Initial Vorticity Distribution Research Articles

Unsteady Circulatory Flow about a Circular Cylinder with Suction or Injection

An exact solution of the Navier-Stokes equations is presented for an incompressible circulatory flow about an infinite circular cylinder with uniform suction or injection. An asymptotic evaluation of the solution for large time shows that, starting from a typical initial state, the flow ultimately becomes irrotational for R ≦1; for 1 2, the initial vorticity distribution changes with the circulation at infinity remaining constant: Here R = V a /ν, with V , a and ν being the suction velocity, radius of the cylinder and the kinematic viscosity, respectively. The torque on the cylinder has been evaluated for both small and large times.

Navier-Stokes calculations for unsteady three-dimensional vortical flows in unbounded domains

Finite-difference Navier-Stokes calculations for unsteady, three-dimensional, incompressible, viscous flows induced by initial vorticity distributions are presented and discussed in this paper. The initial vorticity distributions are assumed to be embedded in a flow field of infinite extent that is quiescent at infinity. These vorticity distributions are typical of vortex rings and other closed vortical tubes or structures. Such structures are important elements in fluid flows such as jets, atmospheric convection and the far-field wakes of aircraft; studies of their interaction may aid in an understanding of complex fluid flows. The calculations employ a method recently proposed by Ting to approximate the infinite-domain boundary value problem with a finite boundary computational domain, and this method is shown to yield accurate three-dimensional results for reasonable expenditures of computer time. Because of the efficiency of the boundary condition technique and the resulting Navier-Stokes code, a 16-bit minicomputer with virtual memory was capable of performing the calculations for the unsteady motion of two obliquely colliding vortex rings. The results of these calculations are presented in the paper.

Starting flow for an obstacle moving transversely in a rapidly rotating fluid

The initial-value problem for Taylor columns is considered for the oceanographically relevant case of slow flow over obstacles of small slope, and horizontal scale of order of the fluid depth or larger. In such flows depth variations cause, in the neighbourhood of the obstacle, a gradient of background potential vorticity. Topographic Rossby waves, forced in the starting flow, propagate across the gradient, cycling the obstacle in times of order of the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the obstacle height. The linear inertialwave equation and the quasigeostrophic equations are related by considering the case where the inertial period is small compared with the topographic time, which is in turn small compared with the advection time.When viscosity is present the waves decay to a steady motion in a time of order of the viscous spin-up time. In inviscid flow the waves do not decay. The flow oscillates about steady, irrotational, zero-circulation flow. The drag and lift on the obstacle oscillate with almost constant frequency and amplitude equal to the Coriolis force on a body of fluid whose volume matches that of the obstacle. Particles above a right cylinder move in closed orbits of diameter 1/S, where S is the topographic parameter introduced by Hide (1961). Particles away from the cylinder move irrotationally, but asymmetrically, from upstream to downstream. A shear layer is present at the boundary of the cylinder throughout the motion. The initial vorticity distribution for flow over a paraboloid is everywhere finite. However, regions of positive and negative vorticity interlace ever more closely during the motion, causing at large time a similar shear layer at the column boundary. By this time the vorticity is increasing without bound above the obstacle and neglected nonlinear, horizontal viscous, or vertical effects are important.

Continuous Mode in the Theory of the Barotropic Instability of Zonal Flows

Evolution of the two-dimensional small disturbance superimposed on a viscous parallel shear flow is investigated under the beta-plane approximation of revolution. Formal solution is obtained for an arbitrarily localized initial distribution of vorticity. The eigenfunctions of the normal mode are proved to compose a complete set. In the region where the basic flow is uniform the eigenfunction of the continuous mode behaves as the Rossby wave. The interaction between the shear and the Rossby wave is found to be an inherent character of the eigen function of the continuous mode. The properties of the critical latitude of the Rossby wave are attributable to those of the friction layer in the eigenfunction.

On the application of the integral invariants and decay laws of vorticity distributions

Unsteady three-dimensional incompressible viscous flow fields induced by initial vorticity distributions are studied. Relevant invariants and decay laws of the moments of vorticity distributions are presented and shown to be useful in the numerical calculation of flow fields in two ways. First, the moments determine the leading terms of the far-field velocity, which can be employed as boundary conditions for the numerical calculation. Secondly, the deviations of the numerical results from the invariants and the decay laws can be used to measure the error of the numerical solution.

