One-dimensional unsteady motion of a gas initially at rest and the dam-break problem

1954 ◽  
Vol 50 (1) ◽  
pp. 131-138 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Initial Conditions ◽  
Real Variable ◽  
Initial Distribution ◽  
Unsteady Motion ◽  
Dam Break ◽  
One Dimensional ◽  
Integral Methods ◽  
Countable Sequence ◽  
The One ◽  
Hydraulic Analogy

ABSTRACTThe object of this paper is to discuss the one-dimensional unsteady adiabatic motion of a gas which is initially at rest with a prescribed density distribution such that the specific entropy is uniform. The contour integral methods which Copson developed recently for even analytic functions are extended to apply to general analytic initial conditions. The solution is valid in the range 1 < Υ > 3, where y is the adiabatic index of the gas. Of particular interest, in view of the hydraulic analogy, is the case Υ = 2 for which real variable methods cannot readily be adapted. The motion of the front of a water column Sowing into a dry, horizontal stream bed is discussed. A curious type of solution, corresponding to a particular choice of initial distribution, which was established by Pack for a, countable sequence of values of Υ, is verified to hold over the whole range and is interpreted in terms of the dam-break problem.

On an extension of Riemann's method of integration, with applications to one-dimensional gas dynamics

1952 ◽  
Vol 48 (3) ◽  
pp. 499-510 ◽  
Author(s):  
Geoffrey S. S. Ludford
Keyword(s):  
Gas Dynamics ◽  
Initial Conditions ◽  
Unsteady Motion ◽  
Hyperbolic Type ◽  
Two Dimensional ◽  
Perfect Gas ◽  
Initial Value ◽  
One Dimensional ◽  
Geometrical Properties ◽  
The One

AbstractThe object of this paper is to investigate the influence of the initial values on the behaviour of the one-dimensional unsteady isentropic motion of a perfect gas, and in particular the occurrence of singularities. For this purpose it is expedient to develop an unfolding procedure in the theory of partial differential equations of hyperbolic type. This not only resolves a difficulty in the classical treatment of the initial value problem associated with this type of equation, but also simplifies the presentation.The geometrical properties of the singularities (branch and limit lines, edges of regression, etc.) occurring in two-dimensional steady flow have been well discussed already, and their counterparts in one-dimensional unsteady motion have been indicated by Stocker and Meyer(5). However, here a method for finding the explicit analytic connexion between the occurrence of such singularities and the prescribed initial conditions is developed.Assuming quite general initial conditions, two examples are considered. In the first, a conjecture of Riemann is verified; in the second a special type of periodic solution is discussed, and it is shown that this solution cannot be continued to all values of t > 0. Finally, this latter result is shown to be true for arbitrary periodic initial conditions.

scholarly journals On sound waves of finite amplitude

1953 ◽  
Vol 216 (1127) ◽  
pp. 539-547 ◽  
Keyword(s):  
Integral Equation ◽  
Analytical Solution ◽  
Arbitrary Function ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Sound Waves ◽  
One Dimensional ◽  
The One ◽  
Complex Integral ◽  
Gas Cloud

An analytical solution of Riemann’s equations for the one-dimensional propagation of sound waves of finite amplitude in a gas obeying the adiabatic law p = k ρ γ is obtained for any value of the parameter γ. The solution is in the form of a complex integral involving an arbitrary function which is found from the initial conditions by solving a generalization of Abel’s integral equation. The results are applied to the problem of the expansion of a gas cloud into a vacuum.

scholarly journals 1D and 2D Numerical Modeling for Solving Dam-Break Flow Problems Using Finite Volume Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Szu-Hsien Peng
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equation ◽  
Dam Break ◽  
Flume Experiments ◽  
One Dimensional ◽  
Dam Break Flow ◽  
The One ◽  
Volume Method

The purpose of this study is to model the flow movement in an idealized dam-break configuration. One-dimensional and two-dimensional motion of a shallow flow over a rigid inclined bed is considered. The resulting shallow water equations are solved by finite volumes using the Roe and HLL schemes. At first, the one-dimensional model is considered in the development process. With conservative finite volume method, splitting is applied to manage the combination of hyperbolic term and source term of the shallow water equation and then to promote 1D to 2D. The simulations are validated by the comparison with flume experiments. Unsteady dam-break flow movement is found to be reasonably well captured by the model. The proposed concept could be further developed to the numerical calculation of non-Newtonian fluid or multilayers fluid flow.

scholarly journals Spectral analysis of Hahn-Dirac system

2021 ◽  
Author(s):  
Bilender Allahverdiev ◽  
Hüseyin Tuna
Keyword(s):  
Spectral Analysis ◽  
Boundary Value Problem ◽  
Spectral Properties ◽  
Orthonormal Basis ◽  
Boundary Value ◽  
Dirac System ◽  
One Dimensional ◽  
Countable Sequence ◽  
Greens Function ◽  
The One

In this paper, we study some spectral properties of the one-dimensional Hahn-Dirac boundary-value problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Greens function, the existence of a countable sequence of eigenvalues, eigenfunctions forming an orthonormal basis of L2w,q ((w0. a): E).

