Tomographic reconstruction of stratified fluid flow

A method for imaging a moving fluid is proposed and evaluated by numerical simulation. A cross-section of a three-dimensional fluid is probed by high-frequency acoustic waves from several different directions. Assuming straight-ray geometric acoustics, the time of flight depends on both the scaler sound speed and the vector fluid velocity. By appropriately combining travel times, projections of both the sound speed and the velocity are isolated. The sound speed is reconstructed using the standard filtered backprojection algorithm. Though complete inversion of velocity is not possible, sufficient information is available to recover the component of fluid vorticity transverse to the plane of insonification. A new filtered backprojection algorithm for vorticity is developed and implemented. To demonstrate the inversion procedure, a 3-D stratified fluid is simulated and travel time data are calculated by path integration. These data are then inverted to recover both the scaler sound speed and the vorticity of the evolving flow.     

Preliminary measurements of the velocity distribution in a cylinder filled with liquid He II during spin-up

In a rotating He II-filled cylinder with an aspect ratio of 4.3 the transient evolution of the vorticity fields of the normal fluid and the superfluid component during spin-up are investigated. The acoustical method of measurement used utilizes the change of propagation time of first- and second-sound signals irradiated through a slit in the middle of the cylinder. The results of the first-sound measurements within the range 1.3 T 2.1 K essentially do not depend on temperature and, for the final state, indicate a lack of about 30% of vorticity compared to solid-body rotation. On the other hand, the results of the second-sound measurements are temperature-dependent and show that the superfluid component reaches solid-body rotation, while the normal fluid fraction seems to slip at the wall. Accordingly, the velocity in the inner part of the flow field adopts a value lower than one would expect from the present velocity of the wall, thus confirming the lack of vorticity observed.     

基于表面涡方法的单列圆柱流体诱导振动及流弹性不稳定研究

基于表面涡方法和流固耦合模型研究了Re=2.67×104时的单列圆柱流体诱导振动问题,计算了流体力、振动响应、涡脱落频率等,并给出了涡云图。计算模拟结果很好地重现了刚性单列圆柱在T/D=1.5(小间隙比)下以宽窄尾涡交替和多频为特征的非均匀流态,以及T/D=2.0的涡脱落现象。此外,该文还研究了单列弹性圆柱在T/D=1.5时的流体诱导振动以及流体弹性不稳定问题,计算了SG=1.29时圆柱列的无量纲临界速度。     

Flexural- and capillary-gravity waves due to fundamental singularities in an inviscid fluid of finite depth

Wave motion due to line, point and ring sources submerged in an inviscid fluid are analytically investigated. The initially quiescent fluid of finite depth, covered by a thin elastic plate or by an inertial surface with the capillary effect, is assumed to be incompressible and homogenous. The strengths of the sources are time-dependent. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The perturbed flow is decomposed into the regular and the singular components. An image system is introduced for the singular part to meet the boundary condition at the flat bottom. The solutions in integral form for the velocity potentials and the surface deflexions due to various singularities are obtained by means of a joint Laplace-Fourier transform. To analyze the dynamic characteristics of the flexural- and capillary-gravity waves due to unsteady disturbances, the asymptotic representations of the wave motion are explicitly derived for large time with a fixed distance-to-time ratio by virtue of the Stokes and Scorer methods of stationary phase. It is found that the generated waves consist of three wave systems, namely the steady-state gravity waves, the transient gravity waves and the transient flexural/capillary waves. The transient wave system observed depends on the moving speed of the observer in relation to the minimal and maximal group velocities. There exists a minimal depth of fluid for the possibility of the propagation of capillary-gravity waves on an inertial surface. Furthermore, the results for the pure gravity and capillary-gravity waves in a clean surface can also be recovered as the flexural and inertial parameters tend to zero.     

