Contour Dynamics with Non-uniform Background Vorticity

In this paper it is demonstrated how a contour dynamics method can be used to simulate the behaviour of vortices in the presence of non-uniform background vorticity in general, and on the γ-plane in particular. For standard contour dynamics in case of zero, or uniform background vorticity, the initial continuous vorticity distributions of the vortices are replaced by appropriate piecewise-uniform distributions. Then, the evolution of the contours separating the several regions of uniform vorticity, are followed in time. In the case of non-uniform background vorticity, it is necessary to replace the sum of the (relative) vorticity of the vortices and the background vorticity by a piecewise-uniform distribution. This has several consequences for applying the method of contour dynamics, which are discussed in this paper. The resulting method is tested on some numerical examples. One of them is (qualitatively) compared with laboratory experiments carried out in a rotating tank.     

Numerical study of an inviscid incompressible flow through a channel of finite length

A two‐dimensional inviscid incompressible flow in a rectilinear channel of finite length is studied numerically. Both the normal velocity and the vorticity are given at the inlet, and only the normal velocity is specified at the outlet. The flow is described in terms of the stream function and vorticity. To solve the unsteady problem numerically, we propose a version of the vortex particle method. The vorticity field is approximated using its values at a set of fluid particles. A pseudo‐symplectic integrator is employed to solve the system of ordinary differential equations governing the motion of fluid particles. The stream function is computed using the Galerkin method. Unsteady flows developing from an initial perturbation in the form of an elliptical patch of vorticity are calculated for various values of the volume flux of fluid through the channel. It is shown that if the flux of fluid is large, the initial vortex patch is washed out of the channel, and when the flux is reduced, the initial perturbation evolves to a steady flow with stagnation regions. Copyright © 2008 John Wiley & Sons, Ltd.     

Solution to transient Navier–Stokes equations by the coupling of differential quadrature time integration scheme with dual reciprocity boundary element method

The two‐dimensional time‐dependent Navier–Stokes equations in terms of the vorticity and the stream function are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in space with the differential quadrature method (DQM) in time. In DRBEM application, the convective and the time derivative terms in the vorticity transport equation are considered as the nonhomogeneity in the equation and are approximated by radial basis functions. The solution to the Poisson equation, which links stream function and vorticity with an initial vorticity guess, produces velocity components in turn for the solution to vorticity transport equation. The DRBEM formulation of the vorticity transport equation results in an initial value problem represented by a system of first‐order ordinary differential equations in time. When the DQM discretizes this system in time direction, we obtain a system of linear algebraic equations, which gives the solution vector for vorticity at any required time level. The procedure outlined here is also applied to solve the problem of two‐dimensional natural convection in a cavity by utilizing an iteration among the stream function, the vorticity transport and the energy equations as well. The test problems include two‐dimensional flow in a cavity when a force is present, the lid‐driven cavity and the natural convection in a square cavity. The numerical results are visualized in terms of stream function, vorticity and temperature contours for several values of Reynolds (Re) and Rayleigh (Ra) numbers. Copyright © 2008 John Wiley & Sons, Ltd.     

Existence of Vortex Sheets with¶Reflection Symmetry in Two Space Dimensions

The main purpose of this work is to establish the existence of a weak solution to the incompressible 2D Euler equations with initial vorticity consisting of a Radon measure with distinguished sign in H ^{? 1}, compactly supported in the closed right half-plane, superimposed on its odd reflection in the left half-plane. We make use of a new a priori estimate to control the interaction between positive and negative vorticity at the symmetry axis. We prove that a weak limit of a sequence of approximations obtained by either regularizing the initial data or by using the vanishing viscosity method is a weak solution of the incompressible 2D Euler equations. We also establish the equivalence at the level of weak solutions between mirror symmetric flows in the full plane and flows in the half-plane. Finally, we extend our existence result to odd L ¹ perturbations, without distinguished sign, of our original initial vorticity.     

Extremum problems of boundary control for steady equations of thermal convection

An inverse extremum problem of boundary control for steady equations of thermal convection is considered. The cost functional in this problem is chosen to be the root-mean-square deviation of flow velocity or vorticity from the velocity or vorticity field given in a certain part of the flow domain; the control parameter is the heat flux through a part of the boundary. A theorem on sufficient conditions on initial data providing the existence, uniqueness, and stability of the solution is given. A numerical algorithm of solving this problem, based on Newton’s method and on the finite element method of discretization of linear boundary-value problems, is proposed. Results of computational experiments on solving extremum problems, which confirm the efficiency of the method developed, are discussed.     

