Gas Dynamics: Basic Equations, Waves and Shocks

Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks Susanne Höfner Susanne.Hoefner@fysast.uu.se

Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks 1. Basic Concepts of Fluid Mechanics

Fluid Dynamics: Basic Concepts consider a typical fluid...

Fluid Dynamics: Basic Concepts... and define a fluid element

Fluid Dynamics: Basic Concepts Definition of fluid element: A region over which we can define local variables (density, temperature, etc.) The size of this region (length scale l region ) is assumed to be such that it is (i) small enough that we can ignore systematic variations across it for any variable q we are interested in: l region << l scale q / q (ii) large enough to contain sufficient particles to ignore fluctuations due to the finite number of particles (discreteness noise): nl 3 region >> 1 In addition, collisional fluids must satisfy: (iii) l region >> mean free path of particles

Mean Free Path - Examples Mean Free Path - depends on number density and cross section - elastic scattering cross section of neutral atoms: 10-15 cm Examples gas in density MFP size -3 [cm ] [cm] [cm] typical room HI region (ISM) 1019 10 10-4 10 14 10 3 10 19

Fluid Dynamics: Basic Concepts - frequent collisions, small mean free path compared to the characteristic length scales of the system coherent motion of particles - definition of fluid element: mean flow velocity (bulk velocity) - particle velocity = mean flow velocity + random velocity component

Fluid Dynamics: Basic Concepts coherent motion of particles

Fluid Dynamics: Basic Concepts The fluid is described by local macroscopic variables (e.g., density, temperature, bulk velocity) v The dynamics of the fluid is governed by internal forces (e.g. pressure gradients) and external forces (e.g. gravity) ρ, T v The changes of macroscopic properties in a fluid element are described by conservation laws ρ, T

Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks. Fluid Dynamics: Conservation Laws

Fluid Dynamics: The Conservation Laws Mass conservation:

Fluid Dynamics: The Conservation Laws Mass conservation: d dt V divergence theorem dv = A u n da = V u dv rate of change of mass in V = mass flux across the surface rewrite as: V [ ] u dv = 0 t... but the volume V is completely arbitrary u = 0 t equation of continuity

Fluid Dynamics: The Conservation Laws Momentum conservation: d dt V ui dv = A ui u j n j da A P ij n j da V ai dv rate of change of momentum in V = momentum flux across the surface (fluid flow) - effect of pressure on surface + effect of forces applying divergence theorem (surface to volume integrals): P ui ui u j = ai t xj xi equation of motion

Fluid Dynamics: The Conservation Laws Energy conservation: d 1 1 u e dv = A u e u n da A ui P ij n j da V u a dv V dt rate of change of energy in V = energy flux across the surface (due to fluid flow) - work done by pressure + work done by forces applying divergence theorem (surface to volume integrals): 1 1 u e u e u = P u u a t energy equation

Fluid Dynamics: The Conservation Laws u = 0 t P ui ui u j = ai t xj xi 1 1 u e u e u = P u u a t general form of conservation laws: density of quantity flux of quantity = sources sinks t

Fluid Dynamics: The Conservation Laws u = 0 t P ui ui u j = ai t xj xi 1 1 u e u e u = P u u a t... and what about external forces? a= F m

Fluid Dynamics: External Forces Gravitation: a=g... acceleration where g is given by Poisson's equation g= -4πGρ

Fluid Dynamics: External Forces Radiation pressure: frad = ρ/ c frad κν Fν dν... force per volume, added to equation of motion u frad... work done by radiation pressure, added to energy equation

Fluid Dynamics: The Conservation Laws u = 0 t P i ui ui u j = g i f rad t xj xi 1 1 u e u e u = P u u g t u f rad... including: gravity radiation pressure heating & cooling by radiation

Fluid Dynamics: Limit Cases of Energy Transport u = 0 t P ui ui u j = ai t xj xi Barotropic equation of state: Examples: - ideal gas: * isothermal case * adiabatic case Efficient radiative heating and cooling: ( radiative equilibrium) P = P P T P P = K = 0 T P,T

Astrophysical Dynamics, VT 010 Gas Dynamics: Basic Equations, Waves and Shocks 3. Acoustic Waves and Shock Waves

Fluid Dynamics: The Conservation Laws u = 0 t P ui ui u j = ai t xj xi 1 1 u e u e u = P u u a t general form of conservation laws: density of quantity flux of quantity = sources sinks t

Small-Amplitude Sound Waves u = 0 t P ui ui u j = ai t xj xi replacing energy equation: 1D, no external forces: P=K u = 0 t x P u uu = t x x

Small-Amplitude Sound Waves small-amplitude disturbances in gas which initially is at rest with constant pressure and density P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t replacing energy equation: P=K 1D, no external forces: u = 0 t x P u uu = t x x

Small-Amplitude Sound Waves small-amplitude disturbances in gas which initially is at rest with constant pressure and density insert into equations & linearise: P=P 0 P 1 x,t = 0 1 x,t u= u1 x,t P0 P 1= K = 1 0 1 0 1 1 u1 0 = 0 t x u1 P1 0 = t x

Small-Amplitude Sound Waves homogeneous wave equation: general solution: 1 t a 0 1 x =0 1 = f x a0 t g x a0 t waves propagating with sound speed a0

The Conservation Laws and Burgers' Equation Fluid equations, 1D: u = 0 t x u uu P = 0 t x 1 1 u e u e P u = 0 t x

The Conservation Laws and Burgers' Equation Fluid equations, 1D: reformulate the equation of motion: u u = 0 t x u uu P = 0 t x u u =0 1 1 u e u e P u = 0 u P t u x t u x t x x

The Conservation Laws and Burgers' Equation Fluid equations, 1D: reformulate the equation of motion: u u = 0 t x u uu P = 0 t x u u =0 1 1 u e u e P u = 0 u P t u x t u x t x x u u u =0 t x

The Conservation Laws and Burgers' Equation Fluid equations, 1D: reformulate the equation of motion: u u = 0 t x u uu P = 0 t x u u =0 1 1 u e u e P u = 0 u P t u x t =0 u x t Burgers' equation captures the essential non-linearity of the 1D Euler equation of motion. x x... and neglect pressure gradients: u u u =0 t x

The Conservation Laws and Burgers' Equation Fluid equations, 1D: reformulate the equation of motion: u u = 0 t x u uu P = 0 t x u u =0 1 1 u e u e P u = 0 u P t u x t =0 u x t Burgers' equation captures the essential non-linearity of the 1D equation of motion. x x... and neglect pressure gradients: u u u =0 t x

Burgers' Equation: Steepening of Waves

Fluid Dynamics: Steepening of Sound Waves Waves

Fluid Dynamics: Structure of Shock Waves Waves

Shocks and Conservation Laws 1D, no external forces, stationary flow: u = 0 t x P u uu = t x x 1 1 P u u e u e u = t x x jump conditions: specific enthalpy P P h e = 1 u = 1 u1 u P = 1 u1 P 1 1 1 u h = u 1 h1

Shocks and Conservation Laws 1D, no external forces, stationary flow: u = 0 t x P u uu =0 t x x 1 1 P u = 0 u e u e u t x x jump conditions: specific enthalpy P P h e = 1 u = 1 u1 u P = 1 u1 P 1 1 1 u h = u 1 h1

Shocks and Conservation Laws 1D, no external forces, stationary flow: u = 0 t x u uu P = 0 t x 1 1 P u e u e u = 0 t x jump conditions: specific enthalpy P P h e = 1 u = 1 u1 u P = 1 u1 P 1 1 1 u h = u 1 h1