Vortex dynamics would appear to be exempt from Hardy 's pe ssimi stic verdict On one hand, the evolution of vorticity, and thu s the motion s of vortice s, are essential ingredient s of virtually any real flow Hence vortex dynamic s i s of profound practical importance On the other hand, vortex motion ha s always constituted a mathematically sophi sticated branch of fluid mechanics that continue s to invite the application of novel analyti­ cal techniques Indeed it i s ne ither dull nor commonplace  Thi s central role of vorticity in fluid mechanic s i s not difficult to understand A s we know, any velocity field, v, can be split into a sum of two field s, one with the same divergence a s v, and no curl, and one with the same curl a s v and vani shing divergence Thi s important re sult i s due to Stoke s and to Helmholtz ( 1 858 ; see Sommerfeld 1964) In incompre ssi ­ ble flow , a s we deal with exclu sively here, the fir st part i s irrotational and divergencefree and thu s leads to the linear problem o f po tential flow , The second part, however , derive s directly from the vorticity o f the field ' v In the dynamics of thi s part lie s the essence of the problem

Integrable, chaotic, and turbulent vortex motion in two-dimensional flows

https://www.math.sciences.univ-nantes.fr/~saad/cv/computersfluidsoriginalarticle.pdf

The 2D lid-driven cavity problem revisited

A multistep technique with implicit difference schemes for calculating two- or three-dimensional cavity flows

We consider the flapping stability and response of a thin two-dimensional flag of high extensional rigidity and low bending rigidity. The three relevant non-dimensional parameters governing the problem are the structure-to-fluid mass ratio, μ = ρ s h /(ρ f L); the Reynolds number, Re=VL/ν; and the non-dimensional bending rigidity, K B = EI / (ρfV 2 L 3 ). The soft cloth of a flag is represented by very low bending rigidity and the subsequent dominance of flow-induced tension as the main structural restoring force. We first perform linear analysis to help understand the relevant mechanisms of the problem and guide the computational investigation. To study the nonlinear stability and response, we develop a fluid-structure direct simulation (FSDS) capability, coupling a direct numerical simulation of the Navier-Stokes equations to a solver for thin-membrane dynamics of arbitrarily large motion. With the flow grid fitted to the structural boundary, external forcing to the structure is calculated from the boundary fluid dynamics. Using a systematic series of FSDS runs, we pursue a detailed analysis of the response as a function of mass ratio for the case of very low bending rigidity (K B = to-4) and relatively high Reynolds number (Re=10 3 ). We discover three distinct regimes of response as a function of mass ratio μ: (I) a small μ regime of fixed-point stability; (II) an intermediate μ regime of period-one limit-cycle flapping with amplitude increasing with increasing μ; and (III) a large μ regime of chaotic flapping. Parametric stability dependencies predicted by the linear analysis are confirmed by the nonlinear FSDS, and hysteresis in stability is explained with a nonlinear softening spring model. The chaotic flapping response shows up as a breaking of the limit cycle by inclusion of the 3/2 superharmonic. This occurs as the increased flapping amplitude yields a flapping Strouhal number (St=2Af/V) in the neighbourhood of the natural vortex wake Strouhal number, St ≃ 0.2. The limit-cycle von Karman vortex wake transitions in chaos to a wake with clusters of higher intensity vortices. For the largest mass ratios, strong vortex pairs are distributed away from the wake centreline during intermittent violent snapping events, characterized by rapid changes in tension and dynamic buckling.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112007005307

Flapping dynamics of a flag in a uniform stream

A Pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations

https://iopscience.iop.org/article/10.1016/0169-5983(86)90014-6/pdf

Computation of high Reynolds number flow around a circular cylinder with surface roughness

The motions of a vortex tube with an elliptic cross section and of a vortex sheet of finite length in an inviscid, incompressible fluid are simulated by a number of discrete vortex filaments, each vortex moving under the action of the velocity field of all the other vortices. By the use of this simulation, rotation of the vortex tube and rolling-up of the vortex sheet are investigated numerically as initial value problems. Comparison with exact solutions shows the validity of this method of approximation. In order to improve the results, an artificial viscosity is introduced to the equations of motion. This diminishes randomization of vortices inherent to the treatment without viscosity and thus leads to regular rolling-up of a vortex sheet.

Numerical Studies of Two-Dimensional Vortex Motion by a System of Point Vortices

A new upwind scheme for computation of incompressible flow has been developed. It was found that this scheme works well at high Reynolds number even using limited number of mesh points.

New higher-order upwind scheme for incompressible Navier-Stokes equations

A finite-difference solution of the unsteady Navier-Stokes equations by the explicit use of the pressure and the velocity is presented. The aims are to test the scheme and to estimate the three-dimensional effect of the side walls. A 20×20×20 grid is used and the Reynolds numbers are 0, 100 and 400. The scheme works well except that it needs a very large amount of computation time. It is shown that the effect of the side walls is restricted to a relatively thin layer and the central part of the cavity is almost two-dimensional.

Numerical Study of Three-Dimensional Flow within a Cubic Cavity

Analyse du champ d'ecoulement autour d'une aile oscillante par resolution des equations de Navier-Stokes compressibles bidimensionnelles

Hideo Takami

Papers

Computation of high Reynolds number flow around a circular cylinder with surface roughness

Numerical Studies of Two-Dimensional Vortex Motion by a System of Point Vortices

New higher-order upwind scheme for incompressible Navier-Stokes equations

Numerical Study of Three-Dimensional Flow within a Cubic Cavity

Computation of dynamic stall of a NACA-0012 airfoil