平行涡管三维合并的直接数值模拟

采用拟谱方法求解Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，对两个同向平行涡管的三维合并过程进行直接数值模拟，分析了它的不同于二维合并的“缠绕式”特性，特别地，对在合并过程中产生的垂直于初始涡管方向的非主体涡量的意义和物理机制了探讨。     

Direct measurement of the vorticity field in digital particle images

A new method called direct measurement of vorticity (DMV) for digital particle images is described in this paper. Unlike previous methods for calculating the vorticity in particle image velocimetry (PIV), the vorticity is determined directly from the average angular displacement of rotation between two matched patterns. In order to improve the stability and precision of the angular displacement, polar coordinates are used instead of Cartesian coordinates to depict gray level patterns. The results of a Monte Carlo simulation of an Oseen-vortex flow indicate that the accuracy of the DMV method is independent of the spatial resolution of the velocity sampling, and the errors in the velocity field will not be propagated into the vorticity field as occurs with some finite difference methods. Therefore, the DMV method is a good method to be used to extract the vorticity field from velocity data that has higher levels of uncertainty. Received: 26 November 1999/Accepted: 30 October 2000     

Chebyshev spectral method and Chebyshev noise processing procedure for vorticity calculation in PIV post-processing

Chebyshev spectral method and Chebyshev noise processing procedure are proposed for the calculation of vorticity from PIV experimental data. The Chebyshev spectral method offers superior intrinsic accuracy of derivative calculations. To overcome its noise sensitivity, the Chebyshev noise processing procedure can be applied prior to the derivative calculation to remove the high-frequency noise in the Chebyshev transform space. We compare the Chebyshev spectral method against the least-squares approach and test their performance in the calculation of vorticity with an Oseen vortex and with PIV data of the wake of a trapezoidal mixing tab. It is found that for clean velocity data the Chebyshev spectral method is extremely accurate. However, the Chebyshev spectral method alone is found to be more sensitive to noise than the least-squares method. When the Chebyshev noise processing procedure is applied together with the Chebyshev spectral method it greatly reduces the error and makes the Chebyshev spectral method more accurate than the least-squares method for a wide range of vorticity values. A special requirement imposed by the Chebyshev spectral method is that the PIV velocity processing must be carried out on special grids such as Gauss–Lobatto points.     

Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains

We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L^p for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Approximate simulation of stratospheric flow from aircraft data

We show how stratospheric data collected by an aircraft along its flight path can be used to initialize steady state approximate simulations of the stratospheric flow. To this end we reformulate the initial value problem for Boussinesq equations as a large system of stiff ordinary differential equations (using the method of lines). Initial conditions for this system are derived from the aircraft data. As a result we are able to compute the Brunt–Vaisala frequency, Richardson number and the vorticity in the vicinity of the flight path. Copyright © 2003 John Wiley & Sons, Ltd.     

On the Two-Dimensional Euler Equations with Spatially Almost Periodic Initial Data

We consider the nonstationary Euler equations in \mathbbR²{\mathbb{R}}^2 with almost periodic unbounded vorticity. We show that a unique solution is always spatially almost periodic at any time when the almost periodic initial data belongs to some function space. In order to prove this, we demonstrate the continuity with respect to initial data which do not decay at spatial infinity. The proof of the continuity with respect to initial data is based on that of Vishik’s uniqueness theorem.     

Numerical simulation of unsteady interaction of centrifugal impeller with its diffuser using Lagrangian discrete vortex method

In this study, an advanced Lagrangian vortex- boundary element method is applied to simulate the unsteady impeller-diffuser interactions in a diffuser pump not only for design but also for off-design considerations. In velocity calculations based on the Biot-Savart law we do not have to grid large portions of the flow field and the calculation points are concentrated in the regions where vorticity is present. Lagrangian representation of the evolving vorticity field is well suited to moving boundaries. An integral pressure equation shows that the pressure distribution can be estimated directly from the instantaneous velocity and vorticity field. The numerical results are compared with the experimental data and the comparisons show that the method used in this study can provide us insight into the complicated unsteady impeller-diffuser interaction phenomena in a diffuser pump.     

Vortex stretching and filaments

This paper presents a new technique to produce controlled stretched vortices. Intense elliptical vortices are created by stretching of an initial vorticity sheet. The initial vorticity comes from a laminar boundary layer flow and the stretching is parallel to the vorticity vectors. This low velocity flow enables direct observation of the formation and destabilization of vortices. Visualizations are combined with quasi-instantaneous measurements of a full velocity profile. The velocity profile is obtained with an ultrasonic pulsed Doppler velocimeter. The evolution of the central diameter of the vortices is related to the stretching. It is observed that destabilization occurs by pairing of two vortices, by hairpin deformation, and by breakdown of vortices into a “coil shape”.     

