A parametric study of self-similar blast waves

1972 ◽  
Vol 52 (4) ◽  
pp. 657-682 ◽  
Author(s):  
A. K. Oppenheim ◽  
A. L. Kuhl ◽  
E. A. Lundstrom ◽  
M. M. Kamel
Keyword(s):  
Phase Plane ◽  
Integral Curve ◽  
Blast Waves ◽  
Uniform Density ◽  
Integral Curves ◽  
Comprehensive Examination ◽  
Self Similar ◽  
Two Parameters ◽  
The Waves ◽  
Shock Fronts

The paper presents a comprehensive examination of self-similar blast waves with respect to two parameters, one describing the front velocity and the other the variation of the ambient density immediately ahead of the front. All possible front trajectories are taken into account, including limiting cases of the exponential and logarithmic form. The structure of the waves is analysed by means of a phase plane defined in terms of two reduced co-ordinatesF≡ (t/rμ)uandZ≡ [(t/rμ)a]2, wheretandrare the independent (time and space) variables, μ ≡dlnrn/dIntnthe subscriptndenoting the co-ordinates of the front, anduandaare, respectively, the particle velocity and the speed of sound. Loci of extrema of the integral curves in the phase plane are traced and loci of singularities are determined on the basis of their intersections. Boundary conditions are introduced for the case when the medium into which the waves propagate is at rest. Representative solutions, pertaining to all the possible cases of blast waves bounded by shock fronts propagating into an atmosphere of uniform density, are obtained by evaluating the integral curves and determining the corresponding profiles of the gasdynamic parameters. Particular examples of integral curves for waves bounded by detonations are given and all the degenerate solutions, corresponding to cases where the integral curve is reduced to a point, are delineated.

Self-similar viscous gravity currents: phase-plane formalism

1990 ◽  
Vol 210 ◽  
pp. 155-182 ◽  
Author(s):  
Julio Gratton ◽  
Fernando Minotti
Keyword(s):  
Scaling Laws ◽  
Phase Plane ◽  
Gravity Current ◽  
Classical Problem ◽  
Gravity Currents ◽  
Steady Flows ◽  
Two Phase ◽  
Integral Curves ◽  
Self Similar ◽  
Similar Solutions

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Self-similar explosion waves of variable energy at the front

1980 ◽  
Vol 99 (4) ◽  
pp. 841-858 ◽  
Author(s):  
G. I. Barenblatt ◽  
R. H. Guirguis ◽  
M. M. Kamel ◽  
A. L. Kuhl ◽  
A. K. Oppenheim ◽  
...  
Keyword(s):  
Energy Deposition ◽  
Density Ratio ◽  
Blast Waves ◽  
Ambient Atmosphere ◽  
Pressure Profiles ◽  
Integral Curves ◽  
Self Similar ◽  
Strong Explosion ◽  
Similar Solutions ◽  
Variable Energy

A set of self-similar solutions for blast waves associated with the deposition of variable energy at the front is presented. As a consequence of self-similarity, the results are applicable when the ambient atmosphere into which the wave front propagates is at a negligibly low pressure and temperature. Besides the class of (1) blast waves associated with energy gain that covers a regime bounded on one side by the well-known solution for adiabatic strong explosion waves (ASE) and, on the other side, by the solution for waves having the Chapman–Jouguet condition established immediately behind the front, included within the scope of our analysis are two others: (2) blast waves associated with energy loss that occupy a regime between the ASE solution and the case of infinite density ratio across the front, and (3) a non-unique class of solutions for blast waves associated with energy deposition that may have either locally sonic or supersonic flow immediately behind the front, extending over the regime between the waves headed by the Chapman-Jouguet detonation and the case of infinite rate of energy deposition. Specific results for a number of representative cases are expressed in terms of integral curves on the phase plane of reduced blast wave co-ordinates, as well as in the form of particle velocity, temperature, density, and pressure profiles across the flow field.

scholarly journals Mathematical Theory of Cylindrical Isothermal Blast Waves in a Magnetic Field

