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
Top Related