Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

2020 ◽  
Vol 15 (3-4) ◽  
pp. 212-216
Author(s):  
R.Kh. Bolotnova ◽  
V.A. Korobchinskaya
Keyword(s):  
Gas Dynamics ◽  
Numerical Solutions ◽  
Similarity Theory ◽  
Compressible Medium ◽  
Initial Moment ◽  
Point Explosion ◽  
Perfect Gas ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Similar Solutions

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

Self-similar solutions in gas dynamics with exponential time dependence

1997 ◽  
Vol 9 (5) ◽  
pp. 1462-1469 ◽  
Author(s):  
V. Simonsen ◽  
J. Meyer-ter-Vehn
Keyword(s):  
Time Dependence ◽  
Gas Dynamics ◽  
Exponential Time ◽  
Self Similar ◽  
Similar Solutions

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

On self-similarity of waves in tropical cyclones

2021 ◽  
Author(s):  
Maria Yurovskaya ◽  
Vladimir Kudryavtrsev ◽  
Bertrand Chapron
Keyword(s):  
Wave Field ◽  
Tropical Cyclones ◽  
Numerical Solutions ◽  
Parametric Model ◽  
Linear Wave ◽  
Wave System ◽  
Large Computer ◽  
State Assignment ◽  
Self Similar ◽  
Similar Solutions

<p>Wave fields generated by tropical cyclones (TC) are of strong interest for marine engineering, navigation safety, determination of coastal sea levels and coastal erosion. Considerable efforts have been made to improve knowledge about the surface waves in TC, both from measurements and numerical experiments. Full sophisticated spectral wave models certainly have the capability to provide detailed wave information, but they require large computer power, precise well-resolved surface winds and/or needs to consider large ensembles of solutions. In this context, more simplified but robust solutions are demanded.</p><p>This work is based on 2D-parametric model of waves evolution forced by wind field varying in space and time, non-linear wave interactions and wave breaking dissipation [submitted to J. Geoph. Res., see also preprint DOI: https://doi.org/10.1002/essoar.10504620.1]. Numerical solutions of model provide efficient visualization on how waves develop under TC and leave it as swell. Superposition of wave-rays exhibits coherent spatial patterns of wave parameters depending on TC characteristics, - maximal wind speed (um), radius (Rm), and translation velocity (V).</p><p>In this presentation we demonstrate how solutions of 2D-parametric model can be described analytically through self-similar functionsusing proper scaling involving the main TC parameters: um, Rm, and V. These self-similar solutions can be treated as TC-wave Geophysical Model Function (TC-wave GMF), to help analytically derive azimuthal-radial distributions of the primary wave system parameters (SWH, wavelength, direction) under TC characterized by arbitrary sets of um, Rm and V conditions. Self-similar solutions describe the main properties of wave field under TC, in particular: right-to-left half asymmetry of wave field under TC; strong dependence of wave energy and wavelength on V, um and Rm caused by group velocity resonance; division of TCs on “slow” and “fast” when TC-induced waves outrun TC and form wake of swell trailing TC.</p><p>Comparisons between self-similar solutions and measurements of TC-generated waves reported in the literature, demonstrate excellent agreement to warrant their use for research and practical applications.</p><p>The core support for this work was provided by the Russian Science Foundation through the Project №21-47-00038 at RSHU. The support of the Ministry of Science and Education of the Russian Federation under State Assignment No. 0555-2021-0004 at MHI RAS, and State Assignment No. 0736-2020-0005 at RSHU are gratefully acknowledged.</p>

scholarly journals Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents

2014 ◽  
Vol 747 ◽  
pp. 218-246 ◽  
Author(s):  
Zhong Zheng ◽  
Ivan C. Christov ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Numerical Solutions ◽  
Gravity Currents ◽  
Similar Solution ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Self Similar Solution ◽  
Theoretical Predictions ◽  
Self Similar ◽  
Similar Solutions

AbstractWe report experimental, theoretical and numerical results on the effects of horizontal heterogeneities on the propagation of viscous gravity currents. We use two geometries to highlight these effects: (a) a horizontal channel (or crack) whose gap thickness varies as a power-law function of the streamwise coordinate; (b) a heterogeneous porous medium whose permeability and porosity have power-law variations. We demonstrate that two types of self-similar behaviours emerge as a result of horizontal heterogeneity: (a) a first-kind self-similar solution is found using dimensional analysis (scaling) for viscous gravity currents that propagate away from the origin (a point of zero permeability); (b) a second-kind self-similar solution is found using a phase-plane analysis for viscous gravity currents that propagate toward the origin. These theoretical predictions, obtained using the ideas of self-similar intermediate asymptotics, are compared with experimental results and numerical solutions of the governing partial differential equation developed under the lubrication approximation. All three results are found to be in good agreement.