Diffusion-controlled growth of multi-component gas bubbles

Theoretical solutions for some important problems involving diffusion-controlled growth of gas bubbles in liquids in conditions of spherical symmetry are presented. It is shown that bubbles in systems containing several independently diffusing gases always approach an asymptotic composition and a parabolic relation between size and time. Solutions for this asymptotic regime have been obtained analytically for growth from zero size and numerically for growth from finite initial size; the two solutions agree well for sufficiently large sizes. The numerical methods can deal with transient growth from finite size, including the behaviour of bubbles of non-equilibrium initial composition. The differences between initial and equilibrium compositions make it easy to understand why the transient behaviour of bubbles can involve an initial period of shrinkage before the asymptotic regime is established.     

Shear flow over a particulate or fibrous plate

Simple shear flow over a porous plate consisting of a planar array of particles is studied as a model of flow over a membrane. The main objective is to compute the slip velocity defined with reference to the velocity profile far above the plate, and the drift velocity induced by the shear flow underneath the plate. The difference between these two velocities is shown to be proportional to the thickness of the plate. When the geometry of the particle array is anisotropic, the directions of the slip and drift velocity are generally different from the direction of the overpassing shear flow. An integral formulation is developed to describe flow over a plate consisting of a periodic lattice of particles with arbitrary shape, and integral representations for the velocity and pressure are developed in terms of the doubly-periodic Green's function of three-dimensional Stokes flow. Based on the integral representation, asymptotic expressions for the slip and drift velocity are derived to describe the limit where the particle size is small compared to the inter-particle separation, and numerical results are presented for spherical and spheroidal particles of arbitrary size. The asymptotic results are found to be accurate over an extended range of particle sizes. To study the limit of small plate porosity, the available solution for shear flow over a plane wall with a circular orifice is used to describe flow over a plate with a homogeneous distribution of circular perforations, and expressions for the slip and drift velocity are derived. Corresponding results are presented for axial and transverse shear now over a periodic array of cylinders arranged distributed in a plane. Streamline pattern illustrations confirm that a negative drift velocity is due to the onset of eddies between closely-spaced particles.     

Direct numerical simulations of small disturbances in the classical Stokes layer

Direct numerical simulations are used to investigate the stability characteristics of the classical semi-infinite Stokes layer that is generated by an oscillating flat plate. The calculations are based upon a velocity–vorticity formulation of the governing equations which allows efficient tracking of the time evolution of disturbances. The neutral-stability curve for two-dimensional disturbances was computed and found to agree well with the predictions of earlier semi-analytical studies. Further previously determined properties of two-dimensional disturbances in a Stokes layer were also compared to the results obtained by direct numerical simulation. The two-dimensional neutral stability results were then used to validate the extension of the direct numerical solution code to three-dimensional disturbances in the form of oblique waves.     

圆柱绕流近壁面处气动噪声源识别研究

对物体高速行驶下的气动噪声现象的认识和描述一直以来都是气动声学领域探索的基本问题和难点问题,尤其对物体近壁面处声源的产生及其声辐射缺乏有效的描述手段。该研究以圆柱绕流为研究对象,结合数值仿真手段,基于涡声方程的声源项描述圆柱绕流近壁面处的声源特性,建立声源识别方法。研究表明,该方法描述的声源存在不该有声源的位置出现声源的现象。研究进一步基于质点振速的矢量波动方程,将不能辐射噪声的源分离,较为准确地识别出了圆柱绕流气动噪声源的大小和位置。该研究在探索识别圆柱绕流气动噪声源方法的同时,也为准确识别气动噪声源特征提供了有效的方法。     

Heat and mass transfer as a means of flow mode management in a supersonic boundary layer

