Showing papers on "Streamlines, streaklines, and pathlines published in 1973"

A new model for granular porous media: Part II. Numerical solution of steady state incompressible Newtonian flow through periodically constricted tubes

Abstract: A numerical method for the solution of the problem of steady state, incompressible Newtonian flow through periodically constricted tubes is developed. All terms of the Navier-Stokes equation are retained, including the nonlinear inertia terms. Sample calculations for a uniform periodically constricted tube, the geometry of which is connected with the modeling of a packed bed of sand are given, including streamlines, axial and radial velocity profiles, pressure profiles, and the dimensionless pressure drop versus Reynolds number relation. The effect of some geometric characteristics of periodically constricted tubes on their friction factor is investigated numerically, and comparison of some existing experimental data with calculated ones is made.

Turbulent dispersion from sources near two-dimensional obstacles

Abstract: Publisher Summary This chapter extends the statistical theory of turbulent diffusion to the problem of turbulent diffusion from a point or line source in an inhomogeneous straining flow, such as the flow around a two-dimensional body The new effects that have to be considered are the convergence and divergence of mean streamlines, the inhomogeneity of the mean and turbulent velocities, and the presence of a boundary The theory predicts the mean square dispersion of the plume about the mean streamline through the source in terms of the Lagrangian statistics of the turbulence The diffusion from various sources in potential flows over two-dimensional obstacles assuming constant (or variable) eddy diffusivity is analyzed and the results are compared with those of the more rigorous statistical analysis for sources in the region of the obstacle Unlike the statistical analysis, this eddy diffusivity analysis can also be extended to calculate diffusion from sources placed some distance upwind of an obstacle An example in the chapter shows how this analysis may be applied to calculating concentration on hills due to distant sources

Inviscid Surface Streamlines and Heat Transfer on Shuttle-Type Configurations

Abstract: A method is developed which calculates laminar, transitional, and turbulent heating rates on arbitrary blunt-nosed three-dimensional bodies at angle of attack in hypersonic flow. The geometry of the body may be specified analytically, or generated from a doubly cubic spline fit to coordinate points. Inviscid surface streamlines are calculated from Euler's equation using a prescribed pressure distribution. Laminar and turbulent heating rates are determined along a streamline by applying the axisymmetric analog to solutions of the axisymmetric boundary-layer equations. The location of the transition region may be specified optionally by geometric location, momentum thickness Reynolds number, or integrated unit Reynolds number along a streamline. Transitional heating rates are then calculated as a weighted average of the local laminar and turbulent values. Either ideal gas or equilibrium air properties may be used. Results are presented for blunted circular cones, and a typical delta-wing space shuttle orbiter at angle of attack. In comparison with experimental data, the present method was found to yield accurate laminar heating rates and reasonably accurate transitional and turbulent heating rates. The computer program developed to calculate the results presented herein requires only a few seconds of computing time per streamline on the CDC 6600 computer.

Flow of a newtonian fluid through a sudden contraction

Abstract: The full Navier-Stokes equations describing flow through a sudden contraction are solved by an explicit finite-difference method. Streamlines, vorticity distributions, velocity profiles, excess pressure drops, and entrance lengths are calculated as functions of Reynolds number and radius ratio. The results are compared with existing experimental data and the limited theoretical work available.

Transverse motion of an elastic sphere in a shear field

Abstract: The forces acting on an elastic particle suspended in a shear field, and moving relative to it, are found for the case in which there are small deformations from an initially spherical shape. The deformation is the result of the viscous stresses acting on the particle. Of principal interest is that there is a component of the force perpendicular to the free-stream direction, so that the particle will migrate across the undisturbed streamlines.

