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![Page 1: AMS 691 Special Topics in Applied Mathematics Lecture 5 James Glimm Department of Applied Mathematics and Statistics, Stony Brook University Brookhaven.](https://reader036.fdocuments.us/reader036/viewer/2022070412/5697bf801a28abf838c84e67/html5/thumbnails/1.jpg)
AMS 691Special Topics in Applied
MathematicsLecture 5
James Glimm
Department of Applied Mathematics and Statistics,
Stony Brook University
Brookhaven National Laboratory
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Today
Viscosity
Ideal gas
Gamma law gas
Shock Hugoniots for gamma law gas
Rarefaction curves fro gamma law gas
Solution of Reimann problems
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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
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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
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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
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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
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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
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Incompressible Navier-Stokes Equation (3D)
( )
0
dynamic viscosity
/ kinematic viscosity
density; pressure
velocity
t v v v P v
v
P
v
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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
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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",}
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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2
2
/
/ // /
1 1
T
T
u cA
u
c ccu cA c u
c u
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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
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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