On small-time expansion of nonlinear free surface problems

The small-time expansion of the 2-D problem of a heaving semi-submerged circular cylinder, starting from rest, in analyzed. The first three terms of the small-time expansion of the outer solution (away from the cylinder) are developed. A logarithmic singularity is found for the second derivative in time of the surface elevation at the intersection point. This singularity is of second order in the heaving velocity of the cylinder, and the inner expansion developed for the linear wavemaker problem therefore does not cover this case. The problem of the horizontal motion of the semi-submerged cylinder is analyzed as well. For the case with the initial condition for the velocity potential chosen as zero on the free surface, the logarithmic singularity found for the wavemaker is recovered. Selecting the initial condition as a vanishing normal velocity on the free surface leads to a solution where the singularity appears in the fifth derivative in time of the free-surface elevation. The solution of this problem shows the characteristic behaviour of formation of asymmetry, which finally would lead to formation of lee waves. The inner solution of the problem of the heaving cylinder is discussed, showing that the oscillatory behaviour reported for the wavemaker problem appears in the present solution as well.     

A fully nonlinear implicit model for wave interactions with submerged structures in forced or free motion

The purpose of this work is to develop advanced numerical tools for modeling two-way fully nonlinear interactions of ocean surface waves (irregular waves in the general situation) with submerged structures undergoing large amplitude motion, that could represent Wave Energy Converters (WECs). In our modeling approach, an existing two-dimensional Numerical Wave Tank (NWT), based on potential flow theory, is extended to include a submerged horizontal cylinder of arbitrary cross-section. The mathematical problem and related numerical solution are first introduced. Then, conservation of volume and conservation of energy are checked, respectively, in the case of a circular cylinder in a prescribed large amplitude motion and in the case of a circular cylinder in a free motion. Interactions between waves and a submerged circular cylinder computed by the model are then compared to mathematical solutions for two situations: a cylinder in prescribed motion and a freely moving cylinder.     

An experimental study of frost formation on a horizontal cylinder under cross flow

Frost layers formed on the front and rear surfaces of a horizontal cylinder during cross flow are found to be thicker than those at the top and bottom surfaces where the flow separation is nearly initiated. This observation was obtained in an experimental study carried out to examine frost formation on a horizontal cylinder given a cross flow condition. The thickness of the frost layer and the temperature distribution in the cylinder were measured for various experimental conditions. The local heat flux around the cylinder and the effective thermal conductivity of the frost layer were likewise evaluated, while thickness and surface temperature of the frost layer around the cylinder were measured periodically. These measurements were obtained by varying the Reynolds number, temperature, and humidity. The dew point temperature of the inlet air, however, was kept below the freezing point throughout the experiment. Results also reveal that inlet air velocity, temperature, and humidity affect thickness and thermal conductivity of the frost layer.     

Separating inviscid flows

Summary The separating flow of an inviscid fluid is not only a limit solution of the steady separating, laminar fluid flow at high values of the Reynolds number but it is also part of its structure (Smith [1], [2]). This work aims at reexamining the separating flow of inviscid fluid past a bluff body which is fixed in an otherwise uniform stream of fluid. For the purpose of this paper we will assume that the bluff body is a circular cylinder but the theory is applicable to bodies of any shape. It is further assumed that the fluid is in steady two-dimensional motion and is inviscid and of constant density. The flow structure is assumed to consist of a separated flow region, caviting flows in which there exists a free surface on which the pressure is constant, and a wake. A twin spiral vortex model is used in order to determine the shape of the free streamline. Based on the free streamline theory the problem reduces to solving a mixed boundary value problem and a Hilbert solution for the inverse problem in the auxiliary plane is obtained. When we consider the flow in the physical plane the problem is transformed into a direct problem in which the geometry of the solid body is given in advance. We assume that the separation is smooth and thus the curvature of the free streamline at the point of free detachment be equal to that of the body surface. A numerical method for solving the two-dimensional potential flows past arbitrarily shaped curved bluff bodies is developed.When the cavitation number is zero the angle of separation is approximately 55° and the computed results predict that the position of the separation point will move backward as the cavition number increases. The relationships between the drag coefficient, and the width and length of the cavity is determined and is found to be in very good agreement with the predictions of Smith [1].     