Analytical method for predicting the pressure distribution about a nacelle at transonic speeds

Abstract: The formulation and development of a computer analysis for the calculation of streamlines and pressure distributions around two-dimensional (planar and axisymmetric) isolated nacelles at transonic speeds are described The computerized flow field analysis is designed to predict the transonic flow around long and short high-bypass-ratio fan duct nacelles with inlet flows and with exhaust flows having appropriate aerothermodynamic properties The flow field boundaries are located as far upstream and downstream as necessary to obtain minimum disturbances at the boundary The far-field lateral flow field boundary is analytically defined to exactly represent free-flight conditions or solid wind tunnel wall effects The inviscid solution technique is based on a Streamtube Curvature Analysis The computer program utilizes an automatic grid refinement procedure and solves the flow field equations with a matrix relaxation technique The boundary layer displacement effects and the onset of turbulent separation are included, based on the compressible turbulent boundary layer solution method of Stratford and Beavers and on the turbulent separation prediction method of Stratford

Steady separated flow in a linearly decelerated free stream

Abstract: Numerical methods are used to investigate the separated flow over a finite flat plate when the flow at large distances is given by the stream function ψ_∞ = -xy and the plate is situated on the x axis from - 1 to 1. The range of nominal Reynolds number is 10-800. Reduced-mesh calculations are used for fine resolution of the flow field in the immediate vicinity of the separation point. Streamlines, equivorticity lines, and shear stress and pressure gradient at the plate surface illustrate the overall structure of the flow. In each case the streamwise pressure gradient is less than that for undisturbed potential flow and the position of separation is consequently downstream of that predicted by classical boundary layer theory. The boundary-layer structure in the vicinity of the separation point shows a direct transition between the regular upstream behaviour and Dean’s (1950) solution right at separation with no sign whatever of intermediate singular behaviour of the type predicted by Goldstein (1948). The implications of these results for the structure of high Reynolds number, steady, laminar flow are discussed.

Nonlinear wave propagation on an arbitrary steady transonic flow

Abstract: Here, we have studied the propagation of an arbitrary disturbance bounded in space on an arbitrary two- or three-dimensional transonic flow. First we have presented a general theory valid for an arbitrary system of n first-order quasi- linear partial differential equations and then used the theory for the special case of gasdynamic equations. If a disturbance is created in the neighbourhood of a sonic point, only a part of the disturbance stays in the transonic region and it is bounded by a wave front perpendicular to the streamlines. This part of the disturbance is governed by a very simple partial differential equation and the problem essentially reduces to the discussion of one-dimensional waves. The disturbance decays in the neighbourhood of the points where the flow acceler- ates from a subsonic state to a supersonic state and it attains a steady state where the flow is decelerating.

Boundary Layer Electron Profiles for High-Altitude Entry of a Blunt Slender Body

Abstract: Theme A KEY factor in prediction of radio blackout during entry into a planetary atmosphere is determination of the magnitude and distribution of electron concentration (Ne) in the plasma layer. Several of the contributions made in the past ten years to improve this prediction capability can be credited to the RAM (Re-entry Attenuation Measurement) project—a carefully coordinated program of laboratory tests, flight experiments, and theoretical analysis carried out at the Langley Research Center. Because a complex chemistry system is necessary to calculate accurately trace species concentrations, computation of chemical changes along streamlines in a previously-calculated flowfield proved to be satisfactory and more economical than a complete flowfield calculation with nonequilibrium chemistry. Except for minor displacement effects boundary layer has little influence on Ne profiles for blunt nosed bodies at the lower altitudes of interest, since the entropy layer is relatively thick and the Ne maxima fall in the inviscid portion. At higher altitudes, where boundary layers are no longer thin relative to the entropy layer and often include the peak of the Ne profile, a method was devised for following streamlines into the boundary layer. The approach was useful, but, because ambipolar diffusion of charged particles could not be included, the results are invalid above a threshold altitude. In this paper the boundary-layer portions of previously published Ne profiles are calculated by means of a nonequilibrium, multicomponent diffusion, boundary-layer program, and the results are compared with previously published profiles obtained by following streamlines into the boundary layer. Comparisons are made with experimentally measured profiles and also with profiles calculated for nonequilibrium fully viscous flow.

