AMS 691
Special Topics in Applied
Mathematics
Lecture 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
x (t )  particle streamline
v (t )  dx (t ) / dt  velocity
D
 Lagrangian time derivative
Dt
= derivative along streamline


 v
t
x
Now consider Eulerian velocity v  v ( x , t ).
On streamline, v  v ( x (t ), t )
Dv v
v

v
 acceleration of fluid particle
Dt t
x

Euler’s Equation
Forces = 0
Dv

  inertial force
Dt
Pressure = force per unit area
Force due to pressure =
 Pds    Pdx
S
V
Dv

 P  other forces  0
Dt
Conservation form of equations
Conservation of mass

 v

0
t
x
Conservation of momentum
 v
v



v
t
t
t
 

  v 
   v
 v  P  
v
x 

 x 
 v
 v  v

 P  other forces
t
x
Momentum flux
U
 F (U )  0; F  flux of U
t
 v  v  P    flux of momentum
 ik   ik P   vi vk   ik   vi vk
  stress tensor
Now include viscous forces. They are added to 
 ik  P ik   'ik
 'ik  viscous stress tensor
Viscous Stress Tensor
 ' depends on velocity gradients, not velocity itself
 ' is rotation invariant; assume  ' linear as a function of velocity gradients
Theorem (group theory)
 vi vk 2 vi 
vi
 '  

  ik
   ik
xi
 xk xi 3 xi 
Corollary: Incompressible Navier-Stokes eq. constant density
 v  v  v

 P  v
t
x
Incompressible Navier-Stokes
Equation (3D)
 t  v    (v  v)  P  v
 v0
  dynamic viscosity
 /  kinematic viscosity
  density; P  pressure
v  velocity
Two Phase NS Equations
immiscible, 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
Ideal gas:
PV  RT
R  R0 / (molecular weight)
R0  universal gas constant
For an ideal gas, e  e(T )
Tabulated values: e(T ) is a polynomial in T
and polynomial coefficients are tabulated (NASA tables).
Different gasses have different tabulated polynomials.
Polytropic (also called Gamma law) gas:
e  cvT ; cv  specific heat at constant volume
For gamma law gas, cv is independent of T . Also
P  P (  , S )  A  ; A  a ( S )
Derivation of ideal EOS
de  TdS  PdV  P  e( S ,V )V , T  e( S ,V ) S
R  PV / T  RT  ( P )V  0 
ReS  Ve V  0
ODE for e in S , V . Solution:
e  h(VH ); H  exp( S / R). Conclusion: e depends on VH only.
Res  RVh ' H (1/ R);VeV
 VHh '; h ' arbitrary
Substitute and check; ODE has unique solution for given initial data. We define
1
h '(VH )VH
R
Thus T depends on VH only. T
T  es  
as function of VH . (This is a thermodynamic
hypothesis.) Thus es is invertable; VH  VH (T ); e  h(VH )  h(VH (T )).
Thus we write e as a function of T . Also
P  eV  h '(VH ) H  h '(  1H ) H .
This is the ideal EOS.
Gamma
The sound speed, by definition, is c with
1
dP
(

,
S
)
dh
'(

H)
2
c 

H
d
d
c   acoustic impedence
For an ideal gas,
h '( H /  )
c (V , S )  
H  h ''(VH )V 2 H 2

2
dT

 1  R
de


 RT   (T ) RT , where

dT
de
R
 (T )  1  R
; also

 cV 
de
dT  (T )  1
specific heat at constant volume
dT

c (T )  h ''(VH )V H  1  R
de

In fact:
2
2
2

 RT

h '(VH )
eV  RTV   P 
H  RT / V
V
  h ''(VH )VH 2 so
c 2  h ''(VH )V 2 H 2  VeV  VRTV  VP  VRTV
T e
T 
T
 RT  VR
 RT  PVR
 1  R
e V
e 
e

 RT

dT

c  1  R
de

Proof
2

 RT

e  h(VH )
(1)
eV  Hh '(VH )
1
h '(VH )VH
R
RTV   h '(VH ) H  h ''(VH )VH 2
T 
(2)
eV  RTV   h ''(VH )VH 2 by (1,2)
c 2  h ''(VH )V 2 H 2  V (eV  RTV )
dT


T

e
 V
V 
de


dT
 VeV  VR
eV
de
dT
 VP  VRP
de
dT
 RT (1  R
)
de
Polytropic = gamma law EOS
Definition: Polytropic: e = cV T is proportional to T ;
dT
 (T )  1  R
 1  RcV1  const.;
de
e  cV T  h(VH );   1  RcV1    1
1
1
T   h '(VH )VH  e  h(VH )  cV
h '(VH )VH
R
R
R
 h '(VH )  h(VH )
cV VH
 1
 (  1)
 VH 
h '(VH )   h(VH )
h