1981 ◽  
Vol 34 (3) ◽  
pp. 279 ◽  
Author(s):  
I Lerche
Keyword(s):  
Magnetic Field ◽  
Supernova Remnants ◽  
Magnetic Pressure ◽  
Blast Waves ◽  
Physical Domain ◽  
Fluid Flow Velocity ◽  
Magnetic Tension ◽  
Self Similar ◽  
The Magnetic Field ◽  
Similar Flows

An investigation is made of the self-similar flow behind a cylindrical blast wave from a line explosion (situated on r = 0, using conventional cylindrical coordinates r, 4>, z) in a medium whose density and magnetic field both vary as r -w ahead of the blast front, with the assumption that the flow is isothermal. The magnetic field can have components in both the azimuthal B(jJ and longitudinal B, directions. It is found that: (i) For B(jJ =f:. 0 =f:. B, a continuous single-valued solution with a velocity field representing outflow of material away from the line of explosion does not exist for OJ OJ > 0 the governing equation possesses a set of movable critical points. In this case it is shown that the fluid flow velocity is bracketed between two curves and that the asymptotes of the velocity curve on the shock are intersected by, or are tangent to, the two curves. Thus a solution always exists in the physical domain r ~ o. The overall conclusion from the investigation is that the behaviour of isothermal blast waves in the presence of an ambient magnetic field differs substantially from the behaviour calculated for no magnetic field. These results have an impact upon previous applications of the theory of self-similar flows to evolving supernova remnants without allowance for the dynamical influence of magnetic pressure and magnetic tension.

scholarly journals A PARAMETRIC HURRICANE WAVE PREDICTION MODEL

1988 ◽  
Vol 1 (21) ◽  
pp. 82
Author(s):  
Ian R. Young
Keyword(s):  
Spatial Distribution ◽  
Radiative Transfer Equation ◽  
Synthetic Data ◽  
Wave Model ◽  
Limited Growth ◽  
Forward Movement ◽  
Spectral Wave Model ◽  
Wave Conditions ◽  
Two Parameters ◽  
The Waves

A spectral wave model based on a numerical solution of the Radiative Transfer Equation is used to create a synthetic data base on wave conditions within hurricanes. The results indicate that both the velocity of forward movement and maximum wind velocity within the storm play an important role in determining both the magnitude of the waves generated and also the spatial distribution of these waves. An equivalent fetch for hurricane wave generation which is a function of these two parameters is proposed. This concept, together with the standard JONSWAP fetch limited growth relationships, provide a simple means for estimating wave conditions within hurricanes.

A simple representation of a developing contaminant concentration field

1995 ◽  
Vol 289 ◽  
pp. 141-157 ◽  
Author(s):  
B. L. Sawford ◽  
P. J. Sullivan
Keyword(s):  
Simple Form ◽  
Line Source ◽  
Concentration Field ◽  
Scalar Fields ◽  
Source Distribution ◽  
Additional Parameter ◽  
Wide Range ◽  
Observed Behaviour ◽  
Self Similar ◽  
Two Parameters

Chatwin & Sullivan (1990) have demonstrated that, for a wide range of self-similar scalar fields, the moments of the probability density function of concentration have a very simple form. Here, an extension to this simple form which takes account of the source distribution is developed. This extension has two effects. Firstly it modifies the values of the two parameters appearing in the original theory and in particular explains the observed behaviour of these parameters very near to a line source of heat in grid tubulence. Secondly, it introduces an additional parameter in the description of each moment beyond the second. It is shown that these additional parameters are necessary in order to describe measurements of the first four central moments throughout the concentration field from a continuous line source of heat in grid-generated turbulence.

Effects of Small-World Rewiring Probability and Noisy Synaptic Conductivity on Slow Waves: Cortical Network

2017 ◽  
Vol 29 (3) ◽  
pp. 679-715
Author(s):  
Ramazan Tekin ◽  
Mehmet Emin Tagluk
Keyword(s):  
Critical Role ◽  
Small World ◽  
Cortical Network ◽  
Abnormal Development ◽  
Oscillatory Activity ◽  
Normal Brain ◽  
Brain Functions ◽  
Functional Development ◽  
Two Parameters ◽  
The Waves