A BEM for the prediction of free surface effects on cavitating hydrofoils

 In this paper, a boundary element method (BEM) for cavitating hydrofoils moving steadily under a free surface is presented and its performance is assessed through systematic convergence studies, comparisons with other methods, and existing measurements. The cavitating hydrofoil part and the free surface part of the problem are solved separately, with the effects of one on the other being accounted for in an iterative manner. Both the cavitating hydrofoil surface and the free surface are modeled by a low-order potential based panel method using constant strength dipole and source panels. The induced potential by the cavitating hydrofoil on the free surface and by the free surface on the hydrofoil are determined in an iterative sense and considered on the right hand side of the discretized integral equations. The source strengths on the free surface are expressed by applying the linearized free surface conditions. In order to prevent upstream waves the source strengths from some distance in front of the hydrofoil to the end of the truncated upstream boundary are enforced to be equal to zero. No radiation condition is enforced at the downstream boundary or at the transverse boundary for the three-dimensional case. First, the BEM is validated in the case of a point vortex and some convergence studies are done. Second, the BEM is applied to 2-D hydrofoil geometry both in fully wetted and in cavitating flow conditions and the predictions are compared to those of other methods and of the measurements in the literature. The effects of Froude number, the cavitation number and the submergence depth of the hydrofoil from free surface are discussed. Then, the BEM is validated in the case of a 3-D point source. The effects of grid and of the truncated domain size on the results are investigated. Lastly, the BEM is applied to a 3-D rectangular cavitating hydrofoil and the effect of number of iterations and the effect of Froude number on the results are discussed. Received 6 November 2000     

直径30cm圆柱的气动力参数和绕流特性研究

为研究光滑圆柱的气动力系数和绕流特性,在均匀流中进行不同风速下的测压风洞试验,试验获得了阻力系数、升力系数、表面风压分布、风压相关性系数、斯托罗哈数等随雷诺数的变化特征,并将试验结果与以往结果进行比较。研究表明:升力系数的脉动值大于阻力系数的脉动值,说明涡脱造成的横风向激励比顺风向紊流激励剧烈;雷诺数位于临界区域时,圆柱表面风压分布呈现出对称-不对称-对称的变化过程,反映了由层流分离转化为湍流分离的全过程;在雷诺数为352000时呈现一侧为层流分离、另一侧为湍流分离的临界流态,风压呈现出左右不对称的单边泡形式;获得层流分离和湍流分离时的表面风压相关性分布特征,层流分离时圆柱同一侧的风压测点均呈较强的正相关,而湍流分离时在分离点前的区域相关性较强,分离点之后的区域相关性较弱;层流分离的升力系数谱有显著的峰值,表明尾流是规则的漩涡脱落,而湍流分离的升力系数谱没有明显峰值,表明尾流是随机的漩涡脱落。     

Thermal-wave-induced vorticity in a liquid film

We have theoretically studied the influence of a moving local heat source on the structure of flow in a thin liquid layer on a horizontal substrate. A two-dimensional problem is considered in the boundary layer approximation and a stationary equation that describes deformation of the liquid layer is obtained. Results of numerical calculations for a preset temperature distribution on the free surface are presented that demonstrate the formation of a vortex structure. It is established that, even in the absence of a gravity-driven flow, the motion of a local heat source ensures a stationary flow regime without film breakage and the formation of dry spots. The results justify a new scheme of experiments for the investigation of phenomena in nonisothermal liquid films.     

高精度超高压液相泵等面积平滑补偿控制方法

针对高精度超高压液相泵运行时易在加速度拐点处出现电机抖动的问题,提出了一种等面积平滑补偿控制方法。对液相泵运行曲线进行平滑处理,确保液相泵在初始速度较高的加速度拐点处平稳过渡,不产生抖动;对因平滑处理而减小的位移进行补偿,确保在液相泵单个运行周期结束前补齐位移,使得液相泵的吸入液体总量与排出液体总量不变。最后通过电机运行试验、流量精度检测试验、响应速度检测试验、超高效分离试验来验证所提方法的可行性。试验结果表明:等面积平滑补偿控制方法解决了电机在高初始速度的加速度拐点处抖动的问题,使液相泵输出液体的流量精度从0.081%提高到0.055%。等面积平滑补偿控制方法有效提高了液相泵输出液体的流量精度,具有实时性强、计算精度高等特点,为提高样品检测效率提供了一定的理论依据。     

Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.     

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.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

A coupled BEM and arbitrary Lagrangian–Eulerian FEM model for the solution of two‐dimensional laminar flows in external flow fields

This paper describes a new computational model developed to solve two‐dimensional incompressible viscous flow problems in external flow fields. The model based on the Navier–Stokes equations in primitive variables is able to solve the infinite boundary value problems by extracting the boundary effects on a specified finite computational domain, using the pressure projection method. The external flow field is simulated using the boundary element method by solving a pressure Poisson equation that assumes the pressure as zero at the infinite boundary. The momentum equation of the flow motion is solved using the three‐step finite element method. The arbitrary Lagrangian–Eulerian method is incorporated into the model, to solve the moving boundary problems. The present model is applied to simulate various external flow problems like flow across circular cylinder, acceleration and deceleration of the circular cylinder moving in a still fluid and vibration of the circular cylinder induced by the vortex shedding. The simulation results are found to be very reasonable and satisfactory. Copyright © 2001 John Wiley & Sons, Ltd.     

