Showing papers on "Streamlines, streaklines, and pathlines published in 1974"

Quantum mechanical streamlines. I. Square potential barrier

Abstract: Exact numerical calculations are made for scattering of quantum mechanical particles hitting a square two-dimensional potential barrier (an exact analog of the Goos-Haenchen optical experiments) Quantum mechanical streamlines are plotted and found to be smooth and continuous, to have continuous first derivatives even through the classical forbidden region, and to form quantized vortices around each of the nodal points A comparison is made between the present numerical calculations and the stationary wave approximation, and good agreement is found between both the Goos-Haenchen shifts and the reflection coefficients The time-independent Schroedinger equation for real wavefunctions is reduced to solving a nonlinear first-order partial differential equation, leading to a generalization of the Prager-Hirschfelder perturbation scheme Implications of the hydrodynamical formulation of quantum mechanics are discussed, and cases are cited where quantum and classical mechanical motions are identical

Steady simple shear flow past a circular cylinder at moderate Reynolds numbers: a numerical solution

Abstract: The two-dimensional steady flow of an incompressible viscous fluid past a circular cylinder, placed symmetrically in a simple shear field, has been studied for both the stationary and the freely rotating case by solving numerically the Navier-Stokes equations for values of the Reynolds number R in the range from 0.047 to 70. At R = 0.047, the results obtained are in substantial agreement with the analytic small-R perturbation solution given by Robertson and Acrivos (1970). Inertia effects were found, however, to play a significant role even at R = 1, and hence the calculated flow pattern for R greater than or equal to 1 differs significantly from that of the creeping-flow solution. Specifically, for the freely rotating case, the region of closed streamlines decreases rapidly in extent with increasing R, two symmetrically placed wakes are formed on either side of the cylinder, and the dimensionless rotational speed of the freely suspended cylinder decreases as the reciprocal of the square root of R.

Evolution of a moderately short turbulent boundary layer in a severe pressure gradient

Abstract: The early and intermediate development of a highly accelerated (or decelerated) turbulent boundary layer is analyzed. For sufficiently large accelerations (or pressure gradients) and for total normal strains which are not excessive, the equation for the Reynolds shear stress simplifies to give a stress that remains approximately constant as it is convected along streamlines. The theoretical results for the evolution of the mean velocity in favourable and adverse pressure gradients agree well with experiment for the cases considered. A calculation which includes mass injection at the wall is also given.

Laminar boundary layer over a body of revolution at extremely high incidence

Abstract: Results of exact, full three‐dimensional solutions of the incompressible laminar boundary layer over a prolate spheroid at an extremely high incidence are presented. This extremely high incidence problem is characterized by features reflecting the predominance of the circumferential flow over the meridional flow and, more importantly, by a change of separation pattern from an open type to a closed type. These features include much larger circumferential skin friction than meridional skin friction and that the limiting streamlines are more closely oriented along the parallels.

A numerical solution of the flow around a sphere in a circular cylinder

Abstract: A numerical solution of the Navier-Stokes equations is presented for Poiseuille flow around an axially placed, fixed sphere in a circular cylinder. Streamlines and isovorticity lines are calculated from the governing equations for the strearnfunction and the vorticity. Isobars are calculated from a Poisson equation, derived from the Navier-Stokes equations. The pressure and vorticity distribution on the surface of the sphere, the additional pressure drop and the drag coefficients are presented. Solutions are obtained for Reynolds numbers up to 150 (based on cylinder diameter and mean velocity). The wall effects are examined by comparison with results of previous investigations for an unbounded flow around a sphere.

Two-dimensional analysis of backwater at bridges

Abstract: A two-space dimensional mathematical model of computing steady flow distributions and water levels near contractions in natural waterways (e.g., highway bridges) is developed. The two-dimensional model provides a theoretical basis for such one-dimensional concepts as expansion losses by accounting for all losses by friction along properly located two-dimensional streamlines. The finite element solution algorithm locates these streamlines and integrates energy loss due to friction along them. Stagnation zones downstream from contractions can be predicted and located by the algorithm. A necessary condition for separation is that the velocity head at the point of separation exceeds head loss due to friction along some possible path for a free streamline. The absence of separation which is associated with scouring currents along the downstream side of bridge approach embankments can be predicted where the necessary condition is not satisfied.

