Showing papers on "Streamlines, streaklines, and pathlines published in 1968"

Two-dimensional flow under gravity in a jet of viscous liquid

Abstract: This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.

The axisymmetric convective regime for a rigidly bounded rotating annulus

Abstract: The axisymmetric flow of liquid in a rigidly bounded annular container of height H, rotating with angular velocity Ω and subjected to a temperature difference ΔT between its vertical cylindrical perfectly conducting side walls, whose distance apart is L, is analysed in the boundary-layer approximation for small Ekman number v/2ΩL2, with gαΔTHv/4Ω2L2K ∼ 1. The heat transfer across the annulus is then convection-dominated, as is characteristic of the experimentally observed ‘upper symmetric regime’. The Prandtl number v/k is assumed large, and H is restricted to be less than about 2L. The side wall boundary-layer equations are the same as in (non-rotating) convection in a rectangular cavity. The horizontal boundary layers are Ekman layers and the four boundary layers, together with certain spatial averages in the interior, are determined independently of the interior flow details. The determination of the latter comprises a ‘secondary’ problem in which viscosity and heat conduction are important throughout the interior; the meridional streamlines are not necessarily parallel to the isotherms. The secondary problem is discussed qualitatively but not solved. The theory agrees fairly well with an available numerical experiment in the upper symmetric regime, for v/k [bumpe ] 7, after finite-Ekmannumber effects such as finite boundary-layer thickness are allowed for heuris-tically.

Experiments on Rotating Flows between Noncircular Cylinders

Abstract: Observations have been made on the onset of instability, and on the waveform both at onset and in the finite amplitude region for flows with closed nonrotationally symmetric streamlines. The effect of placing obstacles in the gap between concentric rotating circular cylinders is shown to depend primarily on the extent to which they modify the mean flow. The case of a square inner cylinder and circular outer cylinder is compared with the case of a circular inner cylinder and square outer cylinder. It is shown that the former arrangement, which is time‐dependent, is much less stable than the latter. It is concluded that the shape of the boundaries in the Taylor double cylinder problem do not influence the stability markedly, other than by defining the clearance ratio, unless they make the problem time‐dependent.

Flow of stratified fluid through curved screens

Abstract: A curved screen placed across a two-dimensional channel causes the streamlines to be deflected on passing through because of the variation in pressure drop across the section and the refraction effect at the screen. Uniform flows far upstream and far downstream are required by the boundary conditions. An analytical description is based on the separation of the field into two regions distant from the screen in which viscosity and molecular diffusion are negligible, plus a thin layer along the screen in which energy loss and streamline deflexion are concentrated. These are described by empirical relationships. For linear velocity and density distribution upstream of the screen, equations can be simplified so that algebraic relationships between the variables at the screen surface are obtained. These have been solved numerically for the shape of screen required to produce a specified velocity distribution. An approximate solution is also obtained for general velocity profiles and the screen shape which produces uniform shear is derived. Experimental verification of the analysis is obtained from measurements of the velocity and temperature distributions downstream of the derived screen shapes mounted in a wind tunnel 45.6 cm square. It is also shown that the boundary layers along a tunnel wall are accelerated or retarded by the screen depending on the loss coefficient. This effect is evident in all observations. The case of homogeneous fluid is described by a simplified version of the analysis and several examples of velocity distributions are produced. These are verified by experiment and compared with those predicted by Elder (1959).

Decompositional burning of a droplet in a small Peclet number flow.

Abstract: Asymptotic results for chemical-reaction and forced-convection effects on the quasi-steady adiabatic vaporization of a rigid uniform spherical droplet immersed in an unbounded expanse of gas are obtained by inner-and-outer expansions. The zeroth-order approximation is the radially symmetric solution yielding the classical logarithmic mass transfer rate. The perturbation introduces the corrections arising from 1) first-order decompositional burning for reaction rates small relative to flow rates (small first Damkohler similarity parameter); and 2) the possible wake-generating role of a slight relative flow past the droplet (small Peclet number). For tractable closed-form solution the flow is taken as incompressible with constant properties; the Lewis number is restricted to unity in the perturbational analysis. The first perturbation to the Sherwood number (normalized mass transfer rate) is found to be independent of Schmidt number, but dependent on the reaction rate. The first modification to the Stokes drag (owing to mass transfer) is found to be independent of the reaction rate, but dependent on the Schmidt number. For indefinitely large Schmidt number the Stokes drag is always increased because of mass transfer, and streamlines display no wake; for orderunity Schmidt number entirely different results are anticipated.