This paper continues the studies cycle of flow management simulation methods in boundary layers of compressible gas. The influence of distributed heat and mass transfer on the stability characteristics of supersonic boundary layers is considered at Mach numbers M = 2.0 and 5.35. At high Mach numbers, waves of vortex nature and unstable acoustic oscillations emerge. Resistance to both types of disturbances is studied. Both normal injection, with normal mean velocity, V, being the only nonzero component, and injection at other angles, including tangential with the longitudinal component of mean velocity, U, being the only nonzero component on the wall, are simulated. It is shown that a tangential streamwise injection causes significant flow stabilization in relation to vortex and acoustic modes. This mode management provides thermal protection of the streamlined surface under aerodynamic heating, and is able to expand the laminar flow mode region. Cooled gas injection suppresses vortex disturbances and amplifies acoustic waves, while injected heated gas influences boundary layer stability in the opposite way. The performed studies anticipate that an injection of homogeneous cold gas would be similar to an extraneous heavy gas injection, and that injected heated gas would behave similarly to injected light gas.     

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.     

Initial water impact of a wedge at vertical and oblique angles

This paper examines initial asymmetric wedge-impact flows with horizontal as well as vertical impact velocity. The method of two-dimensional vortex distributions is employed to model the initial-boundary-value problem. The numerical analysis involves discretization of the body surface and an iterative solution technique. Experimental drop tests of a prismatic wedge were performed to gain understanding and provide data for comparison of initial water impact when asymmetry and horizontal impact velocity are present. The experimental investigation of initial flow separation off the wedge vertex (i.e., keel) during impact is described. Initial separation-ventilation of the flow from the vertex due to asymmetric impact or horizontal-vertical impact velocity is examined in relation to the present theory. Agreement between the data and the numerical predictions was demonstrated for small degrees of asymmetry and small ratios of horizontal to vertical impact velocity. The initial flow detachment from the vertex also revealed interesting hydrodynamic characteristics.     

Numerical Study on the Suppression of the Oscillating Wake of a Square Cylinder by a Traveling Wave Wall

Traveling waves generated on the side surfaces of a square cylinder are employed to suppress the oscillating wake for improving the flow behavior around a square cylinder; this method is termed the traveling wave wall (TWW) method. This study aimed to evaluate the influence of the key parameters of TWW on the control of aerodynamic forces and the oscillating wake of the flow around a square cylinder. Unsteady numerical analyses at a low Reynolds number (Re) of 100 were performed using a two-dimensional CFD simulation. First, the grid independence and time step independence tests of the simulation were conducted to verify the rationality of the solving parameter settings, and the validation of flow around the fixed square cylinder at Re =100 was carried out. Subsequently, the lift and drag coefficients and the vortex shedding modes under different combinations of three TWW control parameters, including wave velocity, wave amplitude, and wavenumber, were analyzed in detail. The results show that TWW can remarkably reduce the mean value of drag coefficient and the RMS value of the lift coefficient by more than 12% compared to the method involving a standard square cylinder. Two peaks occur in the lift coefficient spectrum, with the low frequency corresponding to the vortex shedding frequency in the wake of the flow around the square cylinder and the high frequency corresponding to the traveling wave frequency. The vorticity contours show that the alternating vortices in the wake of the square cylinder are not completely suppressed under the selected control parameters.     

A method of domain decomposition for calculating the steady flow past a cylinder

For flow past a cylinder it is known that the vorticity is only significant in a thin boundary-layer adjacent to the surface and within a parabolic wake far from the cylinder. To address this behaviour of the vorticity a numerical method is implemented whereby the flow field is decomposed into two regions: an inner region to deal with boundary-layer phenomena and an outer region to model wake phenomena. This method equally applies to any cylinder cross section. The equations of motion are solved in each region and are matched at the boundary. Numerical solutions have been carried out for the trial case of a circular cylinder and the agreement with existing results is good.     