Physiological rhythms play a critical role in the functional development of living beings. Many biological functions are executed with an interaction of rhythms produced by internal characteristics of scores of cells. While synchronized oscillations may be associated with normal brain functions, anomalies in these oscillations may cause or relate the emergence of some neurological or neuropsychological pathologies. This study was designed to investigate the effects of topological structure and synaptic conductivity noise on the spatial synchronization and temporal rhythmicity of the waves generated by cells in the network. Because of holding the ability of clustering and randomizing with change of parameters, small-world (SW) network topology was chosen. The oscillatory activity of network was tried out by manipulating an insulated SW, cortical network model whose morphology is very close to real world. According to the obtained results, it was observed that at the optimal probabilistic rates of conductivity noise and rewiring of SW, powerful synchronized oscillatory small waves are generated in relation to the internal dynamics of cells, which are in line with the network’s input. These two parameters were observed to be quite effective on the excitation-inhibition balance of the network. Accordingly, it may be suggested that the topological dynamics of SW and noisy synaptic conductivity may be associated with the normal and abnormal development of neurobiological structure.

scholarly journals Mathematical Theory of One-dimensional Isothermal Blast Waves in a Magnetic Field

1979 ◽  
Vol 32 (5) ◽  
pp. 491 ◽  
Author(s):  
I Lerche
Keyword(s):  
Magnetic Field ◽  
Fluid Flow ◽  
Critical Point ◽  
Blast Wave ◽  
Supernova Remnants ◽  
Magnetic Energy ◽  
Flow Speed ◽  
Blast Waves ◽  
One Dimensional ◽  
Self Similar

An investigation is made of the self-similar flow behind a one-dimensional blast wave from a planar explosion (situated on z = 0) in a medium whose density and magnetic field vary with distance as Z-W ahead of the blast front, with the assumption that the flow is isothermal. It is found that; if OJ OJ > 0 the governing equation possesses a set of movable critical points. For a weak, but nonzero, magnetic field it is shown that the value of the smallest critical point does not lie in the physical domain z > O. The post-shock fluid flow then cannot intersect the critical point, and is smoothly continuous. It is shown that to be physically acceptable, the fluid flow speed must pass through the origin. It is also shown that OJ must be less than t for the magnetic energy swept up by the blast wave to remain finite. The overall conclusion from the investigation is that the behaviour of isothermal blast waves in the presence of an ambient magnetic field differs substantially from the behaviour calculated for no magnetic field. These results point to the inadequacy of previous attempts to apply the theory of self-similar flows to evolving supernova remnants without making any allowance for the dynamical influence of magnetic field pressure.

Influence of Mantle Structures on Measurements of Anisotropy in the Inner Core

2020 ◽  
Author(s):  
Sandra Beiers ◽  
Christine Thomas
Keyword(s):  
Seismic Anisotropy ◽  
Inner Core ◽  
Incidence Angle ◽  
Outer Core ◽  
Western Hemisphere ◽  
Travel Times ◽  
Array Seismology ◽  
Two Parameters ◽  
The Waves ◽  
Ray Paths

<p>The seismological exploration of the Earth’s inner core has revealed some structural complexities such as seismic anisotropy and hemispherical separation. Investigating the travel times of PKP waves from at least two different ray paths, a polar and an equatorial one, is one of the commonly used methods to probe the inner core’s anisotropy. Since the waves are traversing anomalous structures in the lowermost mantle before entering the core, these heterogeneities have to be taken into account when investigating anisotropy in the inner core.</p><p>In this study we use data from an equatorial path with events from Indonesia recorded in Morocco and a nearly polar one with earthquakes in New Zealand recorded in England. The two waves used in our study, PKPdf and PKPab, both propagate through mantle and outer core and PKPab additionally traverses the inner core. Within this work, we do not only analyse the travel times of the waves but rather investigate their deviations from the originally assumed path along with their incidence angle. This is done with the methods of array seismology, mainly its two parameters slowness and backazimuth.</p><p>The results of this study reveal opposite deviations of slowness and backazimuth of the polar in contrast to the equatorial path. While the polar waves travel shallower and closer to North, the equatorial waves propagate deeper and farther from North than predicted by ak135. Additionally we observe hemispherical differences between waves that sample the eastern and the ones that sample the western hemisphere for both ray paths, PKPdf and PKPab, which leads us to the assumption that the deviations are not caused by the inner core but are rather due to mantle structures.</p>

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