Chapter 3. Euler Equations
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
One-dimensional Euler Equations
Conservative vector form of 1-D Euler Equations
1 1
2
2 2
3 3
From mass/momentum/energy conservtions neglecting effects of viscosity and heat conductivity, we have
( ) with = , ( ) =
u u f
u u u p f
t x
E u uH f
U F U
0 U F U
2 2
2
, , ( ) .
2 2
Assuming calorically perfect gas/air ( / 1.4, ), 1 ( / 2).
From ,
is a quasi-linear function of . 0
( )
p v p v
u p u
E e H h e
c c c c R p RT E u
t x t A x
A
u p
up
U F U U U
U 0
U
F U U
1 2
2 2 2
2
2 3 2 1 2 1 3 2 1
1 2
3 2 2 1 3 2 1
2 1 3 2 1
0
( 1) / (2 ) or / ( 1)( / (2 ))
( / )( ( 1) / (2 ))
( / )( 1) / (2 )
1 0
( ) 1 or
u u
u u u u u u u u u u
u u u u u u u u u u u u
u p
A uI u
H u
U F
U U U
1 2 3 2
1 2 3 3 2
3 3 2 2
0 1 0
( , , )
( 3) / 2 (3 ) 1
( , , )
( 1) 3( 1) / 2
with ( 1) ( 1) / 2 and 3( 1) / 2 ( 1)
f f f
u u
u u u
u uE E u u
u uE u uH E u H u
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
2 2 2
1,2,3
Euler system is hyperbolic since it can be diagonalizable with a set of real eigenvalues and independent right eigenvactors.
From det( ) 0 ( )( 2 ), , ,
From
i i iA I u u u a u c u u c
A
r r
1 2 3
2
1 2
1 2 3 2
1
1 1 1
, , , .
2
1 1 1
1 4
Using , , with 2 2 2 ,
2 1
1 1 1
0 0
0 0
0 0
u c u u c
H uc u H uc
c c
H u c u
R R c H u
c
c c
H u c u
u c
R AR u
u c
r r r
r r r
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
Conservative form
Differential form obtained by applying integral mass/momentum/energy conservation
Rankine-Hugoniot relation
Conservative form is meaningful only if it is consistent with physical laws of conservation.
2( ) with ,
u
u u p
t x
E uH
U F U
0 U F U
t2 t1 txL xR xL x
( ) S t
t1
state L
state R
2 2
1 1
1 2
2 1
From on [ , ] [ , ],
( , ) ( , ) ( , ) ( , )
( )= ( ) or with /
R R
L L
L R
x x t t
L R
x x t t
L R L R
dxdt x x t t
t x
x t dx x t dx x t dt x t dt
x t S S x t
U F
0
U U F U F U
U U F F F U
2
2 2
2
Isothermal form of 1 D Euler equations with 1,
1-D Euler eqns. become 0
If flow is smooth, momentum equation becomes
(
p v
t x
t x x t x
C C c p RT const
u
u u c
c u
u uu u u
2 2
) 0 but, is not a conserved quantity.
2 ln
x
t x
u u
u u c
0
Primitive form, symmetric and symmetrized form
Measurable quantities for U more intuitive and simpler in smooth flow
2
0
With , conservation laws become with 0 1 .
0
From conservative form with chain rule, we have ( ) .
Let
p p
p p p
p
p p
p p
u
u A A u
t x
p c u
A A
t x t x
U U F
U U 0
U
U U
U U U U
U 0
U U
U
U
1 2 3 1
2 2
1 1
1 0 0 1 0 0
( , , )
0 , 1 0
( , , )
2 1 1 1 2 1 1
Thus, with .
Since is similar to , eigenvalues are the same
p p
p p p p
p
p
u u u
P u P u
u p u u u u
P AP P AP P AP A
t x t x
A A
U U
U U U U
0
and eigenvectors are linearly independent.
1
From det( ) 0, , , , and 1 , 0 , 1 0
If no shocks (or isentropic flow), energy equation becomes
p i i
t
c c
A I u c u u c
c c
s u
r
0
Thus, is possible to have 0
x
p p
p p
s
u A
t x
s
U U
U
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
(Symmetric form) Starting from the primitive form, simple and useful symmetric form can be obtained.