A Fractional Model of Gas Dynamics Equations and its Analytical Approximate Solution Using Laplace Transform

2012 ◽  
Vol 67 (6-7) ◽  
pp. 389-396 ◽  
Author(s):  
Sunil Kumar ◽  
Huseyin Kocak ◽  
Ahmet Yıldırım
Keyword(s):  
Laplace Transform ◽  
Gas Dynamics ◽  
Numerical Solutions ◽  
Mass Conservation ◽  
Momentum Conservation ◽  
Gas Dynamics Equations ◽  
Transform Method ◽  
Conservation Of Energy ◽  
Homotopy Perturbation ◽  
He’S Polynomials

In this study, the homotopy perturbation transform method (HPTM) is performed to give approximate and analytical solutions of nonlinear homogenous and non-homogenous time-fractional gas dynamics equations. Gas dynamics equations are based on the physical laws of conservation, namely, the laws of conservation of mass, conservation of momentum, conservation of energy etc. The HPTM is a combined form of the Laplace transform, the homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed method indicate that the approach is easy to implement and accurate. Some numerical illustrations are given. These results reveal that the proposed method is very effective and simple to perform

Application of nonstationary self-similar variables for solving the three-dimensional problem of the decay of a special discontinuity

2021 ◽  
pp. 52-64
Author(s):  
Сергей Петрович Баутин ◽  
Сергей Львович Дерябин
Keyword(s):  
Cauchy Problem ◽  
Power Series ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Physical Space ◽  
Initial Moment ◽  
Algebraic Equations ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Isentropic Flows

Построение в физическом пространстве решения задачи о распаде специального разрыва, т.е. трехмерных изэнтропических течений политропного газа, возникающих после мгновенного разрушения в начальный момент времени непроницаемой стенки, отделяющей неоднородный движущийся газ от вакуума. В задаче учитывается действие силы тяжести и силы Кориолиса. В систему уравнений газовой динамики введена автомодельная особенность в переменную, которая выводит с поверхности раздела. Для полученной системы поставлена задача Коши с данными на звуковой характеристике. Решение задачи строилось в виде степенных рядов. Часть коэффициентов рядов определялась при решении алгебраических уравнений, а часть из решений - обыкновенных дифференциальных уравнений. Методом мажорант доказана сходимость построенных рядов. Построенное решение позволяет задавать начальные условия для разностной схемы при численном моделировании решений данной характеристической задачи Коши The aim of this study is to construct a solution to the problem of the decay of a special discontinuity in physical space. The problem reduces to finding of three-dimensional isentropic flows of a polytropic gas that occur after the instantaneous destruction of an impermeable wall separating an inhomogeneous moving gas from a vacuum at the initial moment of time. The problem takes into account the forces of gravity and Coriolis. Research methods. In the system of gas dynamics equations, a self-similar feature is introduced in a variable that outputs from the initial interface. For the resulting system, the Cauchy problem is formulated using conditions on the sound characteristic. The solution to this problem is constructed in the form of power series. The coefficients of the series are partly determined by solving algebraic equations, another part can be found as solutions of ordinary differential equations. The convergence of the constructed series is proved by the Majorant method The results obtained in the work. In the form of a convergent power series, solutions to the problem of the decay of a special discontinuity in physical space are constructed. Conclusions. The solution constructed in physical space allows setting the initial conditions for the numerical simulation of this characteristic Cauchy problem using a difference scheme.

CONVERGENCE OF LOCALLY SELFSIMILAR SOLUTIONS TO EXACT NUMERICAL SOLUTIONS OF BOUNDARY LAYER EQUATIONS FOR A PLATE

2021 ◽  
pp. 49-62
Author(s):  
Yu.N. Grigoriev ◽  
◽  
A.G. Gorobchuk ◽  
I.V. Ershov ◽  
◽  
...  
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Linear Stability ◽  
Numerical Solutions ◽  
Linear Stability Theory ◽  
Plane Boundary ◽  
Transfer Coefficients ◽  
Vibrationally Excited ◽  
Self Similar ◽  
Similar Solutions

This paper considers a possibility of using locally self-similar solutions for a stationary boundary layer in linear stability problems. The solutions, obtained at various boundary conditions for a vibrationally excited gas, are compared with finite-difference calculations of the corresponding flows. An initial system of equations for a plane boundary layer of the vibrationally excited gas is derived from complete equations of two-temperature relaxation aerodynamics. Relaxation of vibrational modes of gas molecules is described in the framework of the Landau – Teller equation. Transfer coefficients depend on the static flow temperature. In a complete problem statement, the flows are calculated using the Crank – Nicolson finite-difference scheme. In all the considered cases, it is shown that the locally self-similar velocity and temperature profiles converge to the corresponding profiles for a fully developed boundary-layer flow calculated in a finite-difference formulation. The obtained results justify the use of locally self-similar solutions in problems of the linear stability theory for boundary-layer flows of a vibrationally excited gas.

scholarly journals Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 394
Author(s):  
Alexander B. Kukushkin ◽  
Andrei A. Kulichenko ◽  
Vladislav S. Neverov ◽  
Petr A. Sdvizhenskii ◽  
Alexander V. Sokolov ◽  
...  
Keyword(s):  
Radiative Transfer ◽  
Numerical Solutions ◽  
Dominant Role ◽  
Spectral Lines ◽  
Excitation Density ◽  
Probability Method ◽  
Lévy Walks ◽  
Self Similar ◽  
Similar Solutions

Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.

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