Singularities Of Flows Research Articles

Unsteady singularities of Stokes' flows in two dimensions

The accelerated motion of a body in a two-dimensional Stokes flow of a viscous incompressible fluid is considered. The basic fundamental singular solutions due to an instantaneously introduced point force and point couple are derived. The utility of these building blocks, the unsteady stokeslet (for rectilinear acceleration) and the unsteady rotlet (for rotational acceleration) is demonstrated by constructing (new) solutions to several time dependent plane viscous flow problems in the region exterior to a circular cylinder. It is shown, for example, that the disturbance caused by a circular cylinder moving with a velocity U(t) in a fluid otherwise at rest is produced by a time distribution of the unsteady stokeslets and unsteady inviscid doublets of suitable strengths at the centre of the cylinder. Exact closed form solutions are presented for certain special motions of the cylinder and the force and the torque on the body are analysed with the help of Faxen laws which are modified for the unsteady case.

Oscillatory line singularities of stokes' flows

The fundamental singular velocity and pressure fields generated by the presence of an isolated force and a couple acting at a point in a two dimensional unbounded viscous incompressible medium executing oscillatory motions are obtained. These line singularities, the stokeslet and the rollet, are used to construct new solutions for certain oscillatory plane viscous flow problems in the region exterior to a circular cylinder.

Evolution of singularities of potential flows in collision-free media and the metamorphosis of caustics in three-dimensional space

The authors describe the critical values of the maps at time''t'' and their evolution as ''t'' changes for potential initial velocity fields in general position under the assumption that the force field is potential. The paper is concerned with the structure and evolution of caustics of a general one-parameter family of Lagrangian maps of manifolds of dimension not exceeding three. For each type of evolution, the authors give a detailed geometric description of the structure of the singularity. The investigation required new algebraic information about the manifold of polynomials with multiple roots; these are given in the paper.

Modelling turbulent flows in a plane channel

Two- and three-dimensional non-stationary flows of incompressible viscous fluid in a plane infinite channel are considered. Using efficient computing algorithms based on the Navier-Stokes equations, the flows are modelled in significant time intervals. The classes of two- and three-dimensional non-stationary flows with established integral characteristics are found. Two-dimensional flows only qualitatively reflect the typical singularities of the turbulent flows observed in experiments. Three-dimensional flows are in good agreement (even quantitatively) with turbulent flows. The basic characteristics of turbulent flows (the coefficient of resistance, mean-velocity profile, and the distribution of mean square pulsations of the velocity components) are reproduced with good accuracy in the calculated three-dimensional flows.

Beyond elementary catastrophe theory

We give a brief discussion of the relations between elementary catastrophe theory, general catastrophe theory, singularity theory, bifurcation theory, and topological dynamics. This is intended to clarify the status, and potential applicability, of “catastrophe theory,” a phrase used by different authors and at different times with different meanings. Catastrophe theory has often been criticized for (supposed) applicability only to gradient systems of differential equations; but properly speaking this criticism can apply only to the elementary version of the theory (where it is in any case wrong). Roughly speaking, elementary catastrophe theory deals with the singularities of real-valued functions, general catastrophe theory with singularities of flows. Between these lies singularity theory, which deals with vector-valued functions. All relate strongly to bifurcation theory and topological dynamics. The issue is more subtle than it appears to be, and we describe an example where elementary catastrophe theory has been used to solve a long-standing problem about nongradient flows: degenerate Hopf bifurcation.

Two-dimensional mixed flows of a supersaturated and two-phase medium with nonequilibrium phase transitions

The singularities of two-dimensional interchannel flows of a condensing and damp vapor with nonequilibrium phase transitions which contain gas-dynamic discontinuities are investigated. A through-computation difference method is constructed for such flows. The results of the numerical investigation of steam flows with spontaneous condensation in supersonic plane nozzles containing a break in the walls and flow around a wedge are represented. It is shown that nonequilibrium condensation can result in a qualitative rearrangement of the wave structure of the flow which is impossible to obtain within the framework of the one-dimensional approach.

Investigation of nonequilibrium two-phase flows in axisymmetric laval nozzles

The singularities of two-phase flows in Laval nozzles were investigated within the framework of the model of a two-fluid continuous medium [1, 2] mainly in a quasi-one-dimensional approximation ([3] and the bibliography therein). Two-dimensional computations of such flows were performed only recently by using the method of buildup [4–7]. However, systematic computations to clarify the influence of the second phase on such fundamental nozzle characteristics as the magnitude of the specific impulse, its losses, and discharge coefficient were performed only in the quasi-one-dimensional approximation [8, 9] and only for the supersonic parts of the nozzle in the two-dimensional approximation under the assumption of uniform flow in the throat [10, 3]. Such an investigation is performed in this paper in the two-dimensional case for the nozzle as a whole, including the sub-, trans-, and supersonic flow domains, and a comparative analysis is given of the magnitudes of the loss of a unit pulse obtained in the quasi-one-dimensional approximation [8].

Singularities of supersonic flows around pointed bodies with a detached shock

Results are presented of a numerical investigation of the axisymmetric flow around a family of bodies with a pointing angle for which the shock is detached. It is shown that the supersonic part of the stream remains the same for all bodies of the family for a fixed value of M∞ despite the fact that the shaper of the subsonic zone is related quite strongly to the pointing angle Β*. The dependence of the shock standoff and its radius of curvature on the spreading line on the body shape is studied. Effects inherent in flows around sharply pointing bodies are discussed. A dimensionless parameter characterizing each body of the family under consideration is introduced and used to establish general flow regularities. Data illustrating the possibility of applying such parametrization are analyzed for a wider class of pointing bodies.