical tilt angle. When the vortices become unstable the final state often appears to be a dynamical steady state, which may contain turbulent regions where new vortices are constantly created. These new vortices push other vortices in regions with laminar flow towards the top and bottom ends of the cylinder where they finally annihilate. Experimentally the inclined cylinder could be a convenient environment to create long lasting turbulence with a polarization which can be adjusted with the tilt angle.     

Vortex motion in the early stages of unsteady flow around a circular cylinder

In this paper, processes in the early stages of vortex motion and the development of flow structure behind an impulsively-started circular cylinder at high Reynolds number are investigated by combining the discrete vortex model with boundary layer theory, considering the separation of incoming flow boundary layer and rear shear layer in the recirculating flow region. The development of flow structure and vortex motion, particularly the formation and development of secondary vortex and a pair of secondary vortices and their effect on the flow field are calculated. The results clearly show that the flow structure and vortices motion went through a series of complicated processes before the symmetric main vortices change into asymmetric: development of main vortices induces secondary vortices; growth of the secondary vortices causes the main vortex sheets to break off and causes the symmetric main vortices to become “free” vortices, while a pair of secondary vortices is formed; then the vortex sheets, after breaking off, gradually extend downstream and the structure of a pair of secondary vortices becomes relaxed. These features of vortex motion look very much like the observed features in some available flow field visualizations. The action of the secondary vortices causes the main vortex sheets to break off and converts the main vortices into free vortices. This should be the immediate cause leading to the instability of the motion of the symmetric main vortices. The flow field structure such as the separation position of boundary layer and rear shear layer, the unsteady pressure distributions and the drag coefficient are calculated. Comparison with other results or experiments is also made. This work was presented at the First Asian Congress of Fluid Mechanics, Bangalore in December 1980.     

Flow visualization studies of vortices

An actual vortex in the Kármán vortex street downstream of a circular cylinder has a core of finite dimension which increases downstream. The circulation of the vortex is nearly constant. The ratiob/a which is 0.281 according to the theory of Kármán, grows from 0.2 to 0.4 in the near wake. In the flow about a circular cylinder rotating in a uniform flow, a Kármán vortex street, Görtler-type vortices and Taylor vortices are generated at the same time. In the flow about a circular cylinder impulsively started with a constant velocity, the primary twin vortices behind the cylinder induce secondary twin vortices near the separation point. At the beginning of the motion, the separation does not occur even though a reverse flow is observed in the boundary layer. Mutual slip-through of a pair of vortex rings was achieved by increasing the Reynolds number. A vortex ring rebounds from a plane surface due to the separation of the flow on the surface induced by the vortex ring, and the secondary vortex ring is formed from the separated shear layer.     

大跨屋盖表面旋涡的PIV试验研究

该文通过风洞流场显示试验,观察了大跨平屋盖和马鞍屋盖表面的分离泡和锥形涡现象,给出了不同风向、不同屋盖表面的旋涡流线和涡量场分布;分析了风向角、屋盖曲率对于旋涡形态的影响。试验结果表明,当风向垂直于平屋盖迎风前缘时,屋盖表面将形成典型的分离泡现象,且分离泡的涡核位置恰好对应了涡量场的负向峰值。在斜向风作用下,平屋盖和以高点作为迎风点的马鞍屋盖表面将出现锥形涡。观察旋涡的平均流线和涡量场分布图,发现当来流沿两种屋盖对角线时,锥形涡截面形状接近圆形;当来流偏离屋盖对角线时,在靠近来流的一侧,锥形涡截面形状接近椭圆形;流场内负向涡量分布于壁面上,峰值集中在迎风前缘附近和旋涡周围。在相同的风向角下,曲率较大的马鞍表面锥形涡涡轴与屋盖迎风前缘所成角度较大,曲率较小的马鞍表面锥形涡涡轴与迎风前缘所成角度较小。此外,旋涡的瞬时流线图表明,锥形涡是一种瞬时变化的流体现象,其形态和位置在每个瞬时都不相同。     

马鞍屋盖表面锥形涡流动模型研究

该文基于大跨平屋盖和马鞍屋盖表面锥形涡流动显示试验,对现有的兰金涡模型进行改进,在其涡核区与势流区之间添加过渡区,从而建立了简化的二维锥形涡流动模型,给出了旋涡上部流速、旋涡内部流线曲率以及屋面涡核吸力之间的定量关系。据此流动模型,分析了旋涡强度的影响因素即为旋涡内部流线曲率和旋涡流速:旋涡内部流线曲率越大,旋涡转速越快,旋涡强度越大。基于流动显示试验和测压试验所得平屋盖以及马鞍屋盖表面各旋涡参数之比,以平屋盖表面锥形涡为基准,量化了各风向下马鞍屋盖表面的锥形涡强度。通过加入考虑旋涡效应对传统准定常理论进行修正,给出了锥形涡涡核吸力(均值、峰值)的计算公式,并将计算值与马鞍屋盖刚性模型风洞测压试验数据进行对比,验证了锥形涡流动模型对于预测马鞍表面锥形涡涡核吸力的有效性。     

