Jacob E. Fromm

Oscillatory Drag, Lift, and Torque on a Circular Cylinder in a Uniform Flow

Numerical solutions of the equations governing time‐dependent, viscous, incompressible fluid flow past a circle are presented for Reynolds numbers 100, 400, and 1000. These solutions show the dramatic rise of the drag coefficient during the development of the Karman vortex street and reveal the oscillatory character of the drag, lift, and torque that are experienced by the circle. Contour plots of the vorticity and stream function are compared with histories of the pressure distribution, drag, lift, torque, and separation angles. These comparisons show how the pressure distribution, drag, lift, and torque on the circle are intimately and logically related to the well‐known flow pattern of the Karman street. A new method is described for implementing the infinity conditions. The use of this technique makes it possible to observe the motion of the upstream stagnation streamline and relate this effect to the lift on the circle. The fact that the drag is larger for the oscillatory wake than the symmetric wake is interpreted as a tendency toward an equilibrium state of maximum energy dissipation. Comparisons are made with experimental results. These comparisons suggest that the present results are a valid description of flow past a circular cylinder for Reynolds numbers in the range from 40 to 400.

Coordination number for the Li+–F− ion pair in water

A cluster of 200 molecules of water containing the Li+−F− ion pair has been studied using Monte Carlo techniques. Two temperatures (T = 298 and 500°K) and three different distances for the Li+−F− separation [R (Li−F) = 2.0, 6.0, and 10.0 A] have been considered. The water−water potential is obtained from quantum−mechanical Hartree−Fock−type computations corrected by inclusion of dispersion forces; the ion−water potential has been obtained from Hartree−Fock computations for the Li+−F−−H2O complex. The Monte Carlo simulation provides reproducible information on the cluster shape, on the cluster structure, and on the coordination numbers for the Li+ cation and the F− anion, when in presence of one another.

Laminar Flow Past a Circle in a Shear Flow

A numerical solution of the equations governing time‐dependent, viscous, incompressible fluid flow past a circle is presented for Reynolds number 400. The free‐stream velocity profile corresponds to the time‐averaged flow of a turbulent jet. The location and strength of the jet are adjusted so that the free stream approximates a linear shear flow in the vicinity of the circle. The asymmetry of the free stream causes a shift of the upstream stagnation streamline (the Pitot‐tube displacement effect) and small biases in the lift and torque exerted upon the circle. This lift bias is suggested as a contributing factor in the observed stability of raindrops in vertical wind tunnels. Contour plots of the vorticity and stream function are compared with histories of the pressure distribution, drag, lift, torque, and separation angles. These comparisons show how the asymmetry of the free stream manifests itself in terms of fine details within the flow pattern around the circle.

Practical Investigation of Convective Difference Approximations of Reduced Dispersion

A survey of commonly used approximations to the Eulerian form of the equations of ideal fluid flow is given. Comparisons are made through amplitude and phase properties as determined by linear stability analysis. The unacceptable amplitude damping of first‐order approximations is reiterated. For second‐order approximations the discussion emphasizes numerical dispersion effects and shows that the familiar stable forms do not differ significantly in relative merit. Fourth‐order improvements are discussed with reference to further extensions which minimize dispersion. Conservative forms of the approximations are given along with experimentally determined properties regarding their nonlinear behavior in fluid‐dynamic calculations. Comments relating to fluid‐dynamic instability versus numerical instability are included.

The recurrence of the initial state in the numerical solution of the Vlasov equation

Abstract The approximate recurrence of the initial state, observed recently in the numerical solution of Vlasovs equation by a finite-difference Eulerian model, is shown to be a property of three independent numerical methods. Some of the methods have exponentially growing modes (Dawsons beaming instabilities), and some others do not. The recurrence is in fact a manifestation of the finite velocity resolution of the numerical methods—a property which is independent of the approximation of a plasma by a finite number of electron beams. The recurrence is shown explicitly in the numerical simulation of Landau damping by three different methods: Fourier-Hermite, the recent variational method of Lewis, and the Eulerian finite-difference method.

Electrostatic Oscillations in Plasmas with Cutoff Distributions

When numerically solving the linear Vlasov equation with cutoff electron distributions, the well‐known Landau‐damped oscillation of the electric field was observed first but, after a short transition time, a constant‐amplitude harmonic oscillation with a frequency distinctly higher than Landaus frequency was found. This coexistence of two distinct modes of oscillation—one damped and one undamped—has not been observed before. The behavior is explained quantitatively by comparison of the dispersion equation solution with the results obtained from the numerical solution of Vlasovs equation.

Study of the structure of molecular complexes: XIV. Coordination numbers for selected ion pairs in water

A cluster of 200 molecules of water containing one of the LiF, LiCl, NaF, NaCl, KF or KCl ion pairs has been studied at the temperature T= 298°K using Monte Carlo techniques. The anion-cation internuclear separations considered in this work for any of the above pairs are 6.0 Å, 8.0 Å and 10.0 Å. The water-water potential is obtained from quantum-mechanical Hartree-Fock type computations corrected by inclusion of dispersion forces; the ion-water potentials have been obtained from Hartree-Fock type computations on the single ion-water complex. The computed radii for the first hydration shell are 2.7±0.1 Å, 3.4±0.3 Å, 4.0±0.3 Å, 3.0±0.5 Å, and 3.9±0.4 Å, for Li+, Na+, K+, F− and Cl−, respectively. The computed coordination numbers are 5.4±0.7,6.0±1.1, 7.2±1.2,4.5±0.6 and 5.1±0.8 for the same ions, respectively. The range of the coordination number obtained from compressibility, enthalpy, NMR spectroscopy and other experimental methods is much larger than the error ranges above given. Therefore the Monte Carlo simulation provides reliable information on the cluster shape, cluster structure and on the coordination numbers and hydration shell radii for the cations and anions, when both are present in a water cluster.

Lectures on Large-Scale Finite Difference Computation of Incompressible Fluid Flows

In the present section we shall attempt to describe, through example, the essentials of numerical computation of time-dependent, nonlinear fluid flows. The case in consideration will be that of incompressible flow with viscosity, described in terms of a vorticity and streamfunction. The discussions have been simplified so that the overall view of the computation procedures can be emphasized. Refinements of the individual areas, or subprograms, are presented in succeeding sections where the methods are brought up to date. Along with recommended reading the included material should permit the construction of a working program. Alternatively, the outline provided should lend itself to expansion into other areas of numerical computation of initial-boundary value problems.

Numerical Solutions of Two‐Dimensional Stall in Fluid Diffusers

A description and results are given for finite difference computation of incompressible flow in a flared channel. Stall geometry simulations are discussed for Reynolds numbers up to R =  4000. A fourth‐order convective approximation is used and its implementation in conservative form is described. Empirical downstream boundary conditions are established which limit the calculation region and permit long running times. Provisions for modifying channel flare angle are discussed.