Flow About a Stationary Sphere in a Slowly Rotating, Viscous Fluid

Abstract: Analytical solutions are presented for the flow about a nonrotating sphere immersed in a slowly rotating fluid by the singular perturbation technique valid for small Reynolds numbers. Streamlines are plotted and the torque necessary to hold the sphere stationary is calculated. It is found that the transition from inflow near the equator to outflow near the poles takes place at an angle of 54.50 deg measured from the poles. Flow velocities and the stream function are also calculated for the problem when the sphere rotates with an angular velocity different from that of the fluid.

Aerodynamic heating on 3-D bodies including the effects of entropy-layer swallowing

Abstract: A relatively simple method was developed previously (authors, 1973) for calculating laminar, transitional, and turbulent heating rates on three-dimensional bodies in hypersonic flows. This method was shown to yield reasonably accurate results for laminar heating on blunted circular and elliptical cones and an earlier version of the space shuttle vehicle. As the boundary layer along the surface grows, more and more of the inviscid-flow mass is entrained into the boundary layer, and the streamlines which passed through the nearly normal portion of the bow shock wave are 'swallowed' by the boundary layer. This phenomenon is often referred to as entropy-layer or streamline swallowing, and it can have a significant effect on the calculated heating rates. An approximate, yet simple, method for including the effects of entropy-layer swallowing in the heating-rate calculations is given.

The relationship between the base pressure on a bluff body and the velocity at separation

Abstract: In the modified free-streamline theory for the separated flow past bluff bodies due to Roshko, the velocity on the separated free streamlines is expressed as U v =kU 0 , where U 0 is the velocity of the undisturbed stream and k is a constant. If it is assumed that the pressure on the base of the body is uniform and equal to the pressure on the free streamline, then application of Bernoulli's equation leads to; k =(1-C pb ) ½ , … (1) where C pb is the base pressure coefficient.

Numerical calculations of velocity and pressure distribution around oscillating airfoils

Abstract: An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.

Numerical solution of the Navier--Stokes equations at high Reynolds numbers

Abstract: A numerical method is presented which is designed to solve the Navier- Stokes equations for two-dimensional, incompressible flow. The method is intended for use on problems with high Reynolds numbers for which calculations via finite difference methods have been unattainable or unreliable. The proposed scheme is a hybrid utilizing a time-splitting finite difference method in areas away from the boundaries. In areas neighboring the boundaries, the equations of motion are solved by the newly proposed vortex method by Chorin. The major accomplishment of the new scheme is that it contains a simple way for merging the two methods at the interface of the two subdomains. The proposed algorithm is designed for use on the time-dependent equations but can be used on steady state problems as well. The method is tested on the popular, time-independent, square cavity problem, an example of a separated flow with closed streamlines. Numerical results are presented for a Reynolds number of 10$sup 3$. (auth)

Tornadolike gravity-driven vortex model

Abstract: The buoyancy-induced vorticity concentration produced as the fluid in a vortex accelerates vertically was studied. The boiloff from liquid nitrogen, to which a small amount of initial vorticity was added, provided a source of cool, heavy gas in which a concentration of vorticity took place. Condensation streamers made the flow visible. It is shown that the presence of a surface boundary layer is not necessary for the effective concentration of vorticity. A simple theoretical analysis of the phenomenon was also made. A radial contraction of the flow with vertical position and a characteristic hook shape in the top view of the streamlines were observed in both theory and experiment. The vorticity concentration observed may be similar to that which occurs in tornadoes.

On the Difference between Developments of Two- and Three-Dimensional Turbulent Boundary Layers

Abstract: Computational investigation was made on characteristics of incompressible turbulent boundary layers on a shiplike body and on two kinds of two-dimensional bodies. One of the latter bodies had nearly the same contour as one of the Streamlines on the three-dimensional body. The other had pressure distribution on it the same as along the streamline on the three-dimensional body. Momentum thickness, shape parameter and surface cross-flow angle of the boundary layer were calculated for the three cases and factors contributing to variations of these quantities were analyzed. Comparison between the results for the three cases revealed the following facts. The rate of growth of the momentum thickness in the region of deceleration was promoted mainly by the streamline convergence which was absent in two dimensions. In this case the development of the momentum thickness became larger in the three-dimensional boundary layer than in the two dimension alone. The surface cross-flow angle near the rear stagnation deviated rapidly in the direction of the center of streamline curvature due to the curvature and convergence of the three-dimensional streamline.

