Effect of the Ekman boundary layer on the evolution of vortex formations

In numerical calculations of vortex evolution the effect of nonlinear terms related with the parametrization of the Ekman bottom boundary layer in the barotropic vortex equation is demonstrated. It is shown that taking these terms into account can lead to the coalescence of mesovortices formed as a result of the barotropic-instability-induced breakdown of an annular vortex, as distinct from the conventional parametrization of the bottom friction by a linear term, when a symmetric system of mesovortices, or the so-named vortex crystal, is formed.     

Bifurcation of an elliptic vortex ring

Time-dependent vorticity fields of elliptic vortex rings of aspect ratios 2, 3 and 4 were measured by means of hot-wire anemometry. The time evolution of their vorticity fields was analyzed and the processes of vortex ring formation, advection, interaction and decay, and the mechanism of vortex bifurcation are studied. The following crosslinking model is proposed: A thick vortical region composed of many equivalent vortex filaments with distributed cores is initially formed at the orifice and they behave as inviscid filaments. The elliptic ring deforms and the end parts of its major axis get closer. Then, the vortex filaments interact at the touching point and the ring partially bifurcates. Almost simultaneously, turbulent spot appears at this point, and propagates around the ring cross section, thus preventing further bifurcation. And it becomes a turbulent blob. This model is also supported by numerical simulation by a high-order vortex method and the Navier-Stokes solution.     

Méthode particulaire anisotropeAnisotropic particle method

A particle method has been used to simulate the vorticity transport in a two-dimensional flow of an incompressible inviscid fluid. In this method, not only the location and the circulation of the particle are used but also the moments of the internal vorticity. The transport equation for these moments has been derived from the vorticity transport equation. The method has been compared to the usual particle method as well as to Teng's elliptic vortex model. The test case is that of the evolution of two circular patches of vorticity already used by Teng. To cite this article: A. Beaudoin et al., C. R. Mecanique 330 (2002) 51–56     

VISCOUS FLOWFIELD BASED ON DISCONTINUOUS BOUNDARY ELEMENT METHOD AND VORTEX METHOD

基于非协调边界元方法和涡方法的联合应用, 模拟了二维和三维黏性不可压缩流场. 计算中利用离散涡元对漩涡的产生、凝聚和输送过程进行模拟, 并将整体计算域分解为采用涡泡模拟的内部区域和用涡列模拟的数字边界层区域. 计算域中涡量场的拉伸和对流由Lagrangian涡方法模拟, 用随机走步模拟涡量场的扩散. 内部区域涡元涡量场速度由广义Biot-Savart公式计算, 势流场速度则采用非协调边界元方法计算. 非协调边界元将所有节点均取在光滑边界处, 从而避免了法向速度的不连续现象; 而对于系数矩阵不对称的大型边界元方程组,引入了非常高效的预处理循环型广义极小残余(the generalized minimum residual, GMRES)迭代算法, 使得边界元法的优势得到了充分发挥, 同时, 在内部涡元势流场计算中对近边界点采用了正则化算法, 该算法将奇异积分转化为沿单元围道上一系列线积分, 消除了势流计算中速度及速度梯度的奇异性. 二维、三维流场算例证明了所用方法的正确性, 也验证了该算法可以大幅度提高模拟精度和效率.     

Lateral Spreading Mechanism of a Turbulent Spot and a Turbulent Wedge

We examine the spreading of turbulent spots and wedges into a surrounding laminar Blasius boundary layer. The spreading is not due to the lateral propagation of turbulent eddies but rather to a developing disturbance in the surrounding spanwise vorticity of the laminar boundary layer. We concentrate on the mechanisms for generating streamwise vorticity. In particular, inclined generally streamwise vortex tubes along the spot/wedge boundary tilt mean shear vortex lines either up or down. These lines subsequently tend to either lag back or lead forward. As the leading or lagging vortex lines continue to wrap around and reinforce the causative inclined tube, the lines arch up or down. The outboard portion of the resulting arch must acquire a vertical, ω_y, component of vorticity which induces the rollup of a new inclined tube now outboard of the first. Close to the wall the arching mechanism is inhibited by the no through flow boundary condition while far from the wall the process is inhibited by the lack of sufficient mean spanwise vorticity.     