Separation in a two-dimensional Stokes flow inside an elliptic cylinder

The two-dimensional Stokes flow due to a line rotlet inside a fixed elliptic cylinder is investigated, where it is assumed that the line rotlet intersects the major axis of each elliptical cross-section of the cylinder. For the case in which the line rotlet coincides with the centre-line of the elliptic cylinder, it is shown that the number of eddies in the flow increases in a roughly linear way with the ratio of length to width of a cross-section of the cylinder. Moreover, results obtained by varying the rotlet position for several different fixed boundary shapes suggest that the aforementioned ratio, and not the rotlet position, is the principal determinant of the number of eddies.     

Shear flow past a square cylinder near a wall

A numerical study on the wake behind a square cylinder placed parallel to a wall has been made. The cylinder is considered to be within the boundary layer of the wall, so that the outside flow is taken to be due to uniform shear. Flow has been investigated in the laminar Reynolds number (based on the cylinder height) range. The interaction of wall boundary layer on the vortex shedding at Reynolds number up to 1400.0 has been investigated for cylinder to wall gap height 0.5 and 0.25 times the cylinder height. The gap flow between the cylinder and wall during a period of vortex shedding has been obtained. The governing unsteady Navier–Stokes equations are discretised through the finite volume method on staggered grid system. An algorithm SIMPLE has been used to compute the discretised equations iteratively. Our results show that at the gap height 0.5 times the cylinder height the vortex shedding occurs at a Strouhal number greater than for an isolated cylinder. Vortex shedding suppression occurs and wake becomes steady up to a certain value of Reynolds number at gap height 0.25 time the cylinder height. At higher Reynolds number the formation of a single row of negative vortices behind the cylinder when it is in close proximity to wall is noteworthy. Due to the shear, the drag experienced by the cylinder is found to decrease with the reduction of gap height.     

Modelling with viscous and viscoplastic materials under combining and separating flow configurations

Combining and separating incompressible flow of Newtonian and inelastic Herschel–Bulkley fluids is studied numerically employing a semi-implicit Taylor–Galerkin pressure-correction algorithm, where steady solutions are obtained through a transient finite element procedure. The influence of inertia and fluid rheology is analysed on flow patterns, velocity fields and pressure drops for various flow configurations, with fixed geometric gap width that stimulates the merging and splitting in the flow. For Newtonian fluids and at larger levels of inertia, the appearance of vortices was observed, with an increase in velocity differences and pressure drops across the channel. In this case, the numerical procedure was verified with good agreement against previous numerical and experimental observations. To extend the consideration to non-Newtonian inelastic materials, the material rheological characteristics were approximated with the use of the Herschel–Bulkley fluid model, incorporating the Ostwald–de Waele power-law model and viscoplastic yield stress. Findings for unyielded power-law fluids reveal slight increase in the size of the vortices as power index (m) was decreased. Variation of the consistency index (k) shows strong influence on the streamline patterns with a rapid increase in the vortex formation as k was decreased. For Bingham model solutions, devoid of shear-thinning and increasing yield stress, a higher value of Reynolds number is required for equivalent levels of vortex formation; also one observes the appearance of yielded and unyielded regions. Under Herschel–Bulkley modelling, there was little change noted in the kinematics, but some was apparent in rheological response. Once more, observations reveal the tendency to eliminate vortices at larger yield stress levels, with the appearance of unyielded regions.     

Study on boundary layer transition of a rotating disk

Summary Behaviour of spiral vortices being generated in transition regime of a disk rotating in otherwise undisturbed fluid is experimentally studied in detail. Through visualizations of the transition regime by using close-up camera, new striped flow patterns originating along the axis of spital vortices are found to be ring-like vortices which occur on the surfaces of each spiral vortices. Mechanism of the spiral vortex is clarified by cutting the vortices by strobo slit light. It is also found out experimentally that the phase velocity of the vortices is zero.     

Vortex simulation for non-axisymmetric collision of a vortex ring with solid particles

This study is concerned with the numerical simulation for the non-axisymmetric collision between a vortex ring and solid particles. The vortex ring convects with its self-induced velocity in a quiescent air, and the half part collides with spherical glass particles. The vortex method for gas-particle two-phase flow proposed by the authors in a prior paper is used for the simulation. The Reynolds number of the vortex ring is 2600, and the particle diameter is 50 μm. The Stokes number, defined as the ratio of the particle response time to the characteristic time of the vortex ring, is 0.74. The simulation clarifies that the particles induce the vortices, having an axis parallel to the convection direction of the vortex ring, inside the vortex ring and that pairs of the positive and negative vortex tubes appear. It also highlights that highly organized three-dimensional vortical structures composed of the streamwise vortices yield the rapid deformation and collapse of the vortex ring.     

