AMS 691Special Topics in Applied

MathematicsLecture 5

James Glimm

Department of Applied Mathematics and Statistics,

Stony Brook University

Brookhaven National Laboratory

Today

Viscosity

Ideal gas

Gamma law gas

Shock Hugoniots for gamma law gas

Rarefaction curves fro gamma law gas

Solution of Reimann problems

Total time derivatives

( ) particle streamline

( ) ( ) / velocity

Lagrangian time derivative

= derivative along streamline

Now consider Eulerian velocity ( , ).

On streamline, ( ( ), )

x t

v t dx t dt

D

Dt

vt x

v v x t

v v x t t

Dv

Dt

acceleration of fluid particle

v vv

t x

Euler’s EquationForces = 0

inertial force

Pressure = force per unit area

Force due to pressure =

other forces 0

S V

Dv

Dt

Pds Pdx

DvP

Dt

Conservation form of equationsConservation of mass

0

Conservation of momentum

other forces

v

t x

v vv

t t tv

v v P vx x

v v vP

t x

Momentum flux

( ) 0; flux of U

flux of momentum

stress tensor

Now include viscous forces. They are added to

'

' viscous stress tensor

ik ik i k ik i k

ik ik ik

ik

UF U F

tv v P

P v v v v

P

Viscous Stress Tensor

' depends on velocity gradients, not velocity itself

' is rotation invariant; assume ' linear as a function of velocity gradients

Theorem (group theory)

2'

3

C

i k i iik ik

k i i i

v v v v

x x x x

orollary: Incompressible Navier-Stokes eq. constant density

v v vP v

t x

Incompressible Navier-Stokes Equation (3D)

( )

0

dynamic viscosity

/ kinematic viscosity

density; pressure

velocity

t v v v P v

v

P

v

Two Phase NS Equationsimmiscible, Incompressible

• Derive NS equations for variable density• Assume density is constant in each phase with a jump

across the interface• Compute derivatives of all discontinuous functions using

the laws of distribution derivatives– I.e. multiply by a smooth test function and integrate formally by

parts• Leads to jump relations at the interface

– Away from the interface, use normal (constant density) NS eq.– At interface use jump relations

• New force term at interface– Surface tension causes a jump discontinuity in the pressure

proportional to the surface curvature. Proportionality constant is called surface tension

Reference for ideal fluid EOS and gamma law EOS

@Book{CouFri67, author = "R. Courant and K. Friedrichs", title = "Supersonic Flow and Shock Waves", publisher = "Springer-Verlag", address = "New York", year = "1967",}

EOS. Gamma law gas, Ideal EOS

0

0

Ideal gas:

/ (molecular weight)

universal gas constant

For an ideal gas, ( )

Tabulated values: ( ) is a polynomial in

and polynomial coefficients are tabulated (NASA tables).

Different gass

PV RT

R R

R

e e T

e T T

es have different tabulated polynomials.

Polytropic (also called Gamma law) gas:

; specific heat at constant volume

For gamma law gas, is independent of . Also

( , ) ; ( )

v v

v

e c T c

c T

P P S A A a S

Derivation of ideal EOS

( , ) , ( , )

/ ( ) 0

0

ODE for in , . Solution:

( ); exp( / ). Conclusion: depends on only.

' ( 1/ ); '; ' arbitrary

Substitute and check; O

V S

S V

s V

de TdS PdV P e S V T e S V

R PV T RT P V

Re Ve

e S V

e h VH H S R e VH

Re RVh H R Ve VHh h

DE has unique solution for given initial data. We define

1'( )

Thus depends on VH only. as function of . (This is a thermodynamic

hypothesis.) Thus is invertable; ( ); ( ) (

s

s

T e h VH VHR

T T VH

e VH VH T e h VH h

1

( )).

Thus we write as a function of . Also

'( ) '( ) .

This is the ideal EOS.V

VH T

e T

P e h VH H h H H

Gamma

12

2 2 2

The sound speed, by definition, is with

( , ) '( )

acoustic impedence

For an ideal gas,

'( / )c ( , ) ''( )

1 ( ) , where

( ) 1 ; also ( ) 1

c

dP S dh Hc H

d d

c

h HV S H h VH V H

dTR RT T RTde

dT de RT R

de dT T

specific heat at constant volume

Vc

2 2 2

2

2 2 2

( ) ''( ) 1

In fact:

'( )/

''( ) so

''( )

1

V V

V V V

dTc T h VH V H R RT

de

h VHe RT P H RT V

V

h VH VH

c h VH V H Ve VRT VP VRT

T e T TRT VR RT PVR R RT

e V e e

Proof 2 1dT

c R RTde

2

2

2 2 2

( )

(1) '( )

1'( )

(2) '( ) ''( )