Finite-Element Analysis of Unsteady Incompressible Flow around an Oscillating Obstacle of Arbitrary Shape

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

FORTRAN program for calculating velocities and streamlines on the hub-shroud mid-channel flow surface of an axial-or mixed-flow turbomachine. 2: Programmer's manual

Abstract: A FORTRAN-IV computer program, MERIDL, has been developed that obtains a subsonic or shock-free transonic flow solution on the hub-shroud mid-channel flow surface of a turbomachine. The blade row may be fixed or rotating and may be twisted and leaned. Flow may be axial or mixed, up to 45 deg from axial. Upstream and downstream flow variables can vary from hub to shroud, and provision is made to correct for loss of stagnation pressure. The results include velocities, streamlines, and flow angles on the flow surface and approximate blade surface velocities. Subsonic solutions are obtained by a finite-difference stream-function solution. Transonic solutions are obtained by a velocity-gradient method, using information from a finite-difference stream-function solution at a reduced mass flow.

A Small Cross Flow Theory for Three-Dimensional, Compressible, Turbulent Boundary Layers on Adiabatic Walls

Abstract: A small cross flow integral method for predicting the growth of three-dimensi onal, compressible, turbulent boundary layers on adiabatic walls is presented. The method is based on the two momentum integral equations and the compressible entrainment equation corresponding to an orthogonal, curvilinear coordinate system based on the projections of the outer flow streamlines on the body surface. For small cross flow conditions the main flow momentum integral equation and the entrainment equation reduce to forms analogous to the axisymmetric system of equations employed by Sumner and Shanebrook in their study of flow over bodies of revolution. The cross flow momentum integral equation is integrated by assuming the cross flow profile can be described by a member of the (p., qj) family of hodograph models. The method is applied to the three-dimensional flowfield measured by Hall and Dickens and it is found that good comparisons with data are obtained in the region where Hall and Dickens indicate that small cross flow conditions prevail. Moreover, it is found that, for the conditions of this paper, the present cross flow model is superior to Mager's model in the prediction of cross flow profiles.

Plane couette flow of two immiscible incompressible fluids with uniform suction at the stationary plate

Abstract: The contact layer in the plane couette flow of two immiscible fluids with uniform suction at the stationary plate has been investigated. The effect of suction is to supply an adverse pressure gradient to the fluid which causes back flow near the stationary plate. The streamlines are drawn to show the effect of uniform suction at the stationary plate on the contact layer.

A Numerical Study of the Steady Circulation in an Open Bay

Abstract: The steady-state circulation in a rectangular bay is studied numerically in a model of homogeneous water and vertical coasts. The competing influences of the surface winds and longshore currants flowing by the open side of the bay and the effect of the bathymetry are emphasized. For a wind-stress field that does not vary along the coast but decays linearly inshore from the open side of the bay, the mass-transport streamfunction contours form a gyre rotating in the sense of the wind-stress curl. A uniform continental shelf slope distorts the gyre by creating depth variations that cause vortex stretching. Consequently, the streamlines become more packed to the right of the down-slope direction. The nonlinearity tends to destroy symmetry by crowding streamlines in the direction of the induced current. The influence of large-scale ocean currents along the open side is normally confined to the outer half of the bay. When the wind is blowing against these currents, the influence of the wind creates two...

Finite element analysis of unsteady incompressible flow around an oscillating obstacle of arbitrary shape.

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

Application of the Vortex-Lattice Method to Represent a Jet Exhausting from a Flat Plate into a Crossflowing Stream

Abstract: : A study was conducted to develop a mathematical model of a jet exhausting from a flat plate into a crossflowing stream. The modeling was accomplished using the vortex-lattice method. Analytical streamlines and pressure distributions on the flat-plate surface as well as vector data in the field above the plate are compared with available experimental data. Recommendations are made for further improvement of vortex-lattice jet-modeling techniques.

Entrainment by a rising thermal

Abstract: A theoretical model is proposed to describe the entrainment of ambient air by a rising thermal. The model is appropriate for the first few revolutions of a strongly heated thermal whose motion is dominated by its initial conditions. At sufficiently early times, only the outer edge of the thermal is turbulent. The ambient gas mixes with the thermal gas along the turbulent interface and is convected into the thermal along streamlines. The gas densities and species concentrations on each streamline are obtained by superimposing a turbulent mixing layer on the edge of Hill's spherical vortex. Solutions are obtained in terms of the dimensionless ratio of the mixing length to the thermal radius (l/a ) and the entrainment coefficient is shown to be directly proportional to l/a.

