Channel Of Finite Width Research Articles

Induced Mach wave–flame interactions in laminar supersonic fuel jets

A model problem is proposed to investigate the steady response of a reacting, compressible laminar jet to Mach waves generated by wavy walls in a channel of finite width. The model consists of a two-dimensional jet of fuel emerging into a stream of oxidizer which are allowed to mix and react in the presence of the Mach waves. The governing equations are taken to be the steady parabolized Navier–Stokes equations which are solved numerically. The kinetics is assumed to be a one-step, irreversible reaction of the Arrhenius type. Two important questions on the Mach wave–flame interactions are discussed: (i) how is the flame structure altered by the presence of the Mach waves, and (ii) can the presence of the Mach waves change the efficiency of the combustion processes?

Development of solidification microstructures in the presence of fibers or channels of finite width

Directional solidification studies in a transparent succinonitrile system have been carried out in the presence of single or multiple fibers. The solute-rich region and the solute bands which form as the interface approaches the fibers are discussed. The microstracture formation in small interfiber spacings has been examined and it is shown that the development of microstructure depends on (a) the interfiber spacing, (b) the angle between the two fibers and (c) the orientation of fibers with respect to the growth direction. Experimental studies have also been carried out to examine systematically the microstructural development in well-characterized rectangular channels of controlled widths. Under given experimental conditions of growth rate, temperature gradient and composition, it is shown that the microstructure changes from dendritic to cellular to a nearly planar interface as the width of the channel is reduced. Furthermore, for certain channel widths, half-dendritic or half-cellular structures have also been observed. The various implications of these observations for the fundamental understanding of microstructural evolutions and for the applied aspects of solidification of composite materials are discussed.

Stress-Gradient Coupling in Glacier Flow: II. Longitudinal Averaging in the Flow Response to Small Perturbations in Ice Thickness and Surface Slope

AbstractAs a result of the coupling effects of longitudinal stress gradients, the perturbations ∆u in glacier-flow velocity that result from longitudinally varying perturbations in ice thickness ∆h and surface slope ∆α are determined by a weighted longitudinal average of ϕh∆h and ϕα∆α, where ϕh and ϕα are “influence coefficients” that control the size of the contributions made by local ∆h and ∆α to the flow increment in the longitudinal average. The values of ϕh and ϕα depend on effects of longitudinal stress and velocity gradients in the unperturbed datum state. If the datum state is an inclined slab in simple-shear flow, the longitudinal averaging solution for the flow perturbation is essentially that obtained previously (Kamb and Echelmeyer, 1985) with equivalent values for the longitudinal coupling length l and with ϕh = n + 1 and ϕα + n, where n is the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practice ϕα differs little from n, whereas ϕh can differ considerably from n + 1. The weighting function in the longitudinal averaging integral, which is the Green’s function for the longitudinal coupling equation for flow perturbations, can be approximated by an asymmetric exponential, whose asymmetry depends on two “asymmetry parameters” μ and σ, where μ is the longitudinal gradient of ℓ(= dℓ/dx). The asymmetric exponential has different coupling lengths ℓ+ and ℓ− for the influences from up-stream and from down-stream on a given point of observation. If σ/μ is in the range 1.5–2.2, as expected for flow perturbations in glaciers or ice sheets in which the ice flux is not a strongly varying function of the longitudinal coordinate x, then, when dℓ/dx > 0, the down-stream coupling length ℓ+ is longer than the up-stream coupling length ℓ−, and vice versa when dℓ/dx < 0. Flow-, thickness- and slope-perturbation data for Blue Glacier, obtained by comparing the glacier in 1957–58 and 1977–78, require longitudinal averaging for reasonable interpretation. Analyzed on the basis of the longitudinal coupling theory, with 4ℓ + 1.6 km up-stream, decreasing toward the terminus, the data indicate n to be about 2.5, if interpreted on the basis of a response factor Ѱ + 0.85 derived theoretically by Echelmeyer (unpublished) for the flow response to thickness perturbations in a channel of finite width. The data contain an apparent indication that the flow response to slope perturbations is distinctly smaller, in relation to the response to thickness perturbations, than is expected on a theoretical basis (i.e. ϕα/ ϕh + (n/n + 1) for a slab). This probably indicates that the effective ℓ is longer than can be tested directly with the available data set owing to its limited range in x.

