AMS 691Special Topics in Applied Mathematics

Lecture 3

James GlimmDepartment of Applied Mathematics and

Statistics,Stony Brook University

Brookhaven National Laboratory

Compressible Fluid Dynamics Euler Equation (1D)

1 1

3 3

2

( )

... ; ( ) ...

( )

( ) 0

mass density; momentum density, = pressure;

1 + = total energy density; = internal energy2

t

U F U

U m F U

U E F U

U F U

m P

E mv e e

v

F vv P

Ev vP

Equation of State (EOS)• System does not close. P = pressure is an extra unknown; e = internal

energy is defined in terms of E = total energy.• The equation of state takes any 2 thermodymanic variables and writes all

others as a function of these 2.• Rho, P, e, s = entropy, Gibbs free energy, Helmholtz free energy are

thermodynamic variables. For example we write P = P(rho,e) to define the equation of state.

• A simple EOS is the gamma-law EOS.

• Reference:• author = "R. Courant and K. Friedrichs",• title = "Supersonic Flow and Shock Waves",• publisher = "Springer-Verlag",• address = "New York",• year = "1967

Entropy

• Entropy = s(rho,e) is a thermodynamic variable. A fundamental principle of physics is the decrease of entropy with time.– Mathematicians and physicists use opposite signs

here. Confusing!

Analysis of Compressible Euler Equations

(2 ) (2 ) matrix

acoustic matrix

Governs small amplitude (linear) disturbances

Eigenvalues and eigenvectors of

known by exact formulae (for simple

equations of state), and these are used in s

FA D D

UA

A

ome

modern numerical schemes

}

Compressible Fluid Dynamics Euler Equation

• Three kinds of waves (1D)• Nonlinear acoustic (sound) type waves: Left or right

moving– Compressive (shocks); Expansive (rarefactions)– As in Burgers equation

• Linear contact waves (temperature, and, for fluid concentrations, for multi-species problems)– As in linear transport equation

Nonlinear Analysis of the Euler Equations

• Simplest problem is the Riemann problem in 1D• Assume piecewise constant initial state, constant for x < 0 and

x > 0 with a jump discontinuity at x = 0.• The solution will have exactly three kinds of waves (some may

have zero strength): left and right moving “nonlinear acoustic” or “pressure” waves and a contact discontinuity (across which the temperature can be discontinuous)

• Exercise: prove this statement for small amplitude waves (linear waves), starting from the eigenvectors and eigenvalues for the acoustic matrix A

• Reference: Chorin Marsden

Symmetries for Riemann Solutions

• U(x,t) -> Uax,at) is a symmetry of equations– Change of scale transformation– Require a > 0; otherwise symmetry violates

entropy inequality– Riemann initial conditions are invariant under

scale transformations– So we expect Riemann solutions to be invariant

also• U(x,t) = U(x/t) for Riemann solution

Rarefaction Shocks• Are unstable as solutions of the PDE• Violate entropy• Are excluded from solutions based on many different criteria• Result from disallowed scale transformation a = -1 applied to

a shock wave

Jump Relations for Riemann Solutions

• Assume a simple discontinuity, propagating with speed s in x,t space in 1D

• Apply the differential equation at the discontinuity– Distribution derivatives are needed– Or weak solutions are needed– Or the “pill box” proofs common in physics books are

needed– Solution is a function of x/t alone

Weak solutions

• Since the Euler equations admit discontinuous solutions, we need to be careful in taking derivatives. One way to do this is through the notion of weak solutions. For every space time test function phi(x,t), smooth with compact support,

2

( , ) ( , ) ( , ) ( , ) 0t x

R

U x t x t U x t x t

Rankine-Hugoniot Relations

[ ] [ ( )]

[ ] [ ] (linear waves)

Thus (linear case) is an eigenvector of acoustic matrix

[ ] 0

s U F U

s U AU

s A

s A U

For a gamma law gas EOS, the solutions of the RH relations can be analyzed. Thereare three branches of solutions: left and right shocks and contact waves, just as weexpect from the linear theory.Reference: Chorin Marsden

Pill Box Proof

s = speed = dx/dt[U]dx = [F(U)]dt ors[U] = [F(U)]

( )

'

'

( ( )) ( )

' ( ) [ ] [ ( )]

x

t

t t x

x st

s

U F U dxdt U F U dxdt

sU F U dxdt s U F U

RH Relations, Continued

• For any initial jump condition, the RH relations have a solution with three discontinuities, which yields a solution of the Euler equation as traveling shock waves, contact waves, and rarefaction waves. (gamma law gas EOS).

• Reference: Chorin-Marsden

RH Relations, Continued

• Approximate solutions of Riemann problems are a basic ingredient of many modern numerical algorithms

• Reference

author = "R. LeVeque", title = "Numerical Methods for Conservation Laws", publisher = "Birkh{\"a}user Verlag", address = "Basel--Boston--Berlin", year = "1992"

Picture of Riemann Solution

Rankine Hugoniot Relations

• Are there jump conditions for the derivatives of the solution? [gradP]? [grad rho]? Etc.

• If so, what are they for gas dynamics?

Fluid Transport

• The Euler equations neglect dissipative mechanisms• Corrections to the Euler equations are given by the

Navier Stokes equations• These change order and type. The extra terms

involve a second order spatial derivative (Laplacian). Thus the equations become parabolic. Discontinuities are removed, to be replaced by steep gradients. Equations are now parabolic, not hyperbolic. New types of solution algorithms may be needed.

Fluid Transport

• Single species– Viscosity = rate of diffusion of momentum

• Driven to momentum or velocity gradients

– Thermal conductivity = rate of diffusion of temperature• Driven by temperature gradients: Fourier’s law

• Multiple species– Mass diffusion = rate of diffusion of a single species in a

mixture• Driven by concentration gradients• Exact theory is very complicated. We consider a simple

approximation: Fickean diffusion• Correct for 2 species only