scholarly journals Sonic point and photon surface

2020 ◽  
Vol 102 (4) ◽  
Author(s):  
Masataka Tsuchiya ◽  
Chul-Moon Yoo ◽  
Yasutaka Koga ◽  
Tomohiro Harada
Keyword(s):  
Sonic Point

scholarly journals Outflowing Envelopes From Stars at Arbitrary Optical Depths a New Approach to the Problem

1998 ◽  
Vol 11 (1) ◽  
pp. 381-381
Author(s):  
A.V. Dorodnitsyn
Keyword(s):  
Differential Equations ◽  
Massive Star ◽  
System Of Differential Equations ◽  
Continuous Transition ◽  
Satisfactory Description ◽  
Sonic Point ◽  
New Approach ◽  
Star Evolution ◽  
Matter Flux ◽  
Massive Star Evolution

We have considered a stationary outflowing envelope accelerated by the radiative force in arbitrary optical depth case. Introduced approximations provide satisfactory description of the behavior of the matter flux with partially separated radiation at arbitrary optical depths. The obtained systemof differential equations provides a continuous transition of the solution between optically thin and optically thick regions. We analytically derivedapproximate representation of the solution at the vicinity of the sonic point. Using this representation we numerically integrate the system of equations from the critical point to the infinity. Matching the boundary conditions we obtain solutions describing the problem system of differential equations. The theoretical approach advanced in this work could be useful for self-consistent simulations of massive star evolution with mass loss.

A Cure for the Sonic Point Glitch

2000 ◽  
Vol 13 (2) ◽  
pp. 143-159 ◽  
Author(s):  
J.-M. MOSCHETTA ◽  
J. GRESSIER
Keyword(s):  
Sonic Point

On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex

1998 ◽  
Vol 374 ◽  
pp. 285-303 ◽  
Author(s):  
D. W. MOORE ◽  
D. I. PULLIN
Keyword(s):  
Finite Difference ◽  
Compressible Flows ◽  
Numerical Solutions ◽  
Flow Direction ◽  
Fluid Velocity ◽  
Major Axis ◽  
Sonic Point ◽  
Euler Flow ◽  
Compressible Euler ◽  
Spherical Vortex

We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M∞=U∞/C∞, where U∞, C∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh–Janzen expansion in powers of M2∞ to order M4∞. When M∞>0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M∞≈0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

scholarly journals The charge-exchange induced coupling between plasma-gas counterflows in the heliosheath

2003 ◽  
Vol 21 (6) ◽  
pp. 1289-1294 ◽  
Author(s):  
H. J. Fahr
Keyword(s):  
Solar Wind ◽  
Critical Point ◽  
Charge Exchange ◽  
Plasma Temperature ◽  
Exchange Term ◽  
Momentum Exchange ◽  
Plasma Bubble ◽  
Friction Forces ◽  
Sonic Point ◽  
Interplanetary Physics

Abstract. Many hydrodynamic models have been presented which give similar views of the interaction of the solar wind plasma bubble with the counterstreaming partially ionized interstellar medium. In the more recent of these models it is taken into account that the solar and interstellar hydrodynamic flows of neutral atoms and protons are coupled by mass-, momentum-, and energy-exchange terms due to charge exchange processes. We shall reinvestigate the theoretical basis of this coupling here by use of a simplified description of the heliospheric interface and describe the main physics of the H-atom penetration through the more or less standing well-known plasma wall ahead of the heliopause. Thereby we can show that the type of charge exchange coupling terms used in up-to-now hydrodynamic treatments unavoidably leads to an O-type critical point at the sonic point of the H-atom flow, thus not allowing for a continuation of the integration of the hydrodynamic set of differential equations. The remedy for this problem is given by a more accurate formulation of the momentum exchange term for quasi-and sub-sonic H-atom flows. With a refined momentum exchange term derived from basic kinetic Boltzmann principles, we instead arrive at a characteristic equation with an X-type critical point, allowing for a continuous solution from supersonic to subsonic flow conditions. This necessitates that the often treated problem of the propagation of inter-stellar H-atoms through the heliosheath has to be solved using these newly derived, differently effective plasma – gas friction forces. Substantially different results are to be expected from this context for the filtration efficiency of the heliospheric interface.Key words. Interplanetary physics (heliopause and solar wind termination; interstellar gas) – Ionosphere (plasma temperature and density)

scholarly journals Numerical Simulations of Advective Flows Around Black Holes

1997 ◽  
Vol 163 ◽  
pp. 690-691
Author(s):  
Sandip K. Chakrabarti ◽  
D. Ryu ◽  
D. Molteni ◽  
H. Sponholz ◽  
G. Lanzafame ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Periodic Oscillation ◽  
Compact Objects ◽  
Advective Flow ◽  
Sonic Point ◽  
Quasi Periodic Oscillation ◽  
Accretion Flows ◽  
Advective Flows ◽  
Cooling Effects ◽  
Implicit Code

