Large-Scale Simulations of High

rangebeaverMécanique

22 févr. 2014 (il y a 3 années et 4 mois)

50 vue(s)

Zhaorui Li and Farhad Jaberi

Department of Mechanical Engineering

Michigan State University

East Lansing, Michigan




Large
-
Scale Simulations of High
Speed Turbulent Flows


Develop high
-
fidelity numerical models for high speed turbulent flows


Fundamental understanding of compressible turbulent flows and
shock
-
turbulence Interactions


Numerical experiments
-

Analyses of supersonic/hypersonic
problems for various flow parameters

Objectives:

Approach:


High
-
order numerical methods for LES and DNS of high speed
(supersonic) turbulent flows in complex geometries


Existent low
-
speed SGS models extended and applied to high speed
turbulent flows


DNS and experimental data are employed for validation and
improvement of LES submodels

LES and DNS of High Speed Turbulent Flows




High
-
order numerical schemes for compressible turbulent velocity field in
complex geometries. Needed for LES & DNS of very high speed
supersonic/hypersonic flows.



In LES, large
-
scale compressibility effects are explicitly calculated. So far,
compressible SGS (Dynamic) Gradient, Mixed and MKEV models have been
employed. Work is in progress to develop improved deterministic subgrid
turbulence and wall models for supersonic and hypersonic flows.



High
-
order numerical schemes for the filtered scalar field in supersonic
turbulent flows. In LES, large
-
scale (compressible) mixing are explicitly
calculated. So far, SGS Gradient models have been employed.



Direct evaluation and improvement of subgrid models via DNS data.



Comparisons with DNS and experimental data for validation of numerical
method and SGS models


Application of LES to High Speed Flows

Numerical Methods

1)

High
-
order Compact
-
RK scheme + limiters and/or artificial viscosity
(Rizzetta et al. 2001, Cook & Cabot 2005, Kawai & Lele 2007)


3D code
is developed and tested

2)
High
-
order WENO
-
RK scheme (Shi et al. 2003)
-

3D code is developed
and tested

3)
High
-
order Monotonicity Preserving (MP)
-
RK scheme (Huynh 2007)
-

3D code is developed and tested)

Test Problems

1)
1D Problems: Advection (Wave), Burgers, Lax, Shock Tube, (1D
calculations with 3D codes)

2)
2D Problems: Rayleigh
-
Taylor Instability, Double Mach Reflection,
Isotropic Turbulence (2D calculations with 3D codes)

3)
3D Isotropic Turbulence

4)
Converging
-
Diverging Nozzle


3D LES

5)
Supersonic Boundary Layer with Shock wave


3D DNS and LES

6)
Supersonic Mixing Layer


3D DNS and LES


A Single FD code in
generalized coordinate system


F
ully Compressible Filtered Navier
-
Stokes Equations in Generalized Coordinates System

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η
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ξ
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1
1
2
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(
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0




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Y
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0




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u
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3
3
2
3
4
3
2
33
33
23
23
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(
Re
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)
1
(
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(
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Pr
Pr
Re
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0




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Y
L
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H
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.
0
(
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1
/
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(
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(




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)
(




/
)
(




0





ˆ
2
/
3
kk
i
j
ij
z
y
x
z
y
x
C
u
L
J
Q
Fr
n
w
n
v
n
u
Fr
n
Fr
n
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n
S









ij
k
k
j
i
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u
L
u
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)
~
(
3
2
)
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(
)
~
(
Re
~
~
J
x
x



ˆ












)


(
)
,
,
(
ζ
η
ξ
ζ
η
ξ
ζ
η
ξ
z
z
z
y
y
y
x
x
x
ζ
η,
ξ,
z
y
x
J





F
ully compressible Navier
-
Stokes equations in generalized
coordinates system with transformation

)
,
,
(
)
,
,
(




z
y
x
JS
H
G
F
t
U
J















ˆ
ˆ
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(
)
(
)
(
ˆ
),
(
)
(
)
(
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),
(
)
(
)
(
ˆ
v
z
v
y
v
x
v
z
v
y
v
x
v
z
v
y
v
x
H
H
J
G
G
J
F
F
J
H
H
H
J
G
G
J
F
F
J
G
H
H
J
G
G
J
F
F
J
F













































