FOUR NORMAL SIZE LIMIT CYCLES IN TWO-DIMENSIONAL QUADRATIC SYSTEMS

The existence criterion of three normal size limit cycles in quadratic systems with a weak focus of first order is obtained. Further, giving a finite disturbance for weak focus, the fourth normal size limit cycle is obtained. Bifurcation of appearance of two limit cycles via semistable cycle is given.

Prediction of Pressure Pulsation for the Reciprocating Compressor System Using Finite Disturbance Theory

Pressure pulsations in the piping system of the reciprocating compressor produce excessive noise and even lead to damage in piping and machinery. Therefore, it is very important to predict precisely the pressure pulsation with large amplitude in the piping system. In this paper, the finite disturbance theory is used to solve the nonlinear partial differential equations that describe the unsteady one-dimensional compressible flow in the complex piping system. The solution is then compared with experimental results. The comparison shows that the finite theory fits the large pressure disturbance more precisely than the acoustic theory.

On Symmetric Buckling of a Finite Flat-Lying Heavy Sheet

An elastic sheet with non-negligible density and finite length lies horizontally on the ground. The ends are clamped and subjected to compressive forces. Depending on the force, the sheet may be regarded as “long” or “short” with different characteristics. The critical buckling load, redefined as the force below which the sheet will always return to the horizontal state under any finite disturbance, is higher than the Euler buckling load of a weightless sheet. When deflections are small and finite the sheet is stable for given end displacement, but is unstable for given force. Approximate analytic solutions compare well with the results of exact numerical integration.

The nonlinear behaviour of a constant vorticity layer at a wall

The so-called ‘water-bag’ method is used to study the behaviour of a two-dimensional inviscid layer of constant vorticity ω and of mean thickness δ adjacent to a wall with slip at the wall. A nonlinear initial-value equation is derived which describes the motion of the material interface separating the rotational fluid within the layer from the irrotational free stream, for the case where this interface is subject to streamwise cyclic disturbances to its undisturbed shape. A linearized solution to this equation shows that a sinusoidal disturbance of wavelength λ propagates as one mode of a neutrally stable Kelvin-Helmholtz wave with velocity ωλ[1 − exp (−4πδ/λ)]/4π relative to the fluid at infinity. Numerical solutions of the full nonlinear equation for a range of wavelengths and finite disturbance amplitudes indicate different behaviour. For sufficiently large amplitude the interface valleys evolve into long re-entrant wedges of irrotational fluid which are ‘entrained’ into the layer and which are separated from the free stream by lobes or bulges of rotational fluid. This single-mode nonlinear interfacial distortion could be generated over a broad wavelength range with no indication of preferential scaling based on δ. It is suggested that the interface behaviour bears distinct resemblance to flow features observed at the interface between turbulent and non-turbulent fluid in recent smoke-in-air flow-visualization studies of the outer part of a constant pressure turbulent boundary layer. The calculated rotational fluid lobe velocities, which are not very different from the equivalent linearized wave velocities, are found to be in reasonable agreement with the few existing measurements of the velocity of bulges at the turbulent–nonturbulent fluid interface, while the computed velocity field in the lobe is in qualitative agreement with the general flow pattern observed in experiments. In the absence of a preferred scale or range of scales for the development of the interfacial distortion, however, it is concluded that the present results cannot be interpreted as supporting the hypothesis of the presence of largescale coherent motions in the outer part of the layer.

Some Remarks on the Progress of Cavity Flow Studies

Theoretical developments on cavity flow studies are briefly reviewed. Physical and mathematical difficulties involved in cavity flow problems are discussed. Particular attention, with regard to practical applications, is given to the development of linearized theories. Based on the existing analyses, efforts to develop simple approximate expressions for the force coefficients of supercavitating hydrofoils (including the effects of free surface, cascade, aspect ratio, and finite disturbance) are made. Numerical results calculated from these expressions are compared with existing experimental data. Special problems involving unsteady cavity flows, such as pulsation of the finite cavities, are also discussed.

Pressure-wave propagation through a separated gas-liquid layer in a duct

Pressure-wave propagation through a separated gas-liquid layer at rest in a duct of constant rectangular cross-section and infinite length is considered. Such a system is dispersive, possessing an infinite number of modes which depend on the ratios of the densities, thicknesses and sound speeds of the two phases. The transitional variation of an infinitesimal disturbance initially having a step profile is investigated analytically and numerically. In addition, it is shown that a weak but finite disturbance is described asymptotically by the solution of the Korteweg-de Vries equation.