Observational results of compact objects are best understood using advective accretion flows (Chakrabarti, 1996, 1997). We present here the results of numerical simulations of all possible types of such flows.Two parameter (specific energy ε and specific angular momentum λ) space of solutions of inviscid advective flow is classified into ‘SA’ (shocks in accretion), ‘NSA’ (no shock in accretion), ‘I’ (inner sonic point only), ‘O’ (outer sonic point only) etc. (Fig. 1 of Chakrabarti, 1997 and references therein). Fig. 1a shows examples of solutions (Molteni, Ryu & Chakrabarti, 1996; Eggum, in preparation) from ‘SA’, ‘I’ and ‘O’ regions where we superpose analytical (solid) and numerical simulations (short dashed curve is with SPH code and medium dashed curve is with TVD code; very long dashed curve is with explicit/implicit code). The agreement is excellent. In presence of cooling effects, shocks from ‘SA’ oscillate (Fig. 1b) when the cooling timescale roughly agrees with postshock infall time scale (Molteni, Sponholz & Chakrabarti, 1996). The solid, long dashed and short dashed curves are drawn for T1/2 (bremsstrahlung), T0.4 and T0.75 cooling laws respectively. In the absence of steady shock solutions, shocks for parameters from ‘NSA’ oscillate (Fig. 2) even in the absence of viscosity (Ryu et al. 1997). The oscillation frequency and amplitude roughly agree with those of quasi-periodic oscillation of black hole candidates. When the flow starts from a cool Keplerian disk, it simply becomes sub-Keplerian before it enters through the horizon. Fig. 3a shows this behaviour where the ratio of λ/λKeplerian is plotted. When the flow deviates from a hot Keplerian disk, it may develop a standing shock as well (Fig. 3b) (Molteni et al. 1996).

scholarly journals Parameter Space of Shock Formation in Adiabatic Flows

1997 ◽  
Vol 159 ◽  
pp. 72-73
Author(s):  
Ju-Fu Lu ◽  
K.N. Yu ◽  
F. Yuan ◽  
E. C. M. Young
Keyword(s):  
Black Hole ◽  
Angular Momentum ◽  
Parameter Space ◽  
World Line ◽  
Rest Mass ◽  
Specific Enthalpy ◽  
Shock Formation ◽  
Characteristic Functional ◽  
Kerr Geometry ◽  
Sonic Point

We study shock formation in a stationary, axisymmetric, adiabatic flow of a perfect fluid in the equatorial plane of a Kerr geometry. For such a flow, there exist two intrinsic constants of motion along a fluid world line, namely the specific total energy, E = −hut, and the specific angular momentum, l = −uφ/ut, where the uμ’s are the four velocity components, h is the specific enthalpy, i.e., h = (P + ε)/ρ, with P, ε, and ρ being the pressure, the mass-energy density, and the rest-mass density, respectively.As shown in Fig. 1 (Fig. la is for a Schwarzschild black hole, i.e. the hole’s specific angular momentum a = 0; Fig. lb is for a rapid Kerr hole, i.e. a = 0.99M, where M is the black-hole mass, and prograde flows: and Fig. 1c is for a = 0.99M and retrograde flows), in the parameter space spanned by E and l there is a strictly defined region bounded by four lines: three characteristic functional curves lk(E), lmax(E), and lmin(E), and the vertical line E = 1. Only such a flow with parameters located within this region can have two physically realizable sonic points, the inner one rin, and the outer one rout. In between there is still one more, but unrealizable, sonic point, rmid. The region is divided by another characteristic functional curve lc(E) into two parts: in region I (= Ia + Ib) only τout is realized in a shock-free global solution (i.e., that joining the black-hole horizon to large distances), while in region II (= IIa + IIb) only rin is realized.

scholarly journals On transonic viscous–inviscid interaction

2002 ◽  
Vol 470 ◽  
pp. 291-317 ◽  
Author(s):  
E. V. BULDAKOV ◽  
A. I. RUBAN
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Body Surface ◽  
Inviscid Flow ◽  
Boundary Layer Separation ◽  
Viscous Sublayer ◽  
Interaction Region ◽  
The Body ◽  
Sonic Point ◽  
Similar Solutions

The paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Kármán–Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dpw/dx ∼ (−x)−1/3 of the distance (−x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Kármán–Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region.

Study of self-similar and steady flows near singularities II. A case of multiple characteristic velocity

1971 ◽  
Vol 322 (1548) ◽  
pp. 45-62 ◽  
Keyword(s):  
Differential Equations ◽  
Fluid Mechanics ◽  
Linear Partial Differential Equation ◽  
Characteristic Velocity ◽  
Multiple Characteristic ◽  
Partial Differential ◽  
Sonic Point ◽  
First Order ◽  
Self Similar ◽  
Steady Solutions

We consider here a system of first-order quasilinear partial differential equations in two independent variables: t , time and x , spatial coordinate. In many physically realistic problems in fluid mechanics, a singularity of the system of ordinary differential equations representing the steady solutions represents a critical state where one of the characteristic velocities vanishes (e.g. sonic point in fluid mechanics). Kulikovskii & Slobodkina (1967) have shown that the stability of all the steady solutions near a singularity can be studied with the help of a simple first-order quasi-linear partial differential equation. The simplicity of their method lies in the fact that all the results can be deduced from the phase-plane of the steady equations. The analysis of Kulikovskii & Slobodkina is valid for any system of equations, totally hyperbolic or mixed type with the only assumption that the characteristic velocity under consideration is real and not multiple. We have earlier (1970, to be referred to as part I) extended their treatment to self-similar flows. In this paper we have shown that in the case of a characteristic velocity of multiplicity s ( s > 1), it is still possible to approximate the system provided there exists exactly s linearly independent eigenvectors corresponding to this characteristic velocity. The approximate system consists of s quasi-linear equations and we have to consider the s + 1 dimensional phase-space of the steady equations. In the end we have also discussed two illustrative examples.

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