)
,
,
,
,
(
E
w
v
u
U







)
,
,
(
/
)
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,
(






z
y
x
J
Solution vector

Transformation Jacobian

Sixth
-
order Compact scheme

Eighth
-
order implicit filter

)
(
)
(
2
2
1
1
1
1















i
i
i
i
i
i
i
f
f
c
f
f
b
f
f
f


)
36
/(
1

),
9
/(
7

,
3
/
1
h
c
h
b














4
0
1
1
)
(
2
)
(
ˆ
ˆ
ˆ
k
k
i
k
i
i
f
i
i
f
f
f
k
a
f
f
f


)
14
7
(
32
1
)
2
(

),
18
7
(
16
1
)
1
(

),
70
93
(
128
1
)
0
(
f
f
f
a
a
a










)
2
1
(
128
1
)
4
(

),
2
1
(
16
1
)
3
(
f
f
a
a











WENO5

MP5/MP7

Eigensystem in generalized coordinates

0

x

Here only the format eigenvector for are given


















































2
b

)
2
~
2
b
(
-

)
2
~
2
b
(
-

)
2
~
2
b
(
-



2
~
2
b

0

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1


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-


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~
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1
~

0

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1


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w
~
~
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~
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b
(
-

)
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~
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b
(
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)
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2
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2
b

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1
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1
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w
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ξ
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u-(
ξ
ξ
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ξ
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u-(
ξ
b
w
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c
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1


1

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y
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4
3
2
2
2
2
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2
2
2
1
,
,


1
1
2
2
1
2
2
2
/
1

,

,
/
)
1
(

,
2
/
b
q
H
q
b
b
c
b
w
v
u
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/
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and


,
~
~
~

~

2
2
2
z
y
x
x
x
z
y
x
w
v
u
















h
F
F
U
F
j
j
J
/
)
ˆ
ˆ
(
)
(
ˆ
2
/
1
2
/
1









(1) Lax
-
Friedrichs flux splitting

(2) Compute left and right eigenvectors of Roe’s mean matrix, transform all quantities needed for
evaluating the numerical flux to local characteristic field.

U
d
U
F
d
U
U
F
U
F
U
F
U
F
U
F
U









/
)
(
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max

),
)
(
ˆ
(
)
(
ˆ

),
(
ˆ
)
(
ˆ
)
(
ˆ










2
/
1
ˆ

j
F
2
/
1
2
/
1
2
/
1
2
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1
)
(

)
/
)
(
ˆ
(
)
(






j
j
C
j
j
C
R
U
d
U
F
d
L


)
(
ˆ
)
(
)
(
ˆ
2
/
1
j
j
C
c
j
U
F
L
U
F




(3) Calculate numerical flux in characteristic field

k
j
k
k
c
j
F
F
2
/
1
3
1
2
/
1
ˆ
)
ˆ
(







6
/
)
(
ˆ
6
/
)
(
ˆ
5
3
/
)
(
ˆ
ˆ
3
/
)
(
ˆ
6
/
)
(
ˆ
5
6
/
)
(
ˆ
ˆ
6
/
)
(
ˆ
11
6
/
)
(
ˆ
7
3
/
)
(
ˆ
ˆ
2
1
2
2
/
1
1
1
2
2
/
1
1
1
2
1
2
/
1
c
j
c
j
c
j
j
c
j
c
j
c
j
j
c
j
c
j
c
j
j
U
F
U
F
U
F
F
U
F
U
F
U
F
F
U
F
U
F
U
F
F





























3
.
0

,
6
.
0

,
1
.
0

where
)
(
~

,
~
~
3
2
1
3
1


















k
k
k
i
i
k
k
4
/
)
)
(
ˆ
)
(
ˆ
4
)
(
ˆ
3
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
4
/
)
)
(
ˆ
)
(
ˆ
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
4
/
)
)
(
ˆ
3
)
(
ˆ
4
)
(
ˆ
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
2
2
1
2
2
1
3
2
1
1
2
1
1
2
2
1
2
2
1
2
1
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F

















































(4) Calculate numerical flux in physical space

c
j
j
C
j
F
R
F
)
ˆ
(
)
(
ˆ
2
/
1
2
/
1
2
/
1






(*) a mirror imagine (with respect to j+1/2)

