Francis H. Harlow

Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface

A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired.

Transport Equations in Turbulence

Turbulence transport equations, describing the dynamics of transient flow of an incompressible fluid in arbitrary geometry, have been derived in such a manner as to incorporate the principles of invariance (tensor and Galilean) and universality. The equations are described in detail and their applicability is demonstrated by comparison of solutions with experiments on turbulence distortion and on the turbulence in the flow between flat plates.

A numerical fluid dynamics calculation method for all flow speeds

Abstract The ICE technique for numerical fluid dynamics has been revised considerably, and generalized in such a way as to extend the applicability to fluid flows with arbitrary equation of state and the full viscous stress tensor. The method is useful for the numerical solution of time-dependent fluid flow problems in several space dimensions, for all Mach numbers from zero (incompressible limit) to infinity (hypersonic limit). This new version is considerably less complicated than the original form. The present description does not assume a familiarity with the previous one.

Numerical calculation of multiphase fluid flow

Abstract A new computing technique is described for the solution of fluid flow problems in which several fields interpenetrate and interact with each other. An implicit coupling for each field between mass transport and equation of state allows for calculations in all flow-speed regimes, from far subsonic (incompressible) to far supersonic. In addition, the momentum transport between fields is implicit, allowing for all degrees of coupling, from very loose to completely tied together. Phase transitions permit interchange of mass, momentum and energy between fields, each of which is composed of several components. Considerable generality is present, to permit application to a wide scope of complicated problems, for example, the fluidized dust bed, the flow of a liquid with entrained bubbles, and atmospheric condensation with the fall of precipitation.

The Splash of a Liquid Drop

The full Navier‐Stokes equations are solved numerically in cylindrical coordinates in order to investigate the splash of a liquid drop onto a flat plate, into a shallow pool, or into a deep pool. Solution is accomplished with the Marker‐and‐Cell technique using a high‐speed computer. Results include data on pressures, velocities, oscillations, droplet rupture, and the effects of compressibility. They also show how the technique can be applied to a wide variety of other complicated fluid flow problems involving the transient behavior of a free surface.

Numerical Study of Large‐Amplitude Free‐Surface Motions

The marker and cell method for high‐speed computer was used to investigate the motion of a fluid whose free surface has been perturbed by an impulsive sinusoidal pressure. For gravity pointing out of the fluid, the resulting motion exhibits Rayleigh‐Taylor instability, whose progress is followed from low amplitude to the asymptotic bubble and spike stage. The effects of small and large viscosity are contrasted. For gravity pointing into the fluid, the large‐amplitude oscillatory motion exhibits a shift of resonance frequency and other effects of nonlinearity. Comparisons of results are presented wherever previous analytical or experimental work was available, and many new aspects of these types of flow are discussed. The calculations are based upon the full, time‐dependent, nonlinear Navier‐Stokes equations for an incompressible fluid.

Numerical calculation of almost incompressible flow

A new method is presented for the numerical solution of time-dependent problems in several space dimensions. The technique is applicable to low-speed (incompressible) flows, to high-speed (supersonic) flows, and to all flow speeds in between, thereby bridging the gap through the almost-incompressible regime in which previously-described techniques break down.

Turbulence Transport Equations

A generalized eddy viscosity function σ, is introduced in order that the Reynolds stress in an incompressible fluid be expressible as a linear combination of the Kronecker and rate‐of‐strain tensors. A transport equation for the eddy viscosity is derived from the general turbulence energy equation, thereby introducing two additional functions, the specific turbulence kinetic energy q, and a scale variable s. To determine the three variables, a transport equation for s is postulated, and a modified Prandtl—Wieghardt relation among the three variables is accepted. The theory is expressed in universal, invariant form, and validity is demonstrated by application to several problems.

Numerical Solution of the Problem of Vortex Street Development

A method is described for the solution of time‐dependent problems concerning the flow of viscous incompressible fluids in several space dimensions. The method is numerical, using a high‐speed computer for the solution of a finite‐difference approximation to the partial differential equations of motion. The application described here is to a study of the development of a vortex street behind a plate which has impulsively accelerated to constant speed in a channel of finite width; the Reynolds‐number range investigated was 15 ≤ R ≤ 6000. Particular attention was given to those features for which comparison could be made with experiments, namely, critical Reynolds number for vortex shedding, drag coefficient, Strouhal number, vortex configuration, and channel‐wall effects. The nature of the early stages of flow‐pattern development was also investigated.

Effects of Diffusion on Interface Instability between Gases

Experiments are described concerning the interface instability which arises when an argon‐bromine mixture falls under gravity into air or into helium. Existing theories which include viscosity effects fail to explain the observed amplitude growth rate, but it is shown that approximate inclusion of diffusion effects allows calculation of all features of early perturbation growth accurately to within experimental error.