Solution of three-dimensional viscous flows using integral velocity–vorticity formulation

In this paper, a numerical approach is presented to solve the velocity–vorticity integro-differential formulations for three-dimensional incompressible viscous flow. Both the velocity and pressure are solved in integral formulations and the general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so-called generalized Biot–Savart formula combined with a fast multipole algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well-known fractional step approaches are used to solve the vorticity transport equation. No-flux boundary conditions on solid objects are satisfied as vorticity Helmholtz equation is solved. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential-flow boundary condition. As an application example, the impulsively started flow through a sphere with different Reynolds numbers is computed using the method. The calculated results are compared with the experimental data and other numerical results and show good agreement.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

Oscillation of a floating body in a viscous fluid

The nonlinear viscous-flow problem associated with the heaving motion of a two-dimensional floating cylinder is considered. It is formulated as an initial-boundary-value problem in primitive variables and solved using a finite-difference method based on boundary-fitted coordinates. A fractional-step procedure is used to advance the solution in time. As a case study, results are obtained for a rectangular cylinder oscillating at a Reynolds number of 10³. The nonlinear viscous forces are compared with those of linear potential theory. An assessment on the importance of viscous and nonlinear effects is made. The solution technique is sufficiently robust that extensions to consider other single and coupled modes of motion are possible.     

Effects on vortex breakdown due to an abrupt change in the rotation of endwall in a disk-cylinder system

Summary The transient flow in a cylindrical enclosure with fixed sidewall is studied numerically. The initial motion due to uniform rotation of the lower endwall is disturbed by setting the top endwall to corotate impulsively with a small angular velocity. The flow parameter values are chosen so as to induce a vortex breakdown in the initial steady state. The unsteady rotationally-symmetric Navier-Stokes equations are solved iteratively using a combination of second-order and fourth order compact difference schemes. At higher values of the Reynolds number, upwind differencing is used in the convective terms. The breakdown bubble in the initial steady state occurs between the stationary top end wall and the mid-plane. The numerical soulution shows that at the early stage of the transient flow the breakdown bubble enlarges in size, and in subsequent time the bubble occurs between the mid-plane and the faster rotating endwall. The role of the azimuthal component of vorticity in the breakdown phenomena is analysed. The torque on the lower endwall is obtained at several values of non-dimensional time during the transient flow.     

Non-linear finite element analysis of large amplitude sloshing flow in two-dimensional tank

This paper is concerned with the accurate and stable finite element analysis of large amplitude liquid sloshing in two-dimensional tank under the forced excitation. The sloshing flow is formulated as an initial-boundary-value problem based upon the fully non-linear potential flow theory. The flow velocity field is interpolated from the velocity potential with second-order elements according to least square method, and the free surface conditions are tracked by making use of the direct time differentiation and the predictor–corrector method. Meanwhile, the liquid mesh is adapted such that the incompressibility condition is strictly satisfied. The accuracy and stability of the numerical method introduced are verified from the comparison with the existing reference solutions. As well, the numerical results are compared with those obtained by the linear theory with respect to the liquid fill height and the excitation amplitude. Copyright © 2004 John Wiley & Sons, Ltd.     

Laminar flow of a non-Newtonian fluid in channels with wall suction or injection

The one-dimensional approximate equation in the rectangular Cartesian coordinates governing flow of a non-Newtonian fluid confined in two large plates separated by a small distance of h, with the upper plate stationary while the lower plate is uniformly porous and moving in the x-direction with constant velocity, is derived by accounting for the order of magnitude of terms as well as the accompanying approximations to the full-blown three-dimensional equations by using scaling arguments, asymptotic techniques and assuming the cross-flow velocity is much less than the axial velocity. The one-dimensional governing equation for a power-law fluid flow confined between parallel plates, with the upper plate is stationary and the bottom plate subjected to sudden acceleration with a constant velocity in the x-direction and uniformly porous, is solved analytically for a Newtonian fluid case (n = 1) and numerically for various values of power-law index to determine the transient velocity and thus the overall transient velocity distribution. The effects of mass suction/injection at the porous bottom plate on the flow of non-Newtonian fluids are examined for various values of time and power-law index. The results obtained from the present analysis are compared with the data available in the literature.