Procedure to that in step(3) is used to calculate



2
/
1
ˆ
j
F
(*) only difference between WENO and MP only in step (3) is described in the following

(a) Calculate

original

interface value

60
/
)
)
(
ˆ
3
)
(
ˆ
27
)
(
ˆ
47
)
(
ˆ
13
)
(
ˆ
2
(
)
ˆ
(
2
1
1
2
2
/
1
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
F
















420
/
)
)
(
ˆ
4
)
(
ˆ
38
)
(
ˆ
214
)
(
ˆ
319
)
(
ˆ
101
)
(
ˆ
25
)
(
ˆ
3
(
)
ˆ
(
3
2
1
1
2
3
2
/
1
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
U
F
U
F
F























7
tt

order scheme

5
tt

order scheme

(b) Determine discontinuity









)
)
ˆ
(
)
ˆ
)((
)
ˆ
(
)
ˆ
((
2
/
1
,
,
2
/
1
,
c
MP
m
c
j
m
c
j
m
c
j
m
F
F
F
F

)
(
ˆ

of
component

one

is

)
5
1
(

ˆ
,
j
j
m
U
F
m
F



(*) limiter needed



)
)
ˆ
(
)
ˆ
((
,
)
ˆ
(
)
ˆ
(
minmod
)
ˆ
(
)
ˆ
(
1
,
,
,
1
,
,
c
j
m
c
j
m
c
j
m
c
j
m
c
j
m
c
MP
m
F
F
F
F
F
F







Where

(c) Limiting procedure

)]
ˆ
,
ˆ
,
)
ˆ
max((
),
ˆ
,
)
ˆ
(
,
)
ˆ
min[max((
ˆ
)]
ˆ
,
ˆ
,
)
ˆ
min((
),
ˆ
,
)
ˆ
(
,
)
ˆ
max[min((
ˆ
3
/
4
)
)
ˆ
(
)
ˆ
((
5
.
0
)
ˆ
(
ˆ

,
5
.
0
ˆ
ˆ

),
)
ˆ
(
)
ˆ
((
5
.
0
ˆ

),
)
ˆ
(
)
ˆ
((
)
ˆ
(
ˆ
)

,

,
4

,
4
(
minmod

,
)
ˆ
(
2
)
ˆ
(
)
ˆ
(
,
1
,
,
max
,
1
,
,
min
4
2
/
1
1
,
,
,
4
2
/
1
1
,
,
1
,
,
,
1
1
1
4
2
/
1
,
1
,
1
,
LC
m
UL
m
c
j
m
MD
m
c
j
m
c
j
m
m
LC
m
UL
m
c
j
m
MD
m
c
j
m
c
j
m
m
M
j
c
j
m
c
j
m
c
j
m
LC
m
M
j
AV
m
MD
m
c
j
m
c
j
m
AV
m
c
j
m
c
j
m
c
j
m
UL
m
j
j
j
j
j
j
M
j
c
j
m
c
j
m
c
j
m
j
F
F
F
F
F
F
F
F
F
F
F
F
F
F
d
F
F
F
F
d
F
F
F
F
F
F
F
F
F
d
d
d
d
d
d
d
F
F
F
d




































)
ˆ
,
ˆ
,
)
ˆ
((
median
)
ˆ
(
max
min
2
/
1
,
2
/
1
,
m
m
c
j
m
c
j
m
F
F
F
F





X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
3
0
.
6
0
.
9
1
.
2
E
x
a
c
t
S
o
l
u
t
i
o
n
U
n
l
i
m
i
t
e
d
u
(
x
)
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
3
0
.
6
0
.
9
1
.
2
E
x
a
c
t
S
o
l
u
t
i
o
n
E
N
O
3
u
(
x
)
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
3
0
.
6
0
.
9
1
.
2
E
x
a
c
t
S
o
l
u
t
i
o
n
W
E
N
O
5
u
(
x
)
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
3
0
.
6
0
.
9
1
.
2
E
x
a
c
t
S
o
l
u
t
i
o
n
M
P
5
u
(
x
)

1D Advection

or Wave Eq.