High Pressure Diesel Engine Modeling

2003 ◽  
Author(s):  
Riccardo Baudille ◽  
Gino Bella ◽  
Rossella Rotondi
Keyword(s):  
Liquid Fuel ◽  
Initial Conditions ◽  
Injection Velocity ◽  
Dimensional Model ◽  
Flow Conditions ◽  
One Dimensional ◽  
Engine Modeling ◽  
Fuel Jet ◽  
Initial Drop ◽  
The One

In multi hole Diesel injectors, cavitation can offer advantages in the development on the fuel spray, because the primary atomization of the liquid fuel jet can be improved due to the enhanced turbulence. Several multi dimensional models of cavitating nozzle flow have been developed in order to provide information about the flow at the exit of a cavitating orifice. In this paper an analytical one-dimensional model, by Sarre et al. [1], to predict the flow conditions at the exit of a cavitating nozzle, is analyzed. The results obtained are compared with the ones obtained using the multi dimensional code Fluent in order to investigate the predictive capability of the one-dimensional code. The model provides initial conditions for multidimensional spray modeling: the effective injection velocity and the initial drop or injected liquid ‘blob’ size. The simulations were performed using an improved version of the KIVA3V code, in which an hybrid break up model, developed by the authors, is used and the results in terms of penetrations and global SMD are compared with the experimental ones. The one dimensional model predicts reasonable discharge coefficient for sharp injector geometry. Where the r/d ratio increases and the cavitation effects appear not clearly marked there are same discrepancies between the one dimensional and the multidimensional approach.

scholarly journals General solutions of the nonlinear PDEs governing the erosion kinetics

2002 ◽  
Vol 8 (1) ◽  
pp. 69-85
Author(s):  
D. E. Panayotounakos ◽  
K. P. Zafeiropoulos
Keyword(s):  
Partial Differential Equations ◽  
General Solution ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pde ◽  
Nonlinear Pdes ◽  
General Solutions ◽  
One Dimensional ◽  
Full Complement ◽  
The One

We present the construction of the general solutions concerning the one-dimensional (1D) fully dynamic nonlinear partial differential equations (PDEs), for the erosion kinetics. After an uncoupling procedure of the above mentioned equations a second–order nonlinear PDE of the Monge type governing the porosity is derived, the general solution of which is constructed in the sense that a full complement of arbitrary functions (as many as the order) is introduced. Afterwards, we specify the above solution according to convenient initial conditions.

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

2018 ◽  
Vol 28 (14) ◽  
pp. 1850176 ◽  
Author(s):  
Hegui Zhu ◽  
Wentao Qi ◽  
Jiangxia Ge ◽  
Yuelin Liu
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Function System ◽  
Topological Transitivity ◽  
One Dimensional ◽  
Devaney’S Chaos ◽  
Chebyshev Map ◽  
Sensitive Dependence ◽  
Sine Map ◽  
The One

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

On the theory of magneto-sound double simple waves

2008 ◽  
Vol 74 (4) ◽  
pp. 455-471 ◽  
Author(s):  
DAVY D. TSKHAKAYA ◽  
HOMAYOON ESHRAGHI
Keyword(s):  
Magnetic Field ◽  
Wave Solution ◽  
Blow Up ◽  
Initial Conditions ◽  
Fluid Velocity ◽  
Simple Wave ◽  
Initial Distribution ◽  
Relativistic Plasmas ◽  
The One ◽  
The Magnetic Field

AbstractA two-dimensional double simple wave solution is given for both weakly and highly magnetized non-relativistic plasmas moving across the magnetic field. The dependence of the density and the magnetic field on the two independent phases, namely, components of the fluid velocity, is derived. It is shown that initial spatial distributions must satisfy a definite equation whose solution determines a special category for initial conditions. The time of blow up for any fixed value of the pair phase is found. A large general class of solutions for initial distributions is obtained. For any chosen initial distribution, the physical plane of flow at any instant of time splits into two regions, one forbidden and the other permitted. These regions are obtained numerically at a typical time for a special initial distribution. For this double wave solution, differential equations for streamlines and fluid trajectories are derived. Only for the simplest cases can the corresponding curves be completely integrated and these are given in this paper. The results are qualitatively similar to the one-dimensional case derived by Stenflo and Shukla.

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