2
2
0 1 / 1
with 1 , , , , / , 0 , / .
0 0 0 0
p i i
u p s
A c u p s u c u u c c c
u c
r
2
log( / )
( )/ ...
0 1 0
1 0
p p s p const
p ds dp c d p
u p u p u
A u c
t x t x x c t c x x
u u p
t u x x
U U
0
2 2 2
2
2
0
0 ( ) 0
/ /
By introducing or ( / )
/
s
u c p u
t c x u x
p u p p p
c u c u c
t x x t t x x
dp c dp c
d du c du
dp c d dp c
w
2
1
2 1
2
0
, with 0
0 0
Now, from , we have
with
1 0 1 / 2
, / 1/ 0
/ 2
s s
s s
s s s s
s s
s s
s s
s
u c
A A c u
t x
d u
A A P AP
t x t x t x
u u
P u c u P u c c
H cu u
w w
0
w w w w
U U U U
0 0
w w
U w
1 2
2
, , and ( 1) / .
( )
s s s
P AP A c
u H u
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
Symmetric form (cont’d)
1
1
1 2 3
1 1 1
Since is symmetric, we have orthonomal eigenvectors with , where 1/ 2 0 1 / 2
, , 1/ 2 0 1/ 2 ,
0 1 0
From , we have ( ) ( ) .
It is noted that s
s s s s
T
s s s
s s s s s s s s s
A R A R
R R R
P AP A R R P R A P R
r r r
1 1
ymmetric form can be utilized to obtain symmetrized hyperbolic form with any set of dependent variables.
(Conservative variables) From ,
we ha
s s s s
s s s s s
A A P A P
t x t x t x
w w w U w U U U
0 U U
1 1 1 1 1
1 1
ve , where ( ) is symmetric and
positive definite, and is symmetric.
(Primitive variables) From
T T T
s s s s s sc s s
T
sc s s s
s s s p s
s s
p
P P P A P Q A Q P P
t x t x
A P A P
A A
t x t
U U U U
0
w w w U w
0 U
2
2
,
0 0 1/
we have , where 0 / 0 ,
1 0 1/
is symmetric and positive definite, and is symmetric.
p p p
s p
p p p p
T T
s sp
T T
sp s
M A M
x t x
c
M M M A M N A M c
t x t x
c
N M M A M A M
U U U
U
U U U U
0
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
Symmetric form (cont’d)
(Entropy variables) By introducing a suitable form of entropy function/flux/variables, symmetrized hyperbolic
( ) ( )
form can be obtained. Motivated by s us 0 obtained from energy and continu
t x
ity equations, consider the generalized entropy function, , and entropy flux, .
(The case of SCL: ( ) 0 ( ) ( ) 0 with and 0) For , we seek ( ) and ( ) suc
s s
t x t x
s s
U G
u f u U u F u U f F U
A U G
t x
U U
0 U U h that = with being a convex function of .
From 0, we obtain the generalized entropy conservation law of 0.
Define entropy variable vector as
s s
s
s s s
T
U G
U
U U G
t A x t x
F U
U U U
U U
U
S , and scalar functions ( ) ( ), ( ) ( )
i) ( ) is symmetric.
ii)
s
s s
T s T i j
s
j j i i j i
T s
U q U r G
U u u
q q q
S S S S S S N
G r
T T
T
T
= S = S U U S = S F U
U
U U U
U S U
S S U S S
F U U
F S
S U S U
( ) is symmetric.
Thus, from , we have a symmetrized form of .
T i j
j j i i j i
s
s s
f r r f
S S S S S S
AN
A A N AN
t x t x t x
S F
F F U
S U S
U U U S U S S S
0 0
S S
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
Symmetric form (cont’d)
Characteristic form
Wave nature of the Euler equations can be best analyzed by utilizing characteristic form.
If A(U) = const. (locally or globally),
The convexity of ( ) can be treated as a generalized energy since is unbounded as is unbounded. It can be shown that is a convex function of , and thus we take
log
1 1
s s
s
U U
s U s
U
U U
2
2
2 2
log as a generalized entropy function.
