Charles Zemach

A continuum method for modeling surface tension

Abstract A new method for modeling surface tension effects on fluid motion has been developed. Interfaces between fluids of different properties, or “colors,” are represented as transition regions of finite thickness, across which the color variable varies continuously. At each point in the transition region, a force density is defined which is proportional to the curvature of the surface of constant color at that point. It is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero. The continuum method eliminates the need for interface reconstruction, simplifies the calculation of surface tension, enables accurate modeling of two- and three-dimensional fluid flows driven by surface forces, and does not impose any modeling restrictions on the number, complexity, or dynamic evolution of fluid interfaces having surface tension. Computational results for two-dimensional flows are given to illustrate the properties of the method.

Methods for numerical conformal mapping

Abstract Nonlinear integral equations for the boundary functions which determine conformal transformations in two dimensions are developed and analyzed. One of these equations has a nonsingular logarithmic kernel and is especially well suited for numerical computations of conformal maps including those which deal with regions having highly distorted boundaries. Numerical procedures based on interspersed Gaussian quadrature for approximating the integrals and a Newton-Raphson technique to solve the resulting nonlinear algebraic equations are described. The Newton-Raphson iteration converges reliably with very crude initial approximations. Numerical examples are given for the mapping of a half-infinite region with periodic boundary onto a half plane, with up to nine-figure accuracy for values of the map function on the boundary and for its first derivatives. The examples include regions bounded by “spike” curves characteristic of Rayleigh-Taylor instability phenomena. A differential equation is derived which relates changes in the map function to changes of the boundary. This is relevant to potential problems for regions with time-dependent boundaries. Further nonsingular integral formulas are derived for conformal mapping in a variety of geometries and for application to the boundary-value problems of potential theory.

Rayleigh-Taylor instability and the use of conformal maps for ideal fluid flow

Abstract Potential theory permits ideal fluid dynamics to be formulated in terms of boundary motion. In two dimension, the flow can then be found using conformal mapping. The evolution of some Rayleigh-Taylor instabilities is calculated well into the large amplitude nonlinear regime. The Rayleigh-Taylor calculation for Atwood ratio unity is used as a prototype for a system of theoretical and numerical techniques exploiting complex variable theory and high-order quadrature methods.

A spectral model applied to homogeneous turbulence

Because a spectral model describes distributions of turbulent energy and stress in wave‐number space or, equivalently, in terms of a distribution of length scales, it can account for the variation of evolution rates with length scale. A spectral turbulence model adapted from a model introduced by Besnard, Rauenzahn, Harlow, and Zemach is applied here to homogeneous turbulent flows driven by constant mean‐flow gradients and to free decay of such flows. To the extent permitted by the experimental data, initial turbulent spectra are inferred, and their evolutions in time are computed to obtain detailed quantitative predictions of the spectra, relaxation times to self‐similarity, self‐similar spectrum shapes, growth rates, and power‐law time dependence of turbulent energies and dominant‐eddy sizes, and integral data, such as the components of the Reynolds stress tensor and the Reynolds stress anisotropy tensor. The match to experimental data, within the limits of experimental uncertainties, is good. Some qual...

Unstable normal mode for Rayleigh–Taylor instability in viscous fluids

The character of the growth rates of the normal modes for Rayleigh–Taylor instability of superposed incompressible, viscous fluids is analyzed in terms of appropriately scaled dimensionless parameters and a particularly simple representation of the Rayleigh–Taylor dispersion relation. The chief feature that emerges is that the scaled growth rate is remarkably insensitive to the values of fluid densities and viscosities. To within a few percent, the physical growth rate depends only on the surface tension, the density‐weighted average viscosity, and the effective acceleration. Approximate formulae for the most unstable wavenumber and the corresponding maximum growth rate are given.

Symmetries and the approach to statistical equilibrium in isotropic turbulence

The relaxation in time of an arbitrary isotropic turbulent state to a state of statistical equilibrium is identified as a transition to a state which is invariant under a symmetry group. We deduce the allowed self-similar forms and time-decay laws for equilibrium states by applying Lie-group methods (a) to a family of scaling symmetries, for the limit of high Reynolds number, as well as (b) to a unique scaling symmetry, for nonzero viscosity or nonzero hyperviscosity. This explains why a diverse collection of turbulence models, going back half a century, arrived at the same time-decay laws, either through derivations embedded in the mechanics of a particular model, or through numerical computation. Because the models treat the same dynamical variables having the same physical dimensions, they are subject to the same scaling invariances and hence to the same time-decay laws, independent of the eccentricities of their different formulations. We show in turn, by physical argument, by an explicitly solvable a...

Schwarz-Christoffel mappings: A general approach

Abstract A general method is developed for constructing equations of the Schwarz-Christoffel type. These equations define conformal transformations between regions of the complex plane, each of whose boundaries may consist of straight or continuously curved line segments and corners. The more familiar types of Schwarz-Christoffel equations are rederived; and some new types are obtained.

Initial value problem for Rayleigh--Taylor instability of viscous fluids

The initial value problem associated with the development of small amplitude disturbances in Rayleigh–Taylor unstable, viscous, incompressible fluids is studied. Solutions to the linearized equations of motion which satisfy general initial conditions are obtained in terms of Fourier–Laplace transforms of the hydrodynamic variables, without restriction on the density or viscosity of either fluid. When the two fluids have equal kinematic viscosities, these transforms can be inverted explicitly to express the fluid variables as integrals of Green’s functions multiplied by initial data. In addition to normal modes, a set of continuum modes, not treated explicitly in the literature, makes an important contribution to the development of the fluid motion.

A conformal map formula for difficult cases

Abstract A simple approximate formula is derived for the boundary map function of a conformal map which provides a remarkably good fit for mappings to regions with highly distorted boundaries. The approximation follows from the reduction of a nonlocal integral equation for the map function to a local equation based on the nature of the ‘crowding’ associated with distorted regions. Numerical examples are given.

Schwarz-Christoffel methods for conformal mapping of regions with a periodic boundary

Abstract Numerical conformal mapping methods for regions with a periodic boundary have been developed. These methods are based on the generalized Schwarz-Christoffel equation and can deal with boundary curves of arbitrary forms, i.e., made up of one or more rectifiable Jordan curves. High-order quadrature rules have been implemented in order to increase accuracy of the mapping. This is of particular relevance to highly accurate grid generation techniques required by, for example, implementation of high-order compact finite-difference discretization schemes.