''( ) by (1,2)

''( ) ( )

(1 )

V

V

V V

V V

V V

V V

e h VH

e Hh VH

T h VH VHR

RT h VH H h VH VH

e RT h VH VH

c h VH V H V e RT

dTT e

de

dTVe VR e

dedT

VP VRPdedT

RT Rde

Polytropic = gamma law EOS

1

1

( 1)

0

0

Definition: Polytropic: = is proportional to ;

( ) 1 1 .;

( ); 1 1

1 1'( ) ( ) '( )

'( ) ( )

1'( ) ( )

V

V

V V

V

V

e c T T

dTT R Rc const

de

e c T h VH Rc

T h VH VH e h VH c h VH VHR R

Rh VH h VH

c VH

VHh VH h VH h

VH H

H

additive constant in the entropy S

10( )'( ) ( 1) vc S SP h VH H e

0

0

( 1)

0

( 1)

0

( 1)

0

/ 1

( )/

( )/ 1

( 1) ( )

( ) ( 1)

; 1

( 1) V

V

V

S RV

S S c

S S c

VHe h

H

HP e V A S

H

HA S

H

H e Rc

P e

e e

Specific Enthalpy i = e +PV

2

1

For adiadic changes, 0,

.

For ideal gas, is a function of .

( ) ( ) (1 ( 1))

1 1

= specific heat at constant pressure .

; 1 ;

P

V V V

di VdP Tds

dS

dPdi VdP V d c Vd

d

i T

di d e PV d e RT R R

dT dT dT

c

dec Rc c R

dT

/ ( 1)

/ / ratio of specific heats (assuming ideal gas)1 1 P V

R Rc c

Enthalpy for a gamma law gas

( 1) ( 1)

21

2 1

1

( )1 1

( )

i e PV

AV AV

cA S

dPc A S

d

Hugoniot curve for gamma law gas

0 0

00 0 0

( )/ ( )/2 1

2 0 00 0

2 20 0 0

Recall

( , ) ( , ) ( , ) ( ) 0;2

1 1; define . ; ( 1)

1 1

12 ( , ) 2 2 ( )( )

1 1 1

( ) ( )

V VS S c S S c

P PH V P V P V P V V

PV e P e

PVPVH V P P P V V

V V P V V P

Rarefaction waves are isentropic, so to study them we studyIsentropic gas dynamics (2x2, no energy equation). is EOS.( )P P

Characteristic Curves

1

A conservation law ( ) 0 or

0; / is hyperbolic

if ( ) has all real eigenvalues

A curve ( ), ( ) in 1D space + time is characteristic

if its speed = / ( / )( / ) is an eige

t

t x

U F U

U AU A F U

A A U

x s t s

dx dt dx ds dt ds

nvalue of .

This definition depends on the solution and should hold

on the entire curve. Along the curve,

( , , ) ( , , )

For a characteristi

t x x x

A

U

dU dt dx dt dx dx dtU U A x t U U A x t U I U

ds ds ds ds ds dt ds

c curve, and for = an eigenvector, is a constant.

In general, one component of is constrained by equation along a charactersitic.xU U

U

Isentropic gas dynamics, 1D

2 2

2 2

0

Rewrite first equation as

where '( ) and '( )

0;/ /

Eigenvalues of :

State space: , : 0

Characteristic curv

xt x

t x x

xt x x x

t x

Pu uu

u u

u uu c P P c P

u uA

u c u u c u

A u c

u

es (there are two families for 2x2 system):

/: ; Eigenvectors of = transpose =

1T cdx

C u c A Adt

2

2

/

/ // /

1 1

T

T

u cA

u

c ccu cA c u

c u

Riemann Invariants

Theorem: is a constant on each curve

Proof:

.

/But = = = left eigenvector of for ei

1

t x

x

cu d C

d dU dt dxU U

ds U ds U ds ds

dx dtA I U

U dt ds

cA

U

u

genvalue .

So result is zero if .

Definition: simple wave (= rarefaction wave): is constant inside that wave.

In a simple wave, both of the 's are constant on a charactersitic,

u c

dxu c

dt

C

thus

= constant in a simple wave on a characteristic.

Equation for a simple wave: = constant, 0.

U C

dS

Centered Simple WaveA rarefaction whose straight caracteristics ( for right/left rarefaction)

all meet at a point, is called centered. Asuming that this point is the origin,

. This is a simple wave, in that =

C

xu c

t

1

constant. These two equations

define the solution at each space-time point.

For a gamma law gas, and we compute

( ) 2 2.

1 1

Starting from a right state with sound speed

r r

dPc A

d

cu d u c u c

, velocity , we have

two equations to determine , at each point. These equations define the

rarefaction wave curve.

r rc u

u c