5
th

Order upwind
3
rd

order ENO

5
th

order WENO,
5
th

order MP
Schemes


Solutions after 10
Periods


NX=200

CFL=0.4


LES and DNS of High Speed Flows


High
-
Order Numerical Methods for Supersonic
Turbulent Flows

5
th

order MP

5
th

order WENO

3
rd

order ENO

5
th

order Upwind

1D Burgers Eq.



5
th

order WENO
and

5
th

order MP
Schemes


Initial condition:

X
0
0
.
5
1
1
.
5
2
1
.
9
8
1
.
9
9
2
2
.
0
1
2
.
0
2
E
x
a
c
t
S
o
l
u
t
i
o
n
W
E
N
O
5
M
P
5
N
X
=
1
0
0
C
F
L
=
0
.
4
u
(
x
,
t
)
X
0
0
.
5
1
1
.
5
2
1
.
9
8
1
.
9
9
2
2
.
0
1
2
.
0
2
E
x
a
c
t
S
o
l
u
t
i
o
n
W
E
N
O
5
M
P
5
N
X
=
2
0
0
C
F
L
=
0
.
4
u
(
x
,
t
)
)
(
sin
2
)
0
,
(
9
x
x
u




1D Shock
-

Tube Problem



5
th

order WENO
and

5
th

order MP
Schemes


Initial Condition

X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
2
0
.
4
0
.
6
0
.
8
1
E
x
a
c
t
S
o
l
u
t
i
o
n
W
E
N
O
5
P
W
E
N
O
5
N
X
=
1
0
0
,
c
f
l
=
0
.
4
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
2
0
.
4
0
.
6
0
.
8
1
E
x
a
c
t
S
o
l
u
t
i
o
n
M
P
5
P
M
P
5
N
X
=
1
0
0
,
c
f
l
=
0
.
4
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
2
0
.
4
0
.
6
0
.
8
1
E
x
a
c
t
S
o
l
u
t
i
o
n
M
P
5

M
P
5
N
X
=
1
0
0
,
c
f
l
=
0
.
4
X
-
1
-
0
.
5
0
0
.
5
1
0
0
.
2
0
.
4
0
.
6
0
.
8
1
E
x
a
c
t
S
o
l
u
t
i
o
n
W
E
N
O
5

W
E
N
O
5
N
X
=
1
0
0
,
c
f
l
=
0
.
4
)
1
.
0

,
0
.
0

,
125
.
0
(
)
,
,
(

R
R
R
p
u


High
-
Order Numerical Methods for Supersonic
Turbulent Flows


High
-
Order
Numerical Methods
for Supersonic
Flows

2D Inviscid
Rayleigh
-
Taylor
Instability Problem




5
th

order WENO

and

5
th

order MP
Schemes

Density Contours

Density Contours


High
-
Order Numerical
Methods for Supersonic Flows

2D Viscous Rayleigh
-
Taylor Instability
Problem

5
th

order WENO

5
th

order MP

Density Contours

Re=25,000

5
th

order WENO

5
th

order MP

Re=50,000

0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
2
4
0
N
Y
=
4
8
0
N
Y
=
9
6
0
R
e
=
2
5
0
0
0
W
E
N
O
5
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
4
8
0
N
Y
=
9
6
0
N
Y
=
1
9
2
0
R
e
=
5
0
0
0
0
W
E
N
O
5
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
2
4
0
N
Y
=
4
8
0
N
Y
=
9
6
0
R
e
=
2
5
0
0
0
M
P
5
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
2
4
0
N
Y
=
4
8
0
N
Y
=
9
6
0
R
e
=
2
5
0
0
0
M
P
7
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
4
8
0
N
Y
=
9
6
0
N
Y
=
1
4
4
0
R
e
=
5
0
0
0
0
M
P
5
0
0
.
0
5
0
.
1
0
.
1
5
0
.
2
0
.
2
5
0
.
8
1
1
.
2
1
.
4
1
.
6
1
.
8
2
2
.
2
N
Y
=
4
8
0
N
Y
=
9
6
0
N
Y
=
1
4
4
0
R
e
=
5
0
0
0
0
M
P
7

High
-
Order
Numerical Methods
for Supersonic
Flows

2D Viscous
Rayleigh
-
Taylor
Instability Problem




5
th

order WENO

5
th

order MP


7
th

order MP

Schemes


Density

X (Y=0.6)