Irrotational Flow with Irregular Boundary Shapes

Abstract: An algorithm for the numerical solution of two-dimensional irrotational flows based on an inverse transformation is presented as an efficient means of solving flows with irregular boundary shapes. Using the intersections of a set of streamlines and equipotential lines as computational nodal points, the algorithm determines the coordinates of these intersections. The streamlines and equipotential lines can be arbitrarily selected to concentrate the nodal points in areas of high velocity flow so as to increase the accuracy in these areas. The algorithm is particularly well suited to modern high speed computers, and FORTRAN statements are given. Since the program operates on many different boundary shapes with no internal alterations, it is suitable for use by persons with limited background in numerical procedures.

A Method for Calculating Three-Dimensional Turbulent Boundary Layer by Using Streamline Co-ordinates

Abstract: A method for calculating a three-dimensional turbulent boundary layer by numerically solving the boundary layer equations given in the integral form is presented. By approximating the equipotential lines perpendicular to the streamlines with small line segments, the equipotential lines and the partial derivatives in the directions of these lines are obtained more easily and accurately. The streamwise momentum thickness, the streamwise shape parameter and the angle formed by the potential flow streamline and the surface (limiting) streamline are calculated by using these quantities. These results are compared with experimental results, and in general good agreement is obtained. An excellent agreement is obtained in particular for the streamwise momentum thickness.

Effect of geometry on the nose-region flow-field of shuttle entry-configurations

Abstract: In order to determine the convective heat-transfer distribution for the nose region of the space shuttle entry configurations, a three-dimensional flow-field is described which may include extensive regions of separated flow. Because of the complexity of the flow field for the nose region, experimental data are needed to define the relation between the nose geometry and the resultant flow field. According to theoretical solutions of the three-dimensional boundary layer, the boundary layer separates from the leeward generator of a blunted cone at an alpha equal to the cone half-angle. Separation results from the transverse pressure gradient, i.e., the velocity derivative due to crossflow. The boundary layer limiting streamlines converge toward the singular point of sep aration. The separated region is bounded by an ordinary line of separation.

Numerical solution of the converse problem for transonic flow in a curved channel

Abstract: THE converse problem on transonic flow in a curved channel is solved by using spline smoothing of the streamlines, and of the flow parameters along them obtained in the course of the computations. This device regularizes the process of solving the Cauchy problem in the subsonic zone. We pose the problem of constructing the flow in a curved channel, given the shape of one streamline and the flow parameters along it. Given the analytic form of the streamline and the analytic data along it, there exists in a neighbourhood of the initial streamline, by Kovalevskii's theorem [1], a unique solution of the relevant Cauchy problem. Having constructed this solution, we can find the adjacent streamline and the flow parameters on it, and continue the solution further, while retaining its analytic properties. In the general case, when the streamline is not analytic, the Cauchy problem is incorrectly posed in the subsonic (elliptic) region, so that the method of solution outlined above cannot be used. Notice that, even if the initial data are analytic in the subsonic region, when obtaining a numerical solution by a finite-difference method the approximation errors and the rounding-off errors may lead to the analytic property being destroyed; this shows itself in the solution surging and deviating away from the true solution. We therefore have to provide a procedure whereby the discrete solution obtained is reduced to an analytic form, i.e. a procedure for regularization on each streamline. As this procedure we propose to use spline smoothing both of the streamline and of the parameters along it. Notice that, in the case of straight channels, a method was devised in [2] for computing flows involving a transition through the velocity of sound, on the basis of a given distribution of the parameters along the channel axis, and a suitable finite-difference method was described. The elements of the regularization lie in using a non-uniform mesh and in the method for evaluating the derivatives. This method cannot be used in the case of substantially curved channels.