Film depth and concentration banding in free-surface Couette flow of a suspension

The film depth of a free-surface suspension flowing in a partially filled horizontal concentric-cylinder, or Couette, device has been studied in order to assess its role in the axial concentration banding observed in this flow. The flow is driven by rotation of the inner cylinder. The banding phenomenon is characterized by particle-rich bands which under flow appear as elevated regions at the free surface separated axially by regions dilute relative to the mean concentration. The concentric cylinders studied had outer radius R(o) = 2.22 cm and inner radii R(i) = 0.64, 0.95 and 1.27 cm; the suspension, of bulk particle volume fraction phi = 0.2 in all experiments described, was composed of particles of either 250-300 microm diameter or less than 106 microm diameter, with the suspending fluid an equal density liquid of viscosity 160 P. The ratio of the maximum to the minimum particle volume fraction along the axis in the segregated condition varies from O(1) to infinite. The latter case implies complete segregation, with bands of clear fluid separating the concentrated bands. The film depth has been varied through variation of the filled fraction, f, of the annular gap between the cylinders and through the rotation rate. Film depth was analysed by edge detection of video images of the free surface under flow, and the time required for band formation was determined for all conditions at which film depth was studied. The film depth increases roughly as the square root of rotation speed for f = 0.5. Band formation is more rapid for thicker films associated with more rapid rotation rates at f = 0.5, whereas slower formation rates are observed with thicker films caused by large f, f > 0.65. It is observed that the film depth over the inner cylinder grows prior to onset of banding, for as yet unknown reasons. A mechanism for segregation of particles and liquid in film flows based upon 'differential drainage' of the particle and liquid phase in the gravity-driven flow within the film over the inner cylinder is formulated to describe the onset of concentration fluctuations. This model predicts that suspension drainage flows lead to growth of fluctuations in phi under regions of negative surface curvature.     

The initial oscillatory flow past a circular cylinder

This paper presents results obtained from an initial approximation for the flow around a circular cylinder in two-dimensional oscillating flow. The analysis is developed in terms of the scalar vorticity and stream function. An expansion in powers of time from the start of the motion is obtained using an exact analysis which extends the results of boundary-layer theory by taking into account corrections for finite Reynolds number. The time development of the physical properties of the flow are determined both by means of analytical expressions and by an accurate numerical procedure. The surface pressure, drag and surface vorticity are calculated and various estimates of the time of separation and the distance moved in this time are obtained. The phenomenon of steady streaming is not considered in this paper since the time of validity of the expansions is small. The agreement between the analytical and numerical results at small times is excellent.     

行波壁抑制圆柱涡激振动的数值模拟研究

采用CFD数值模拟方法完成了圆柱绕流-涡激振动-行波壁流动控制全过程的数值模拟,重点研究行波壁流动控制方法对低雷诺数下两自由度弹性支撑单圆柱涡激振动的抑制作用。详细分析各阶段圆柱横向和流向位移、质心运动轨迹、升力和阻力系数等随频率比的变化。结果表明:行波壁圆柱的波谷处可以产生一系列稳定的随行波壁运动的小尺度旋涡,有效抑制圆柱表面分离涡的产生,达到消除圆柱绕流尾迹和抑制涡激振动的目的;在计算初始和中途启动的行波壁流动控制方法显著抑制了圆柱横向和流向振动、降低了圆柱升力系数脉动值和阻力系数均值,但阻力系数脉动值则明显增大。     

The initial stage of dam-break flow

The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy–Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.     

Effect of motion of a local heat source on thermocapillary deformation of a thin liquid film flowing down under the action of gravity

The influence of a moving local heat source on the structure of flow in a thin liquid film flowing down on an inclined substrate under the action of gravity has been theoretically studied. Two-dimensional steady-state and conjugated hydrodynamic heat transfer problem has been solved in a long-wave approximation. The characteristics of flow are compared for various regimes: from the liquid film flowing down on a vertical surface with an immobile heat source to the behavior of a horizontal liquid layer under the action of a moving heat source. It is shown that changes in the flow velocity profile related to an increase in the velocity of the heat source motion and a decrease in the substrate slope under other equal conditions (constant flow rate, film thickness, and heat release) lead to a sharply increased thermocapillary deformation of the liquid film.     

Experimental investigation of frost formation on a horizontal cold cylinder under cross flow

In the present work, frost formation on a horizontal cold cylinder was investigated experimentally. An apparatus was designed and carried out on a physical model which was included an air tunnel and a cold cylinder exposed to humid air flow. It was found that the frost layers formed on the front and rear surfaces of the cylinder were thicker than the top surfaces where the flow separation was nearly or wholly initiated. The effects of air flow parameters such as flow Reynolds number, entrained air temperature, absolute air humidity and temperature of cylinder surface on the frost thickness and density formed over the cylinder were studied. In this paper, the dew point temperature of the inlet air was above the freezing point and also the earlier transition time was investigated.