High
-
Order Numerical
Methods for Supersonic Flows

Double Mach Problem

X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
W
E
N
O
5
N
X
=
4
N
Y
=
2
4
0
c
f
l
=
0
.
4
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
W
E
N
O
5
N
X
=
4
N
Y
=
4
8
0
c
f
l
=
0
.
4
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
W
E
N
O
5
N
X
=
4
N
Y
=
9
6
0
c
f
l
=
0
.
4
Initial Condition

Ma=10

X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
5
N
X
=
4
N
Y
=
2
4
0
c
f
l
=
0
.
4
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
5
N
X
=
4
N
Y
=
4
8
0
c
f
l
=
0
.
4
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
5
N
X
=
4
N
Y
=
9
6
0
c
f
l
=
0
.
4
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
7
N
X
=
4
N
Y
=
2
4
0
(
a
)
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
7
N
X
=
4
N
Y
=
4
8
0
(
b
)
X
Y
0
0
.
5
1
1
.
5
2
2
.
5
3
0
0
.
5
1
M
P
7
N
X
=
4
N
Y
=
9
6
0
(
c
)
5
th

order WENO

5
th

order MP

7
th

order MP

Density Contours

Next Slide


High
-
Order Numerical
Methods for Supersonic Flows

Double Mach Problem

X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
W
E
N
O
5
N
X
=
4
N
Y
=
4
8
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
W
E
N
O
5
N
X
=
4
N
Y
=
9
6
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
W
E
N
O
5
N
X
=
4
N
Y
=
1
9
2
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
5
N
X
=
4
N
Y
=
4
8
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
5
N
X
=
4
N
Y
=
9
6
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
5
N
X
=
4
N
Y
=
1
9
2
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
7
N
X
=
4
N
Y
=
4
8
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
7
N
X
=
4
N
Y
=
9
6
0
c
f
l
=
0
.
4
X
Y
2
2
.
2
5
2
.
5
2
.
7
5
0
0
.
1
0
.
2
0
.
3
0
.
4
0
.
5
M
P
7
N
X
=
4
N
Y
=
1
9
2
0
c
f
l
=
0
.
4
Density Contours

5
th

order WENO

5
th

order MP

7
th

order MP


High
-
Order
Numerical
Methods for
Supersonic
Flows


Supersonic
Diverging
Nozzle


Low Back

Pressure


Mach Number Contours

0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
1
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
R
e
=
1
0
0
0
0
W
E
N
O
5
0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
2
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
1
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
1
0
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
2
0
0
0

High
-
Order
Numerical
Methods for
Supersonic
Turbulent
Flows


Supersonic
Diverging
Nozzle


High Back

Pressure

Mach Number Contours

0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
1
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
R
e
=
1
0
0
0
0
W
E
N
O
5
0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
W
E
N
O
5
R
e
=
2
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
1
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
1
0
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
5
0
0
0
0
2
4
6
8
1
0
-
2
-
1
0
1
2
M
P
5
R
e
=
2
0
0
0

3D

Isotropic Turbulence

0
2
4
6
8
1
0
1
2
0
.
8
1
.
6
2
.
4
3
.
2
S
p
e
c
t
r
a
l
M
P
5
W
E
N
O
5
M
P
7

t
(
a
)
0
2
4
6
8
1
0
1
2
0
.
0
3
0
.
0
3
5
0
.
0
4
0
.
0
4
5
0
.
0
5
S
p
e
c
t
r
a
l
M
P
5
W
E
N
O
5
M
P
7
t

(
b
)
1
0
2
0
3
0
4
0
5
0
1
0
-
1
0
1
0
-
9
1
0
-
8
1
0
-
7
1
0
-
6
1
0
-
5
1
0
-
4
1
0
-
3
1
0
-
2
S
p
e
c
t
r
a
l
M
P
5
W
E
N
O
5
M
P
7
E
(
K
)
K
(
a
)
2
0
2
5
3
0
3
5
4
0
4
5
5
0
5
5
1
0
-
1
0
1
0
-
9
1
0
-
8
1
0
-
7
1
0
-
6
1
0
-
5
1
0
-
4
S
p
e
c
t
r
a
l
M
P
5
W
E
N
O
5
M
P
7
K
E
(
K
)
(
b
)