The N-vortex problem on a sphere: geophysical mechanisms that break integrability

We describe the dynamical system governing the evolution of a system of point vortices on a rotating spherical shell, highlighting features which break what would otherwise be an integrable problem. The importance of the misalignment of the center-of-vorticity vector associated with a cluster of point vortices with the axis of rotation is emphasized as a crucial factor in the interpretation of dynamical features for many flow configurations. We then describe two important physical mechanisms which break what would otherwise be an integrable problem—the interactions between the local center-of-vorticity vectors of more than one region of concentrated vorticity, and the coupling between the center-of-vorticity vector and the background vorticity field which supports Rossby waves. Focusing on the Polar vortex splitting event of September 2002, we describe simple (i.e., low dimensional) mechanisms that can trigger instabilities whose subsequent development cause the onset of chaotic advection and global particle transport. At the linear level, eigenvalues that oscillate between elliptic and hyperbolic configurations initiate the pinch-off process of a passive patch representing the Polar vortex. At the nonlinear level, the evolution and topological bifurcations of the streamline patterns are responsible for its further splitting, stretching, and subsequent transport over the sphere. We finish by briefly describing how to incorporate conservation of potential vorticity and the development of a model governing the probability density function associated with the point vortex system.     

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

A parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

Reconnections of vortex filaments

The process of break-down and reconnection of vortex filaments is considered by the method of three-dimensional vortex singularities (vortons) in various situations, including oblique interaction of a vortex ring with a boundary in shear flow, shedding of a vortex ring from a horseshoe vortex, instability of elliptic vortex ring, Crow instability of two perturbed antiparallel vortex filaments, merging and subsequent splitting of vortex rings. Special attention is paid to the global integrals (vorticity, momentum, angular momentum) and to the inviscid dissipation of energy. The visualization of the effective vortex core, created by the interference of the vorticity fields of vortons, is presented. The comparisons with other methods of simulation of three-dimensional vortex interactions and with the observations have been made.     

Vortex sound

Vortex motion is the only source of aerodynamic sound production in low Mach number flow: the unsteady part of the vorticity distribution contributes linearly to the sound field. The following fundamental model flows, which illustrate the vorticity as the predominant sound source in unsteady flows, are discussed: An initially planar elliptic vortex; two identical coaxial initially elliptic vortex rings, where a special case is the leap-frogging of two identical circular rings. For head-on collision of two identical circular vortex rings and for several cases of vortex-body interaction good agreement between theory and experiment exists. If the Mach number is not low, other mechanisms have also to be considered. Here the theory is not yet fully developed. Experimental results for a vortex-airfoil interaction in transonic flow show that local flow separation and boundary layer as well as compressibility effects play a basic role. However, if the motion of vorticity would be known in subsonic flow, essential parts of the sound field could be calculated by the theory. — In addition, it is shown that the general theory is well suited to provide a better understanding of the scattering of sound waves by vortex motion, at least for long wave lengths.     

Polymer flooding of oil formations with bottom water

A quasi-one-dimensional model describing the process of polymer flooding of oil formations underlain by bottom water is proposed. The model is based on the assumption of instantaneous gravitational phase separation along the vertical and is a generalization of the hydrodynamic model previously considered in [1]. The self-similar solutions constructed show that in the case of polymer flooding of an oil formation the presence of bottom water leads to qualitative changes in the saturation and concentration distribution and has an important influence on both the running and the final oil output. The results obtained are consistent with the data of two-dimensional numerical modeling of the process [2] and indicate the possibility of more efficient exploitation of water-oil zones as a result of pumping thickened water into the formation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 84–90, May–June, 1986.     

Experimental investigation of vortex rings impinging on inclined surfaces

Vortex–ring interactions with oblique boundaries were studied experimentally to determine the effects of plate angle on the generation of secondary vorticity, the evolution of the primary vorticity and secondary vorticity as they interact near the boundary, and the associated energy dissipation. Vortex rings were generated using a mechanical piston-cylinder vortex ring generator at jet Reynolds numbers 2,000–4,000 and stroke length to piston diameter ratios (L/D) in the range 0.75–2.0. The plate angle relative to the initial axis of the vortex ring ranged from 3 to 60°. Flow analysis was performed using planar laser-induced fluorescence (PLIF), digital particle image velocimetry (DPIV), and defocusing digital particle tracking velocimetry (DDPTV). Results showed the generation of secondary vorticity at the plate and its subsequent ejection into the fluid. The trajectories of the centers of circulation showed a maximum ejection angle of the secondary vorticity occurring for an angle of incidence of 10°. At lower incidence angles (<20°), the lower portion of the ring, which interacted with the plate first, played an important role in generation of the secondary vorticity and is a key reason for the maximum ejection angle for the secondary vorticity occurring at an incidence angle of 10°. Higher Reynolds number vortex rings resulted in more rapid destabilization of the flow. The three-dimensional DDPTV results showed an arc of secondary vorticity and secondary flow along the sides of the primary vortex ring as it collided with the boundary. Computation of the moments and products of kinetic energy and vorticity magnitude about the centroid of each vortex ring showed increasing asymmetry in the flow as the vortex interaction with the boundary evolved and more rapid dissipation of kinetic energy for higher incidence angles.     

