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

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

Multiphase Systems ◽

10.21662/mfs2020.3.132 ◽

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.

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The importance of Hall effect on the magnetized thin accretion disc

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/stz3550 ◽

2019 ◽

Vol 492 (2) ◽

pp. 1770-1777

Author(s):

Maryam Ghasemnezhad

Keyword(s):

Hall Effect ◽

Equatorial Plane ◽

Radial Direction ◽

Vertical Structure ◽

Accretion Rate ◽

Accretion Disc ◽

Mhd Equations ◽

Self Similar ◽

Similar Solutions

ABSTRACT To study the role of Hall effect on the structure of accretion disc, we have considered a toroidal magnetic field in our paper. To study the vertical structure of the disc, we have written a set of magnetohydrodynamic (MHD) equations in the spherical coordinates (r, θ, ϕ) based on the two assumptions of axisymmetric and steady state. Also, we employed the self-similar solutions in the radial direction to obtain the structure of the disc in the θ-direction. We have solved a set of ordinary differential equations in the θ-coordinate with symmetrical boundary conditions in the equatorial plane. In order to describe the behaviour of Hall effect, we introduced the ΛH parameter that was called the dimensionless Hall Elsasser number. The strength of the Hall effect is measured by the inverse of dimensionless Hall Elsasser number. We have shown that the strong Hall effect decreases the accretion rate or infall velocity and size of inflow part. It has also been found the Hall effect is maximum in the equatorial plane and gets the value close to zero near the boundary, and it has the antidiffusive nature. The results display that the strong Hall effect makes the standard accretion sub-Keplerian disc becomes thinner. Our solutions have shown the Hall effect leads to transport magnetic flux outward in the upper layer of the disc and it produces outflows in the surface of the disc.

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Similarity solutions for viscous vortex cores

Journal of Fluid Mechanics ◽

10.1017/s0022112092001794 ◽

1992 ◽

Vol 238 ◽

pp. 487-507 ◽

Cited By ~ 17

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.

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The Influence of Boundaries on Gravity Currents and Thin Films: Drainage, Confinement, Convergence, and Deformation Effects

Annual Review of Fluid Mechanics ◽

10.1146/annurev-fluid-030121-025957 ◽

2022 ◽

Vol 54 (1) ◽

pp. 27-56

Author(s):

Zhong Zheng ◽

Howard A. Stone

Keyword(s):

Thin Film ◽

Oil And Gas ◽

Gravity Currents ◽

Reduced Order Models ◽

Reduced Order ◽

Self Similar ◽

On self-similarity of waves in tropical cyclones

10.5194/egusphere-egu21-14407 ◽

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

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.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).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 &#8220;slow&#8221; and &#8220;fast&#8221; when TC-induced waves outrun TC and form wake of swell trailing TC.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.The core support for this work was provided by the Russian Science Foundation through the Project &#8470;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.

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

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.148 ◽

2014 ◽

Vol 747 ◽

pp. 218-246 ◽

Cited By ~ 20

Author(s):

Zhong Zheng ◽

Ivan C. Christov ◽

Howard A. Stone

Keyword(s):

Power Law ◽

Numerical Solutions ◽

Gravity Currents ◽

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.

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

Vestnik Tomskogo gosudarstvennogo universiteta Matematika i mekhanika ◽

10.17223/19988621/71/5 ◽

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.

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Numerical Computation of the Heat Transfer and Fluid Mechanics in the Laminar Wall Jet and Comparison to the Self-Similar Solutions

Heat Transfer, Volume 2 ◽

10.1115/imece2004-61701 ◽

2004 ◽

Cited By ~ 2

Author(s):

Johnny Issa ◽

Alfonso Ortega

Keyword(s):

Boundary Conditions ◽

Analytical Solutions ◽

Numerical Solutions ◽

Fluid Dynamic ◽

Wall Jet ◽

Temperature Profiles ◽

Computational Domain ◽

Two Dimensional ◽

Self Similar ◽

Similar Solutions

Despite its importance as a canonical two-dimensional flow, the laminar wall jet has not been extensively studied using modern computational fluid dynamic methods. As in the laminar boundary layer, existence of analytical self-similar solutions make the problem particularly attractive for validating CFD code, yet we have found little archival work in which it has been used for this purpose. In the present study, we present a numerical investigation of the steady, laminar, and two-dimensional plane wall jet with constant properties. A finite-volume approach is used to solve the governing equations using self-similar inlet boundary conditions for the velocity and temperature profiles. The thermal solution is investigated for isothermal boundary condition at the wall. Velocity and temperature profiles are reported at various locations downstream and show an excellent agreement with the similarity solution obtained by Glauert [1] and Schwarz, et al. [2] respectively. In addition, the skin friction coefficient and the Nusselt number are investigated and compared with the analytical solutions presented by Glauert [1] and Mitachi, et al. [3] respectively, and very good agreement is observed. Despite its simplicity, it is shown that proper convergence of the numerical solutions of the wall jet to the expected analytical solutions requires care in specification of the jet inlet conditions, and the boundary conditions on the computational domain boundaries.

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