High
-
Order Numerical Methods
for Supersonic Turbulent Flows

Energy
Spectrum

Energy
Spectrum

Enstrophy

Enstrophy
Dissipation
Rate

Incident shock
-
BL interaction

Compression corner
-
BL interaction

Test Case: Supersonic Laminar Flat
-
Plate Boundary Layer (Anderson, 2000)

Computational Details and Problem Setup



Numerical Scheme:

5
th

order Monotonicity
Preserving scheme for inviscid fluxes and 6
th

order compact scheme for viscous/scalar fluxes



Reference Mach No. :

2.5



Reference Reynolds No. :

582

Shock Wave
-

Boundary
Layer Interactions

Pressure Contours

temperature

Streamwise velocity

Shock
-
Laminar BL Interactions

Expansion Fan

Separation region

Incident shock

β

= 30
o

Compression Waves

Pressure Contours

Pressure Distribution in the streamwise direction

Streamwise Velocity Contours

Computational Details and Problem Setup



Numerical Scheme:

5
th

order Monotonicity
Preserving scheme for inviscid fluxes and 6
th

order
compact scheme for viscous/scalar fluxes



Reference Mach No. :

2.5



Reference Reynolds No. :

582


At
y = 2.0
, discontinuities which satisfy Rankine
-
Hugoniot relations are introduced at the inlet.



High
-
Order
Numerical
Methods for
Supersonic
Turbulent
Flows

2D

Turbulent Mixing
Layer


Shock
Interactions


DNS data for
understanding of
turbulence
-
shock
interactions

and development of
improved SGS models

Density Contours

Pressure Contours


wall



shock



High
-
Order
Numerical
Methods for
Supersonic
Turbulent
Flows

2D

Turbulent Mixing
Layer


Shock
Interactions


Species Mass Fraction Contours

0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
M
P
7
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
M
P
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
DNS of Spatially Developing 3D Supersonic Mixing Layer

Pressure Contours at Z=0.75Lz

Re
δ
/2
=200, Mc=1.2

M1=4.2

M2=1.8

Re
δ
/2
=200, Mc=1.2

M1=4.2

M2=1.8

Spanwise

Vorticity

Contours Z=0.75Lz

DNS of Spatially Developing 3D Supersonic Mixing Layer

DNS of Spatially Developing 3D Supersonic Mixing

Layer

Scalar Mass Fraction Contours Z=0.75Lz

63.5 mm diam
center jet

CARS/
Rayleigh
beams
M=2
vitiated
air jet
Burner/
nozzle
CARS/
Rayleigh
beams
M=2
vitiated
air jet
Burner/
nozzle
Coflow
nozzle
Facility flange
M=2 setup
M=1 setup
SiC
liner
Watercooled
shell
Small
-
scale facility

Large
-
scale facility

Nozzle
(
SiC
)
Water
-
cooled
combustion
chamber
Spark
plug
H
2
fuel
tube
Air+O
2
passage
Coflow
nozzle
Water
-
cooled
injector
10 mm diameter
Center jet

Supersonic Mixing and Reaction
-

Co
-
Annular Jet Experiments

(
Laboratory and Full
-
Scale Models)
Cutler et al. 2007

LES of Co
-
Annular Jet

Grid System for LES


Iso
-
Levels of Mach Number

Pressure

Temperature

LES of Supersonic Co
-
Annular Jet


Non
-
Reacting Flow



3D LES Calculations
with Compact Scheme


Summary and Conclusions




Robust high
-
order finite difference methods (i.e. MP, WENO,
Compact+limiter) are developed and tested for large
-
scale and detailed
calculations of compressible turbulent flows with/without shock waves in
complex geometries



Numerical simulations of various 1D, 2D and 3D flows have been conducted
for assessment of numerical schemes and SGS turbulence models



So far, compressible SGS (Dynamic) Gradient and Mixed LES models have
been employed



DNS data are being generated/analyzed for shock
-
turbulent mixing layer and
shock
-
boundary layer interaction problems



Work is in progress to develop improved compressible subgrid turbulence
models for supersonic flows.



