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Shock Waves and Water Waves: Understanding Small and Large Amplitude Waves

This lecture at Ivane Javakhishvili Tbilisi State University explores the development of shock waves and water waves, including waves of small and large amplitudes. It covers topics such as transverse and longitudinal waves, jump conditions, the Rankine-Hugoniot equation, 1D and 3D shocks, viscous heating, linear spectrum, Fourier analysis, MHD modes, Riemann shock tube problem, Godunov method, and interpolation techniques.

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Shock Waves and Water Waves: Understanding Small and Large Amplitude Waves

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  1. ივანე ჯავახიშვილის სახელობის თბილისის სახელმწიფო უნივერსიტეტი ლექცია 9

  2. Shock Waves

  3. Water Waves Small amplitude waves Large amplitude waves

  4. Shock Development Wave front steepening • Transverse waves • Longitudinal waves

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

  6. Rankine-Hugoniot Equation 1D Euler equations:

  7. Rankine-Hugoniot Equation Jump conditions:

  8. 3D Shocks 3D Euler equation in the conservative form:

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

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

  11. 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

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

  13. MHD shocks

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

  15. Riemann Shock Tube Problem Shock wave Rarefaction wave

  16. Riemann Problem Shocks in Euler equations

  17. Godunov Method Godunov 1959 Grid: Cell interface

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

  19. Riemann stepping

  20. Interpolation

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

  22. Approximations Riemann Solver: - 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

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

  24. Interpolations • Linear (numerical stability) • Parabolic (ppm) • High order

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

  26. end www.tevza.org/home/course/modelling-II_2016/

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