The effect of a semi-contaminated free surface on mass transport

Abstract: The mass transport velocity field is determined for surface waves which propagate from a region with a clean free surface into a region beneath an inextensible surface film. The waves are assumed to be incident normally on the edge of the film. Determination of this velocity field requires the investigation of a mixed boundary value problem for the bi-harmonic equation, the solution of which is obtained using the Wiener–Hopf technique. Streamlines for the mean motion of the fluid particles are thus obtained. It is found that considerable vertical displacement of fluid is possible due to the presence of the surface film.

Centrifugal instability development near critical line of cylinder placed in supersonic flow

Abstract: The mathematical model is derived describing centrifugal instability in compressible flows with the natural streamlines curvature. Usually streamlines curvature is supposed to be known beforehand and determined by the surface curvature. But there are many flows where streamlines curvature is caused for example by positive pressure gradient. Analysis done by Goertler and Hammerlin [1] for centrifugal instability in Hiemenz flow near the critical point revealed growing modes. This paper is devoted to the compressible flow analysis. The flow near critical line of infinite cylinder is analysed. It is supposed that Mach number of external flow tends to infinity and that the specific heat ratio tends to unity. It was shown by Neyland [2] that the flow between the bow shock and the cylinder surface contains in general three distinct regions. The flow in the first region located near the shock is inviscid. Due to large density ratio pressure gradient doesn’t influence the flow in the first region. At the same time there is the second region characterized by small longitudinal velocities where pressure gradient influences the inviscid flow. The third region is characterized by viscosity influence and is located near the surface of the body. Such flow structure exists if Reynolds number is large. As a result of analysis solution may be found for all three regions. Obtained velocity and density profiles are used in subsequent stability analysis. System of linear equations is derived describing centrifugal instability development in compressible flow near the critical line of infinite cylinder. Presented are results of numerical analysis of the linear equations for different similarity parameters values.. The non-linear problem formulation is also discussed.

Sonic and Shock Lines in Transonic Flow past a Wavy Wall

Abstract: An approximate analysis of the nonlinear, mixed-type differential equation for inviscid transonic flows past a sinusoidal wall is made using the nonlinear correction theory, which was proposed by Hosokawa to improve the linearized transonic flow theory. In order to execute analysis for the flow far from the wall, the nonlinear correction theory (considered to be valid only near the wall) is applied for a streamline near the wall considered as a new wall to calculate the flow farther than that and continue this process successively to obtain many streamlines which clarify the entire flow field past the original wall. This method gives a good survey of various transonic flows past the sinusoidal wall, including periodic appearance of sonic and shock lines in the flow field which define supersonic pockets in subsonic main flows as well as subsonic pockets in supersonic main flows.

Analysis of impingement heat transfer for two parallel liquid-metal slot jets

Abstract: An analytical method is developed for determining heat transfer by impinging liquid-metal slot jets. The method involves mapping the jet flow region, which is bounded by free streamlines, into a potential plane where it becomes a uniform flow in a channel of constant width. The energy equation is transformed into potential plane coordinates and is solved in the channel flow region. Conformal mapping is then used to transform the solution back into the physical plane and obtain the desired heat-transfer characteristics. The analysis given here determines the heat-transfer characteristics for two parallel liquid-metal slot jets impinging normally against a uniformly heated flat plate. The liquid-metal assumptions are made that the jets are inviscid and that molecular conduction is dominating heat diffusion. Wall temperature distributions along the heated plate are obtained as a function of spacing between the jets and the jet Peclet number.

Low speed flow visulaization using holography and the development of a rapid data reduction system for holographic interferometry.

Abstract: Flow visualization of low speed flow has been obtained in this thesis by the injection of a thin stream of gas. This injected gas has a refractive index which differs from that of the uniform flow in the wind tunnel. A frozen fringe hologram of the introduction of this gas indicates regions of fringe shift. A pathline is produced by connecting points of maximum fringe shift. Pathlines generated in such a manner around a cylinder agree with the stream functions of potential flow theory. A rapid means has been developed of data reduction of fringe information from a photograph of a reconstructed holographic interferogram. Data points are gathered using a digitizer and a mini-computer. The array of data points so obtained can be examined and corrected if necessary and finally expanded by means of Lagrangian interpolation. The expanded array is transferred to a fortran program which accomplishes the integral inversion and subsequently prints out the density field. By using this method of data reduction and inversion, a large time savings is realized.