Showing papers by "Philip Geoffrey Saffman published in 1982"

Stability of a vortex street of finite vortices

Abstract: The stability of the finite-area Karman ‘vortex street’ to two-dimensional disturbances is determined. It is shown that for vortices of finite size there exists a finite range of spacing ratio κ for which the array is stable to infinitesimal disturbances. As the vortex size approaches zero, the range narrows to zero width about the classical von Karman value of 0·281.

Calculations of laminar viscous flow over a moving wavy surface

Abstract: The steady, laminar, incompressible flow over a periodic wavy surface with a prescribed surface-velocity distribution is found from the solution (via Newton's method) of the two-dimensional Navier–Stokes equations. Validation runs have shown excellent agreement with known analytical (Benjamin 1959) and analytico-numerical (Bordner 1978) solutions for small-amplitude wavy surfaces: For steeper waves, significant changes are observed in the computed surface-pressure distribution (and consequently in the nature of the momentum flux across the interface) when a surface orbital velocity distribution, of the type found in water waves, is included,

Three-dimensional stability of vortex arrays

Abstract: The stability to three-dimensional disturbances of three classical steady vortex configurations in an incompressible inviscid fluid is studied in the limit of small vortex cross-sectional area and long axial disturbance wavelength. The configurations examined are the single infinite vortex row, the Karman vortex street of staggered vortices and the symmetric vortex street. It is shown that the single row is most unstable to a two-dimensional disturbance, while the Karman vortex street is most unstable to a three-dimensional disturbance over a significant range of street spacing ratios. The symmetric vortex street is found to be most unstable to three-dimensional or two-dimensional symmetric disturbances depending on the spacing ratio of the street. Short remarks are made concerning the relevance of the calculations to the observed instabilities in free shear layer, wake and boundary-layer type flows.

Finite-amplitude interfacial waves in the presence of a current

Abstract: Solutions for interfacial waves of permanent form in the presence of a current wcre obtained for small-to-moderate wave amplitudes. A weakly nonlinear approximation was used to give simple analytical solutions to second order in wave height. Numerical methods were usctl to obtain solutions for larger wave amplitudes, details are reported for a number of selected cases. A special class of finite-amplitude solutions, closely related to the well-known Stokes surface waves, were identified. Factors limiting the existence of steady solutions are examined.

An inviscid model for the vortex-street wake

Abstract: An inviscid model for the Karman vortex street, containing vortices of uniform vorticity surrounded by irrotational fluid, is related to the wake behind a bluff body by a global analysis requiring the conservation of momentum, energy and vorticity. Some comparison is made with experimental results reported in the literature. A qualitative procedure is proposed whereby the slow evolution of the wake through viscous effects is approximated. Some comments are made regarding the relevance of the stability properties of the inviscid street. Some calculations are made for the ‘secondary vortex street’ that is observed after breakdown and rearrangement, and comparison is made with experiment.

Instability and confined chaos in a nonlinear dispersive wave system

Abstract: Calculations of a discrete nonlinear dispersive wave system show that as the degree of nonlinearity increases, the system experiences in turn, periodic, recurring, chaotic, transitional, and periodic motions. A relationship between the instability of the initial configuration and the long‐time behavior is identified. The calculations further suggest that the corresponding continuous system will exhibit chaotic motions and energy‐sharing among a narrow band of unstable modes, a phenomenon which we call ‘‘confined chaos.’’

Finite-Amplitude Waves in Inviscid Shear Flows

Abstract: This paper examines the existence and properties of steady finite-amplitude waves of cats-eye form superposed on a unidirectional inviscid, incompressible shear flow. The problem is formulated as the solution of nonlinear Poisson equations for the stream function with boundary conditions on the unknown edges of the cats-eyes. The dependence of vorticity on stream function is assumed outside the cats-eyes to be as in the undisturbed flow, and uniform unknown vorticity is assumed inside. It is argued on the basis of a finite difference discretization that the problem is determinate, and numerical solutions are obtained for Couette-Poiseuille channel flow. These are compared with the predictions of a weakly nonlinear theory based on the approach of Benney & Bergeron (1969) and Davis (1969). The phase speed of the waves is found to be linear in the wave amplitude.