Stress-Gradient Coupling in Glacier Flow: II. Longitudinal Averaging in the Flow Response to Small Perturbations in Ice Thickness and Surface Slope

AbstractAs a result of the coupling effects of longitudinal stress gradients, the perturbations ∆uin glacier-flow velocity that result from longitudinally varying perturbations in ice thickness ∆hand surface slope ∆α are determined by a weighted longitudinal average ofϕh∆handϕα∆α, whereϕhandϕαare “influence coefficients” that control the size of the contributions made by local ∆hand ∆α to the flow increment in the longitudinal average. The values ofϕhandϕα depend on effects of longitudinal stress and velocity gradients in the unperturbed datum state. If the datum state is an inclined slab in simple-shear flow, the longitudinal averaging solution for the flow perturbation is essentially that obtained previously (Kamb and Echelmeyer, 1985) with equivalent values for the longitudinal coupling lengthland withϕh=n+ 1 andϕα+n, wherenis the flow-law exponent. Calculation of the influence coefficients from flow data for Blue Glacier, Washington, indicates that in practiceϕα differs little fromn, whereasϕhcan differ considerably fromn+ 1. The weighting function in the longitudinal averaging integral, which is the Green’s function for the longitudinal coupling equation for flow perturbations, can be approximated by an asymmetric exponential, whose asymmetry depends on two “asymmetry parameters”μand σ, whereμis the longitudinal gradient of ℓ(= dℓ/dx). The asymmetric exponential has different coupling lengths ℓ+and ℓ−for the influences from up-stream and from down-stream on a given point of observation. If σ/μis in the range 1.5–2.2, as expected for flow perturbations in glaciers or ice sheets in which the ice flux is not a strongly varying function of the longitudinal coordinatex, then, when dℓ/dx> 0, the down-stream coupling length ℓ+is longer than the up-stream coupling length ℓ−, and vice versa when dℓ/dx< 0. Flow-, thickness- and slope-perturbation data for Blue Glacier, obtained by comparing the glacier in 1957–58 and 1977–78, require longitudinal averaging for reasonable interpretation. Analyzed on the basis of the longitudinal coupling theory, with 4ℓ + 1.6 km up-stream, decreasing toward the terminus, the data indicatento be about 2.5, if interpreted on the basis of aresponse factor Ѱ+ 0.85 derived theoretically by Echelmeyer (unpublished) for the flow response to thickness perturbations in a channel of finite width. The data contain an apparent indication that the flow response to slope perturbations is distinctly smaller, in relation to the response to thickness perturbations, than is expected on a theoretical basis (i.e.ϕα/ ϕh+ (n/n+ 1) for a slab). This probably indicates that the effective ℓ is longer than can be tested directly with the available data set owing to its limited range inx.

Nonlinear Rossby Waves in a Channel of Finite Width

Nonlinear topographic Rossby waves propagating in a channel of finite width are investigated by the reductive perturbation method. When the channel is sufficiently wide and homogeneous in the outside of a wave guide, the resultant equation takes the form of that obtained by Joseph, and Kubota, Ko and Dobbs. On the other hand, in the presence of weak inhomogeneity, it is shown that a Rossby wave soliton undergoes modification such as change in phase velocity and radiation damping according to the properties of the inhomogeneity.