VH
 H0 
H 0  additive constant in the entropy S
cv1 ( S S0 )
P  h '(VH ) H  ( 1)e
 VH 
eh

 H0 
 (  1)
P  eV  (  1)V
 H 


 H0 

 (  1)
 H 
A( S )  (  1) 

 H0 
H  e  S / R ; RcV1    1 
P  (  1)e( S  S0 )/ cV  
e  e ( S  S0 )/ cV   1
 (  1)
 A( S )  
Specific Enthalpy i = e +PV
di  VdP  Tds
For adiadic changes, dS  0,
dP
di  VdP  V
d   c 2Vd  .
d
For ideal gas, i is a function of T .
di d (e  PV ) d (e  RT ) R(1  (  1))  R




dT
dT
dT
 1
 1
= specific heat at constant pressure  cP .
 de

1
  cV ;   1  RcV ; cV  R / (  1) 
 dT

R R

/
 cP / cV  ratio of specific heats (assuming ideal gas)
 1  1
Enthalpy for a gamma law gas
i  e  PV
A  ( 1)
 (  1)

V
 AV
 1


 1
A( S ) 
 1
2
c

 1
 2 dP
 1 
  A( S )  
c 
d


Hugoniot curve for gamma law gas
Recall
H (V , P )   (V , P )   (V0 , P0 ) 
P0  P
(V  V0 )  0;
2
1
 1
PV ; define  2 
.   e( S  S0 )/ cV   1 ; P  (  1)e( S  S0 )/ cV  
 1
 1
PV
PV
 1
2  2 H (V , P)  2
2 0 0 
( P  P0 )(V  V0 )
 1
 1  1



 (V   2V0 ) P  (V0   2V ) P0
Rarefaction waves are isentropic, so to study them we study
Isentropic gas dynamics (2x2, no energy equation). P  P(  ) is EOS.
Characteristic Curves
A conservation law U t  F (U )  0 or
U t  AU x  0; A  F / U is hyperbolic
if A  A(U ) has all real eigenvalues
A curve x( s ), t ( s ) in 1D space + time is characteristic
if its speed = dx / dt  (dx / ds)(dt / ds) 1 is an eigenvalue of A.
This definition depends on the solution U and should hold
on the entire curve. Along the curve,
dU
dt
dx 
dt dx 
dx  dt

 Ut  U x
   A( x, t ,U )   U x    A( x, t ,U )  I  U x
ds
ds
ds 
ds ds 
dt  ds

For a characteristic curve, and for U x = an eigenvector, U is a constant.
In general, one component of U is constrained by equation along a charactersitic.
Isentropic gas dynamics, 1D
ut  uu x  
Px

t  u  x   u x  0
Rewrite first equation as
x
where Px  P '(  )  x and c 2  P '(  )

  

  u
 u
   2
   0; A   2


u
c
/

u
u
c
/

u
 t 
  x


ut  uu x  c 2
Eigenvalues of A : u  c
State space:   , u :   0
Characteristic curves (there are two families for 2x2 system):
dx
C :
 u  c; Eigenvectors of AT = A transpose =
dt
 c /  
 1 


2

u c /
T
A 

u 

2

c
/


 cu /   c /  
 c /  
T 
A 
   c  u  


c  u
 1  
 1 

Riemann Invariants
c
Theorem:    u   d  is a constant on each curve C

Proof:
d     dU  


ds
U ds
U
dx 
 dt
U

U
x
 t

ds 
 ds
 
U
dx 
dt


A

I
U

 x .
dt 
ds

   
     c /  
 
=
But
= 
 = left eigenvector of A for eigenvalue u  c.
U
     1 


 u 
dx
So result is zero if
 u  c.
dt
Definition:   simple wave (=
rarefaction wave):   is constant inside that wave.

In a   simple wave, both of the 's are constant on a C charactersitic, thus
U = constant in a   simple wave on a C characteristic.
Equation for a   simple wave:   = constant, dS  0.
Centered Simple Wave
A rarefaction whose straight caracteristics (C for right/left rarefaction)
all meet at a point, is called centered. Asuming that this point is the origin,
x
u  c  . This is a  simple wave, in that  = constant. These two equations
t
define the solution at each space-time point.
dP
For a gamma law gas, c 
 A  1 and we compute
d
c(  )
2
2
 u 
d  u
c  ur
cr .

 1
 1
Starting from a right state with sound speed cr , velocity ur , we have
two equations to determine u, c at each point. These equations define the
rarefaction wave curve.