On the interaction of a vortex pair and a vortex ring with a flat shield

The approximate method of calculation of nonstationary flow in the interaction of a vortex pair and a vortex ring with a parallel and respectively perpendicular flat shield is presented. It is shown that these primary vortices induce transverse wall flow on the shield in the ideal-fluid approximation; in this flow, with allowance for the fluid’s viscosity, a boundary layer is generated which represents vortex flow with sign opposite to that of the primary vortices. Boundary-layer separations occur on the portion of the shield with a positive longitudinal pressure gradient. Secondary flows interact with the primary ones due to which the flow is rearranged; the transverse displacement of the initial vortex pair with a loop-shaped trajectory of its motion is observed in the plane problem, whereas the formation of ascending flow along the axis of the vortex ring is observed in the axisymmetric problem. The effect found in the latter case in laminar and turbulent regimes of flow is confirmed for the laminar regime by experiment and by the data of numerical simulation of Navier–Stokes and Reynolds equations.     

驻波场中圆柱形换热管外声流特性的数值研究

基于非线性Navier-Stokes方程推导了一般性的声流控制方程,并利用有限元软件COMSOL对平面驻波声场中单根圆柱形换热管和双圆柱形换热管外包含二阶流场信息的Navier-Stokes方程进行全域求解,得到了一阶和二阶流场信息。研究了雷诺数Re和斯特劳哈尔数Sr对换热管外声流结构的影响规律。研究发现:随Re和Sr的增大,换热管边界层内的涡流区域变小,直至消失,而边界层外的涡流区域逐渐增大;且单换热管外涡流个数由8个减少到4个,双换热管外涡流个数从12个减少到8个。此外,边界层内和边界层外的涡流结构呈反向旋转;边界层外流场沿着振荡方向远离换热管,而在垂直方向流向换热管。     

Edge effects in flow around a plasma actuator

We have studied the structure of flow formed in the boundary layer at the lateral edge of the discharge zone of a dielectric barrier discharge plasma actuator. It is established that a region with nonzero component of tangential velocity exists near the plasma layer boundary. At some distance downstream of the actuator, a concentrated vortex is formed with the axis aligned with the flow direction. In the presence of two closely spaced boundaries of the plasma region, a pair of counter-rotating vortices is formed. Separate microdischarges in the plasma layer also appear to be a source of similar longitudinal vortices with smaller amplitude.

     

Swirl flow studies in diffusers using the flux analysis method with orthogonal curvilinear coordinates

A computational method of flux analysis is applied to the study of the swirling flow of different regimes through a diffuser pipe. The flux analysis method is an iterative procedure to construct a system of orthogonal curvilinear co-ordinates consisting of stream-surface and normals following the flow. Examples of the inviscid swirling flow of an incompressible fluid are calculated, and deformations of vortices in a diffuser pipe are studied. For a simple vortex with rigid rotation the deceleration of flow is remarkable on the axis, where the total energy has the lowest level. For a Burgers type vortex, provided the circulation number exceeds a critical value, the rate of deceleration of flow along the axis become more pronounced for only a slight increase of swirl strength, and ultimately a stagnation point appears on the axis. The present method, however, may not be applied to the flow with stagnation point.     

椭圆管换热器对空调室内机声场影响的数值模拟

椭圆管换热器由于良好的流动和换热特性,在换热设备中有广泛的应用。本文利用计算流体力学方法,对椭圆管翅片换热器应用于空调室内机的流场和声场进行数值模拟,并与圆管作对比。计算结果表明,换热管形状对贯流风机内部偏心涡的形成位置和大小没有影响;与采用相同截面积的圆管翅片换热器的室内机相比,相同条件下,采用椭圆管换热器对于室内机增加风量,改善制冷性能,降低噪声尤其是低频噪声方面有良好的效果。其中,长短轴之比为2的椭圆管可以降低室内机噪声4 d B。     

Vortex shedding suppression for laminar flow past a square cylinder near a plane wall: a two-dimensional analysis

Summary A numerical study on the uniform shear flow past a long cylinder of square cross-section placed parallel to a plane wall has been made. The cylinder is considered to be within the boundary layer of the wall. The maximum gap between the plane wall to the cylinder is taken to be 0.25 times the cylinder height. We investigated the flow when the regular vortex shedding from the cylinder is suppressed. The governing unsteady Navier-Stokes equations are discretized through the finite volume method on staggered grid system. A pressure correction based iterative algorithm, SIMPLER, has been used to compute the discretised equations iteratively. We found that the critical value of the gap height for which vortex shedding is suppressed depends on the Reynolds number, which is based on the height of the cylinder and the incident stream at the surface of the cylinder. At high Reynolds number (Re ≥ 500) however, a single row of negative vortices occurs for wall to cylinder gap height L ≥ 0.2. The shear layer that emerges from the bottom face of the cylinder reattaches to the cylinder itself at this gap hight.