Two exact solutions of the Navier-stokes equations for the concentration of eddies in a viscous liquid

In the final analysis, vorticity in a liquid or gas is broken down by viscosity [1]; however, there are known cases of the appearance and long-term existence of three-dimensional eddies in water, air, and other media. Therefore, the conditions under which vorticity can even rise with viscosity are of interest. For example, with the flow of a liquid out of an opening in the bottom of a rotating cylindrical vessel, the total momentum with respect to the vertical axis of the vessel increases with the time [2, 3]. For some flows, there exist contradictory opinions: In [4, 5] it is asserted that an eddy around a flat sink in a viscous liquid is damped, while, in [6, 7], it is argued that, with determined Reynolds numbers, there is an increase in the vorticity around a sink. The present article gives exact solutions of the Navier—Stokes equations, demonstrating the development of eddies in a viscous liquid.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 77–81, November –December, 1977.     

NUMERICAL PREDICTION OF FORCE ON RECTANGULAR CYLINDERS IN OSCILLATING VISCOUS FLOW

Numerical results are presented for an oscillating viscous flow past a square cylinder with square and rounded corners and a diamond cylinder with square corners at Keulegan–Carpenter numbers up to 5. This unsteady flow problem is formulated by the two-dimensional Navier–Stokes equations in vorticity and stream-function form on body-fitted coordinates and solved by a finite-difference method. Second-order Adams-Bashforth and central-difference schemes are used to discretize the vorticity transport equation while a third-order upwinding scheme is incorporated to represent the nonlinear convective terms. Since the vorticity distribution has a mathematical singularity at a sharp corner and since the force coefficients are found in experiments to be sensitive to the corner radius of rectangular cylinders, a grid-generation technique is applied to provide an efficient mesh system for this complex flow. Local grid concentration near the sharp corners, instead of any artificial treatment of the sharp corners being introduced, is used in order to obtain high numerical resolution. The elliptic partial differential equation for stream function and vorticity in the transformed plane is solved by a multigrid iteration method. For an oscillating flow past a rectangular cylinder, vortex detachment occurs at irregular high frequency modes at KC numbers larger than 3 for a square cylinder, larger than 1 for a diamond cylinder and larger than 3 for a square cylinder with rounded corners. The calculated drag and inertia coefficients are in very good agreement with the experimental data. The calculated vortex patterns are used to explain some of the force coefficient behavior.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

A New Class of Equations for Rotationally Constrained Flows

The incompressible Navier–Stokes equation is considered in the limit of rapid rotation (small Ekman number). The analysis is limited to horizontal scales small enough so that both horizontal and vertical velocities are comparable, but the horizontal velocity components are still in geostrophic balance. Asymptotic analysis leads to a pair of nonlinear equations for the vertical velocity and vertical vorticity coupled by vertical stretching. Statistically stationary states are maintained against viscous dissipation by boundary forcing or energy injection at larger scales. For thermal forcing direct numerical simulation of the reduced equations reveals the presence of intense vortical structures spanning the layer depth, in excellent agreement with simulations of the Boussinesq equations for rotating convection by Julien et al. (1996). Received 30 May 1997 and accepted 4 January 1998     

A revised slug model boundary layer correction for starting jet vorticity flux

The flux of vorticity from a piston-cylinder vortex generator is commonly approximated using a model in which the fluid efflux is treated as a uniform slug of fluid with negligible boundary layer thickness. Shusser et al. (2002) introduced a correction to the slug model that accounts for boundary layer growth within the cylinder. We show that their implemented boundary layer solution contains an error, leading to an underestimate of the calculated boundary layer growth. We present a corrected model that agrees more closely with experimental measurements of starting jet vorticity flux and vortex ring core thickness.     

The shedding of vorticity from a smooth surface

This work is concerned with the development of a numerical model for laminar flow separation from a smooth boundary. The concept of irreversible vorticity generation is used to formulate an algorithm to predict both the location of separation and the shedding rate of vorticity. Results of flow separation from a circular cylinder are presented. The calculated vortex sheets are visualized as streak lines in the wake of the cylinder. Streamline patterns are constructed from these calculations showing location of separation and wake structure.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.