Evolution of axisymmetric vorticity distributions in an ideal incompressible stratified liquid

The Cauchy problem is solved for axisyimmetric vortex perturbations of an exponentially stratified incompressible ideal liquid. The behaviour of vorticity inside the region of its initial location, near the boundary of that region, and away from it in the “wave” zone is studied. A number of examples and analyzed with a specific initial distribution of vorticity, among which are examples of anomalous solution behaviour. It is shown that the initial jump of vorticity in a stratified liquid does not vanish, but oscillates at a frequency which depends on the direction. When passing to the limit of strongly singular initial distributions of the vortex filament and cylindrical vortex layer types, the solution increases with time.

The motion of a large gas bubble rising through liquid flowing in a tube

The theory presented here describes the motion of a large gas bubble rising through upward-flowing liquid in a tube. The basis of the theory is that the liquid motion round the bubble is inviscid, with an initial distribution of vorticity which depends on the velocity profile in the liquid above the bubble. Approximate solutions are given for both laminar and turbulent velocity profiles and have the form \begin{equation} U_s = U_c+(gD)^{\frac{1}{2}}\phi(U_c/(gD)^{\frac{1}{2}}), \end{equation}Us being the bubble velocity, Uc the liquid velocity at the tube axis, g the acceleration due to gravity, and D the tube diameter; ϕ indicates a functional relationship the form of which depends upon the shape of the velocity profile. With a turbulent velocity profile, a good approximation to (1) which is suitable for many practical purposes is \begin{equation} U_s = U_s + U_{s0}, \end{equation}Us0 being the bubble velocity in stagnant liquid. Published data for turbulent flow are known to agree with (2), so that in this case the theory supports a well-known empirical result. Our laminar flow experiments confirm the validity of (1) for low liquid velocities.

A numerical study of 2-D turbulence

A simulation of 2-D turbulence in a square region with periodic boundary conditions has been performed using a highly accurate approximation of the inviscid Navier-Stokes equations to which a modified viscosity has been added. A series of flow pictures show how a random initial vorticity distribution quickly assumes a stringlike pattern which persists as the flow simplifies into a few “cyclones” or “finite area vortex regions”. This trend towards well-defined large-scale structures can make it questionable if the 2-D flow should be described as “turbulent” and it casts some doubts on the concept of inertial range and the relevance of energy spectra. The change in appearance seems to be associated with a buildup of phase correlations in the Fourier representation of the vorticity field. During this initial buildup, the energy spectrum seems to follow a k −3-law, but this behavior does not persist. If there is a power law for steady turbulence the results suggest that is more likely to be a k −4-law.

Fluid flow generated by switching a magnetic field on or off

A uniform magnetic field is switched on at time t = 0 outside a body of conducting fluid. It is assumed that the field strength increases in time in proportion to 1 -e−αt, where α is a constant of the circuit generating the field. Under the assumption of small magnetic Reynolds number and small magnetic Prandtl number the equations governing the diffusion of the field into the fluid are derived and a simple expression is given for the initial vorticity distribution produced in the fluid. The situation in which an initially uniform field is switched off is also considered. It is shown that, for sufficiently symmetrically shaped bodies of fluid, the vorticity generated by the switching-on of the field is the same as that generated by the switching-off. The particular case of an infinitely long circular cylinder of conducting fluid is considered in detail and an explicit expression is derived for the vorticity distribution.

Motion of decaying vortex rings with non-similar vorticity distributions

The motion and decay of circular vortex rings with an inner viscous core is considered by systematic matching of inner and outer asymptotic expansions. The governing Navier-Stokes equations are reduced to a coupled integro-differential system. A method of construction of solutions for the integro-differential system is presented. The initial vorticity distribution may be non-similar. Also presented is a method for introducing a time shift which makes the first term in the series solution for the vorticity to be the “best” approximation. The analysis is then applied to the motion and decay of a pair of coaxial vortex rings.