A derivation of Dupuit solution of steady flow toward wells by matched asymptotic expansions

Abstract: An approximate solution of the steady and shallow free-surface flow toward a well in a layer of infinite extent is obtained by expanding the velocity potential in a small parameter power series This expansion is shown to be valid only in the vicinity of the well and is, therefore, called the inner expansion An outer expansion, which solves the flow problem at large distance from the well, is derived by using the method of matched asymptotic expansions The Dupuit approximation coincides with the zero order term of the potential outer expansion The derivation of a second order outer term makes possible the discussion of the validity of the Dupuit approximation, which tends asymptotically toward the exact solution In the outer zone, the streamlines are parabolic and are not orthogonal to the equipotentials The method is illustrated by two numerical examples

Separation of a three-dimensional boundary layer

Abstract: The line of separation of the three-dimensional boundary layer on an arbitrary curvilinear surface is a singular streamline on the body surface which separates the detachment region and which is a line of confluence for the limiting streamlines. Expressions are derived for the three-dimensional separation criteria on the basis of the condition of zero frictional force in the projection on the normal to the line of separation. The position of the line of separation is determined from the solution of an ordinary differential equation. An analysis is made of various cases of separation on the surface of a yawed cylinder and on the surface of sharp cones at an angle of attack in a supersonic stream. The position of the lines of separation is determined experimentally from the confluence of thin liquid films applied to the surface. It is shown that separation occurs on the sharp cone on the line z=π for values of the parameter K=−0.85.

A uniformly valid solution for the hypersonic flow past blunted bodies

Abstract: The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio e across the shock is very small, (ii) the pressure at a point P of the disturbed flow field is not very small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal through P. Terms O(e) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.

On One-Dimensional Unsteady Flow at Infinite Mach Number

Abstract: The use of the blast-wave analogy, as an aid to the interpretation of experimental data on the motion of a fluid past an obstacle at hypersonic speeds, has led to the theoretical study of its role in an asymptotic expansion of the solution to the governing equations at large distances downstream of the body. In all attempts to set up such an expansion it has proved necessary to divide the flow regime into two parts, an outer part dominated by the blast wave and an inner part consisting of streamlines which, originally, pass close by the body. The matching of these two regions is apparently only possible if a certain integral vanishes. In the present paper a numerical integration, in one particular set of circumstances, is carried out to test the validity of the asymptotic expansion proposed. Formally, an unsteady problem is tackled, for ease of computation, but the steady analogue follows immediately and is of exactly the form discussed m the earlier investigations. It is found that the main results are in line with the theory and that the integral in question is indistinguishable from zero. However, a deeper investigation of the asymptotic expansion shows that, for an expansion of the type envisaged, an infinite set of integrals must each vanish. The next integral does not appear to be zero according to our computations but this result is not believed to be conclusive. Assuming that all the integrals do vanish, then it appears that the inner layer, which although inviscid, has many of the characteristics of a viscous boundary layer, has the additional, surprising property that it can exert no direct influence on the outer flow at large distances downstream of the body.

On the Calculation of Unsteady Nonlinear Three-Dimensional Supersonic Flow Past Wings

Abstract: A comprehensive study is made of factors influencing the accuracy of predicted thickness effects on the flow due to oscillatory motion of a sweptback wing in supersonic flight. For a delta planform with its leading edge swept at or near the Mach angle, the streamlines and velocity pattern due to thickness are found to be highly three-dimensional. Based on this observation, improvements are suggested to previous approximate second-order theories for the unsteady loading. Numerical examples are presented for a 45 deg della which indicate a substantial difference between the linearized and nonlinear pressure distributions. Although a greater volume of computation is needed to guarantee adequate accuracy, practical means are described for the complete numerical solution of problems of this type.

A Method of Predicting Oil Recovery in a Five-Spot Steamflood

Abstract: A method is presented of predicting the recovery and performance of a 5-spot steam injection project, in which a realistic approach to pattern sweepout efficiencies is made. Published methods for radial systems were modified for the 5-spot pattern by approximating the streamlines with straight lines radiating out from the injection well and then converging to the producing well. In each radial segment, the position of the steam front were determined by heat balance equations, which included an estimation of heat losses to surrounding formations. The location of the saturations behind the cold-water front was determined from a Buckley-Leverett solution to the material balance equation. Results from this program show steam recovery to be greater than straight waterflood recovery in a 5-spot pattern but considerably less than that predicted for true linear or radial flow systems.

Flow around a vertical sheetpile embedded in an inclined stratified medium

Abstract: The steady-state flow around a vertical sheetpile embedded in an inclined stratified porous medium underlain by an impervious boundary is studied experimentally. The porous medium is a synthetic ‘sandstone’ composed of alternating thin layers of two sand-epoxy mixtures with coefficients of permeability differing by a factor of 18. The problem is subjected to a coordinate transformation based upon considering the layered system as homogeneous and anisotropic, and a graphical flow net solution is obtained for the streamlines, potential distribution, and flow quantity. By consideration of the porous medium as a semi-infinite half-space and use of the same coordinate transformation, complex variable theory and conformal mapping techniques are used to obtain an approximation for the flow quantity and exit gradient variation. Experimental results on streamlines, potential distribution, flow quantity, and exit gradient were obtained from a model study, and these results were found to exhibit some variations from the theoretically predicted results, especially with regard to the flow quantities. A systematic study of possible experimental errors suggests that these variations cannot be attributed wholly to experimental sources and leads to the implication that perhaps the effective permeability of a stratified porous medium cannot be determined by considering the system as homogeneous and anisotropic.