1
From direct calculation, and is given by
, , 1 ,
1 2
0 0 0
1 0 1
/ 2 0 / 2
s
T s
T s
N
U s u
p u
u E
N u u p uH p u
E uH EH pu u E u
S
S = U
U UU
S .
1
1
1,2,3
From ( ) ( ) ,
by introducing characteristic variables , we have or a family of scalar equations, i i i 0 with , , . Thus,
A R R
t x t x t x
R t x
u c u u c
t x
U F U U U U U
U 0
ω ω
ω U 0
U i i
i
R
ω
rChap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
If A(U) is not locally constant,
By defining differential characteristic variables as 1 , we still have or a family of scalar equations, 0. Here, depends on all characteristic variables.
With
i i
i i
i
R t x
t x
d d
ω ω
ω U 0
1,3 1,3
0, . along each characteristics , yielding limited domain of dependence/influence. The compatibility relation (or 0 along ) is then
0 . al
i i
i i
const dx
t dt
d dx dt
dp dp
d du u const
c c
2 2 2
ong ( ) ,
0 from , . along .
Assuming . everywhere (homentropic flow) and calorically perfect gas, all i become integrable, and three wave equations a
dx u c dt
d d dp d ds s const dx udt
c
s const d
re
obtained. This formulation is quite useful in implementing far field BCs to
minimize wave reflection. With the Riemann invariants of 2 , we have 1
0 along ( ) ,
R u c
R R s s
u c dx u c dt u
t x t
1 3
2
0 along .
( ) and ( ) : equations for acoustic waves with the sonic speed of w.r.t.
( ) : equation for entropy wave with the local flow speed of dx udt x
R R c u
s u
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
x t
D of D D of I
3
dx u c dt
2
dx u dt
1
dx u c dt
Simple waves
Genuinely nonlinear and linearly degenerate field
The flow physics of the 1-D Euler equations is rather complicated by the 'nonlinear interaction' of the three elementary waves. And, overall properties are not the same as those of scalar conservation law. For example,
- Total variation of 1-D Euler equations is not, in general, decreasing.
- From shock/contact interactions, new local maxima/minima can be easily produced.
In order to understand the nature of each wave equation clearly, additional assumption is introduced.
- For each wave with the path of 0 along , the other two waves are assumed to be constant. . (a
k k
k
d dx dt
const
ctually, all !) along the straight characteristic line of . Example of simple waves
( ) ( )
- Simple acoustic waves: ( ) 0 along ( ) . Burgers'
equation leadi
i x kt const
V C V C
V C x V C t const
t x
ng to the formation of shock discontinuity/expansion waves
- Simple entropy wave: 0 along . linear advection equation and contact discontinuity
s s
V x Vt const
t x
Chap. 3-1. Mathematical and Physical Aspects of Euler Eqns.
1 2 3
Linear and nonlinear nature of each wave is often identified in phase space ( , , ) rather than in physical space ( , ).
. along in physical space . along an integral cu
i i i
u u u x t
const dx dt const
rve ( ) in phaseu
Genuinely nonlinear and linearly degenerate field (cont’d)
1 2 3
space such that ( ) ( ( ), ( ), ( )) with some parameter .
Thus, ( ) 0 .
By checking the behavior of along the integral curve ( ), we define -
i
j T T
i i
i i i
j
i
u u u u
d u
d u u
u
r r
2 2
Genuinely nonlinear: ( ) 0 - Linearly degenerate: ( ) 0
This is an extension of convexity condition into Euler systems since, in SCL, 1 0
T T
i
i i i
T T
i i i i
T
d u
d
d u
d
d d f
du du
r r
r .
2
Example: Euler equations in primitive variables of ( , , ) 1
From , , , and 1 , 0 , 1 , 0
0 - linearly degenerate: 1 0
i i
u p
c c
u c u u c
c c
r
2 2
1,3 1,3 1,3
0
2 1
- Genuinely nonlinear: 1 0
2 2 c
c
r
r