Numerical simulation of triple shock behavior of gaseous detonation

A numerical analysis of performed on the nonsteady two-dimensional behavior of self-sustaining detonations. In order to produce realistic two-dimensional structures during propagation in a finite width channel, an initially assumed plane one-dimensional Chapman-Jouguet detonation was perturbed by locating up to five pairs of exothermicity spots in the passage of the detonation. The most interesting feature of the simulation was the number of triple shock waves existing after a certain time of transition from plane to periodically nonsteady two-dimensional structures. In other words, the final number of transverse shock waves becomes mostly two, which are nearly irrelevant to the assumed number of initial exothermicity spots. Details of the disappearance of several transverse waves during their rearrangement were clearly observed and are discussed. In addition, despite considerable deviation of the instantaneous propagation velocity from from the C-J value, the average of such oscillations was close to the C-J value. Thus, the results of the simulation provided the numerous features observed in the present gaseous detonations experiments. The numerical schemes utilized were the explicit first-order Van Leer and second-order MacCormack methods. The effect of artificial viscosity was examined and found to affect the transition to fully developed detonation.

Ondes d'inertie dans un canal infini de section rectangulaire

Inertial waves in a channel of finite width consist of two different systems. In the surface layer a suddenly imposed wind produces the same system of Ekman currents and inertial waves as in the open ocean. Besides that the wind produces cross oscillations and a permanent deflection of the sea surface and due to this a geostrophic current. The onset of this geostrophic current is accompanied by a second system of inertial waves which is out of phase by 1800. In the vertically integrated equations the two systems cancel each other.

Analytical Model of Dispersion in Tidal Fjords

Using the integral transforms technique, the unsteady three-dimensional advective-diffusion equation was solved analytically for the dispersion of pollutants from an instantaneous point source in open channels of finite width. The flow velocity through the channel was the mean current velocity which was assumed to vary sinusoidally with time but linearly with water depth. The effect of first-order reactions is to reduce the concentration level throughout the channel. The spread of pollutants in the direction of mean flow is accelerated as a result of the existence of a vertical shear. The width of the channel is important, as is the boundary in the y direction that profoundly affects the horizontal dispersion when the source is close to the bank, but its effect diminishes as the source is moving farther away from the bank. These findings, as well as others, are examined and illustrated graphically.

Finite-element solution of navier-stokes equations for transient two-dimensional incompressible flow

A method is described for the solution of transient, incompressible viscous flow in two dimensions. The dependent variables, stream function and vorticity, were approximated over each triangular element using linear interpolation functions. This approximation reduces the problem to a set of matrix equations whose term involving derivatives of time is the mass matrix. The lumping of this matrix together with the application of Euler integration scheme produces an efficient method of solution. Once the nodal values of the stream function are known the velocities and pressure can be computed. As an application a study of the vortex street development behind a rectangular obstruction is described. The flow has been impulsively accelerated to a constant speed in a channel of finite width. The Reynolds number range investigated is between 20 and 100.

Jets from two-dimensional symmetric nozzles of arbitrary shape

A unified approach to the problem of jet efflux from symmetrical channels of finite width and possessing a general curvilinear nozzle shape is presented. The nozzle may be composed of polygonal and/or curved-arc segments. Precise nozzle shapes cannot be initially prescribed, however. The solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. The selection of an appropriate curvature function is described; examples show possible applications.

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.

VII. The Kármán street of vortices in a channel of finite breadth

1. The following investigations deal with a Kármán street of vortices, or unsymmetrical double row, in a channel of finite width. A discussion on the symmetrical double row has also been incorporated. For the sake of convenience the work has been divided into two distinct parts—(1) Systems of Line Vortices in a Channel of Finite Breadth, and (II) The Kármán Drag Formula for a Channel of Finite Breadth. Part I has itself been divided into two sections headed Problem I and Problem II, and these sections deal with the stream lines and stability of the unsymmetrical and symmetrical double row respectively. Part II discusses Problem I from the point of view of hydrodynamics and contains a section dealing with the hydrodynamical significance of the stability investigations. Summaries of the investigations have been kept distinct and can be found in paragraphs (1.5) and (1.3) of Part I and Part II respectively.