Influence of convection on current patterns in nonequilibrium plasmas.

Abstract: The phenomenon of convection is investigated in nonequilibrium plasmas where there are substantial nonuniformities in the plasma state. It is shown that a boundary‐value problem can be formulated to solve for the current stream function and the electron temperature including convective effects. The resulting equations are coupled and nonlinear so they are solved by an iteration technique. Two simple examples are chosen to illustrate the procedure. The first consists of a pair of electrodes in a straight channel, and the second is the entrance region of a parallel‐walled solid electrode device. In each case, only an applied electric field was considered. The calculations show the current streamlines and the electron temperature distribution displaced downstream for both uniform and nonuniform flows. It is also shown that as the flow velocity increases the voltage necessary to pass a specified amount of current increases.

A class of three dimensional optimum hypersonic wings

Abstract: A limiting case of hypersonic flow is considered in which Mm —> °o • the flow deflections are small so that hypersonic small-disturbance theory applies. Within this framework there are various, known, exact solutions for flow past axisymmetric bodies. These flows are those for which the shock shape follows a power law rs '~ x*. The idea used in this paper is to construct the compression side of a lifting wing from the known streamlines in the flow behind the power-law shock wave. By considering families of such wings an optimum problem is considered, namely, to find the wing with given lift which produces a minimum wave resistance. The optimum problem is solved by variatioiial methods. Numerical results are are obtained for a range of n from ^ to 10, with y = 1.4.

General Two‐Dimensional Steady Magnetogasdynamic Flows

Abstract: Essential features of general two‐dimensional steady magneto‐gasdynamic flows are shown. For the special class of zero transverse electric field flow, new invariants along streamlines and their intrinsic properties are given.

Theory of Laminar Flows in Rounded Pipe Entrance : 1st Report, In Very Slow Flows When Mass Force Is Negligible

Abstract: For a viscous fluid flowing very slowly near a pipe inlet and exit when the mass forces of Navier-Stokes equations are negligible, the author solved strictly the equations of motion and obtained the stream function, the velocity distribution, the coefficient of skin friction and the pressure gradient. The results are as follows. (1) When the radius of a rounded pipe is large, the velocity distributions at a straight pipe entrance are near-parabolic. When it is small, they are close to the mean velocity profile. Also, when it is zero, i.e. the entrance is square-edge, the streamlines and the velocity distribution agree with Sampson's solution for a thin orifice. (2) The pressure gradient on the rounded pipe surface is large near a straight pipe inlet, and is small at a large distance from it. Either for inflow or for outflow, separation can not occur as the pressure continues to decrease due to viscosity.

Comment on "A Combined Visual and Hot-Wire Anemometer Investigation of Boundary-Layer Transition"

Abstract: NAPP and Roache1 discuss the streakline pattern of the disturbed boundary layer on ogive nose cylinders With regard to the rolling-up process of the streak-lines in their transition region Ri, they referred to the paper of Hama 2 who calculated the streakline pattern of a free shear layer perturbed by the neutral disturbance due to the linearized stability theory Hama found that the streaklines roll up, as if to indicate that the flow develops into vortices Hama assumed that in the perturbed flow there was no vorticity concentration, and it followed that the rolling-up of streaklines cannot constitute a positive identification of the presence of discrete vortices, and Knapp and Roache1 argued similarly as well This statement of Hama, however, was incorrect, since in the perturbed flow used by him, local concentrations of vorticity existed which essentially corresponded to a onerow vortex street configuration as shown in Refs 3-5 For amplified disturbances, which were investigated by Knapp and Roache, the rolling-up process of streaklines has been calculated in Refs 5 and 6 The agreement of the theoretical results with the observed smoke pattern in experiments 7 was good Furthermore, it was found in Refs 5 and 6 that the rolling-up process of the streaklines corresponds to a local concentration of vorticity I therefore suppose that the streakline pattern observed by Knapp and Roache in their region RI will, in fact, indicate a concentration of vorticity, since the streaklines look similar to those observed in free shear layers7 Another question, however, is whether these concentrations of vorticity can be characterized as "discrete vortices" I think that this is a question of definition Surely, these local concentrations of vorticity are not of the type found in a potential vortex or in a Hamel-Oseen vortex But I should prefer to denote an essential local concentration of vorticity as a discrete vortex, since the effect of a local concentration of vorticity is similar to that of a discrete vortex due to the induction law