ივანე ჯავახიშვილის სახელობის

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Multiphase shock relations and extra physics

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Presentation transcript:

ივანე ჯავახიშვილის სახელობის თბილისის სახელმწიფო უნივერსიტეტი ლექცია 9

Shock Waves

Water Waves Small amplitude waves Large amplitude waves

Shock Development Wave front steepening Transverse waves Longitudinal waves

Jumps Discontinuous solutions obeying conservation laws Jumps in (P, Rho, V, T); Continuous (E,M)

Rankine-Hugoniot Equation 1D Euler equations:

Rankine-Hugoniot Equation Jump conditions:

3D Euler equation in the conservative form: 3D Shocks 3D Euler equation in the conservative form:

Shock Consequences Viscous Heating Bow shock produced by a neutron star Supernova envelope Supernova remnant

HD Linear Spectrum Fourier Analysis: ( V, P, r ) ~ exp(i k r – i w t) Dispersion Equation: w2 (w2 - Cs2 k2)=0 Solutions: Vortices: w2 = 0 Sound waves: w2 = Cs2 k2

HD Discontinuities Sound waves -> Shocks (P1,r1,Vn1) -> (P2,r2,Vn2) ; Vt1=Vt2 Vortex -> Contact Discontinuity Vt1 -> Vt2 ; (P1, r1,Vn1)= (P2,r2,Vn2) Vt Vn

MHD modes - Fast magnetosonic waves; - Slow magnetosonic waves; - Alfven waves; - Fast shocks; - Slow shocks; - Contact shocks;

MHD shocks

Riemann Shock Tube Problem 1D: Shock tube problem Method of characteristics:

Riemann Shock Tube Problem Shock wave Rarefaction wave

Riemann Problem Shocks in Euler equations

Godunov Method Godunov 1959 Grid: Cell interface

Godunov scheme Find cell interface jumps ¯ Solve Riemann Problem (for every cell) Find cell center values (conservative interpolation) integration step

Riemann stepping

Interpolation

Conservative Interpolation Rho = Rho(P,S) Rho<0

Approximations Riemann Solver: Interpolation - Linear Riemann solver (ROE) Fast, medium accuracy - Harten-Laxvan-Leer solver (HLL) Problems with contact discontinuities - Two-Shock Rieman Solver Problems with entropy waves Interpolation Linear (numerical stability) Parabolic (ppm) High order

Riemann Solvers Linear Riemann solver (Roe, 1981) Formulate approximate linear problem d/dt Ut + A(UL,UR) d/dx U = 0

Interpolations Linear (numerical stability) Parabolic (ppm) High order

+/- + Best accuracy + Shock capturing + study of the Heating, viscosity, … Slow Turbulence Complicated

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