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.     

On Spatial Decay Estimates for Derivatives of Vorticities with Application to Large Time Behavior of the Two-Dimensional Navier–Stokes Flow

In this paper we establish spatial decay estimates for derivatives of vorticities solving the two-dimensional vorticity equations equivalent to the Navier–Stokes equations. As an application we derive asymptotic behaviors of derivatives of vorticities at time infinity. It is well known by now that the vorticity behaves asymptotically as the Oseen vortex provided that the initial vorticity is integrable. We show that each derivative of the vorticity also behaves asymptotically as that of the Oseen vortex.      

基于FTM方法的二维Kelvin-Helmholtz不稳定性数值模拟

使用界面跟踪法FTM（Front Tracking Method）对二维不混溶、不可压缩流体的K-H（Kelvin-Helmholtz）不稳定性进行数值模拟。研究表明,速度梯度层越厚,界面在水平分量中移动越快,卷起越少;初始水平速度差越大,界面卷起越多,内扰动增长速度越快,K-H不稳定性的特征形式更加明显;此外,在Neumann边界条件（即无滑移边界条件）下界面的扰动发展得比Dirichlet边界条件（即对称边界条件）下的扰动快。由于Dirichlet边界中的边界层,在开始时刻涡量扩展到两侧,影响了K-H不稳定性的生长速率;而在Neumann边界条件下涡量由于初始水平速度差,在界面中心聚集。最后,研究了不同边界条件下各种理查德森数对K-H不稳定性的影响。     

A stream function–vorticity formulation‐based immersed boundary method and its applications

A new stream function–vorticity formulation‐based immersed boundary method is presented in this paper. Different from the conventional immersed boundary method, the main feature of the present model is to accurately satisfy both governing equations and boundary conditions through velocity correction and vorticity correction procedures. The velocity correction process is performed implicitly based on the requirement that velocity at the immersed boundary interpolated from the corrected velocity field accurately satisfies the nonslip boundary condition. The vorticity correction is made through the stream function formulation rather than the vorticity transport equation. It is evaluated from the firstorder derivatives of velocity correction. Two simple and efficient ways are presented for approximation of velocity‐correction derivatives. One is based on finite difference approximation, while the other is based on derivative expressions of Dirac delta function and velocity correction. It was found that both ways can work very well. The main advantage of the proposed method lies in its simple concept, easy implementation, and robustness in stability. Numerical experiments for both stationary and moving boundary problems were conducted to validate the capability and efficiency of the present method. Good agreements with available data in the literature were achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Uncertainty in calculating vorticity from 2D velocity fields using circulation and least-squares approaches

The most common method for determining vorticity from planar velocity information is the circulation method. Its performance has been evaluated using a plane of velocity data obtained from a direct numerical simulation (DNS) of a three dimensional plane shear layer. Both the ability to reproduce the vorticity from the exact velocity field and one perturbed by a 5% random uncertainty were assessed. To minimize the sensitivity to velocity uncertainties, a new method was developed using a least-squares approach. The local velocity data is fit to a model velocity field consisting of uniform translation, rigid rotation, a point source, and plane shear. The least-squares method was evaluated in the same manner as the circulation method. The largest differences between the actual and calculated vorticity fields were due to the filter-like nature of the methods. The new method is less sensitive to experimental uncertainty. However the circulation method proved to be slightly better at reproducing the DNS field.The least-squares method provides additional information beyond the circulation method results. Using the correlation and a vorticity threshold criteria to identify regions of rigid rotation (or eddies), the rigid rotation component of the least-squares method indicates these same regions.The authors thank Dr. Michael Rogers of NASA Ames Research Center for supplying the DNS fields. In addition, Professor Ellen Longmire at the University of Minnesota asked how else might one calculate vorticity other than the circulation method and prompted this investigation.     

Evolution of a Viscous Vortex Ring

The velocity field inside a viscous vortex ring and its overall velocity are obtained on the basis of an existing solution of the Stokes equation for the vorticity in a moving coordinate system. It is shown that these characteristics of the motion relate the results obtained earlier for limiting times and are confirmed by the experimental data. The effect of the initial Reynolds number on the development of the vortex ring is analyzed by constructing entrainment diagrams.