LES data for supersonic co
-
annular jet are being compared with
experimental data


High
-
Order
Numerical
Methods for
Supersonic
Turbulent
Flows

Turbulent Mixing
Layer


Shock
Interactions

DNS data for
understanding of
turbulence
-
shock
-
Combustion interactions


and development of
improved SGS models


Density


Pressure



wall




shock


0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
M
P
7
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
M
P
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)
0
5
0
1
0
0
1
5
0
2
0
0
-
2
0
-
1
0
0
1
0
2
0
W
E
N
O
5
_
C
O
M
P
6
(
N
o
n
-
C
o
n
s
e
r
)

Scalar


Scalar Equation



F
ully compressible Navier
-
Stokes equations in generalized
coordinates system with transformation

)
,
,
(
)
,
,
(




z
y
x
JS
H
G
F
t
U
J















ˆ
ˆ
ˆ

)
(
)
(
)
(
ˆ
),
(
)
(
)
(
ˆ
),
(
)
(
)
(
ˆ
v
z
v
y
v
x
v
z
v
y
v
x
v
z
v
y
v
x
H
H
J
G
G
J
F
F
J
H
H
H
J
G
G
J
F
F
J
G
H
H
J
G
G
J
F
F
J
F













































)
,
,
,
,
(
E
w
v
u
U







)
,
,
(
/
)
,
,
(






z
y
x
J
Solution vector

Transformation Jacobian

6
-
order Compact scheme

Eighth
-
order implicit filter

)
(
)
(
2
2
1
1
1
1















i
i
i
i
i
i
i
f
f
c
f
f
b
f
f
f


)
36
/(
1

),
9
/(
7

,
3
/
1
h
c
h
b














4
0
1
1
)
(
2
)
(
ˆ
ˆ
ˆ
k
k
i
k
i
i
f
i
i
f
f
f
k
a
f
f
f


)
14
7
(
32
1
)
2
(

),
18
7
(
16
1
)
1
(

),
70
93
(
128
1
)
0
(
f
f
f
a
a
a










)
2
1
(
128
1
)
4
(

),
2
1
(
16
1
)
3
(
f
f
a
a














WENO scheme

MP
Scheme


Eigensystem in generalized
coordinates

0

x


Here only the format eigenvector for are
given


















































2
b

)
2
~
2
b
(
-

)
2
~
2
b
(
-

)
2
~
2
b
(
-



2
~
2
b

0

~
~
1


~
~
~


~
-


~
~
~
~
~
1
~

0

~
~
~
-


~
~
1


~
-


w
~
~
~
~
~
1
~



b



b



b




1


2
b

)
2
~
2
b
(
-

)
2
~
2
b
(
-

)
2
~
2
b
(
-



2
~
2
b

1
1
1
1
2
2
2
2
2
1
1
1
1
2
1
1
1
1
2
c
w
c
v
c
u
c
ξ
ξ
-
ξ
v
)w
ξ
ξ
-
u-(
ξ
ξ
ξ
-
ξ
)v
ξ
ξ
-
u-(
ξ
b
w
v
u
b
c
w
c
v
c
u
c
L
z
y
x
x
z
x
z
y
z
x
z
y
x
z
z
x
z
y
x
y
y
x
z
y
x
y
y
z
y
x
C







































































~

~
~

~
~


~
~


~


0


~
~


0


~



~
~


~

~


~
1


0


0


1


1

c
H
u
w
u
v
q
c
H
c
w
w
c
w
c
v
v
c
v
c
u
u
c
u
R
z
x
y
x
z
x
z
y
x
y
x
z
y
x
C
w
v
u
U
c
U
c
U
z
y
x
z
y
x
z
y
x




























4
3
2
2
2
2
5
2
2
2
1
,
,


1
1
2
2
1
2
2
2
/
1

,

,
/
)
1
(

,
2
/
b
q
H
q
b
b
c
b
w
v
u
q










/
~

and


,
~
~
~

~

2
2
2
z
y
x
x
x
z
y
x
w
v
u
















h
F
F
U
F
j
j
J
/
)
ˆ
ˆ
(
)
(
ˆ
2
/
1
2
/
1









(1) Lax
-
Friedrichs flux splitting

(2) Compute left and right eigenvectors of Roe’s mean matrix, transform all quantities needed for
evaluating the numerical flux to local characteristic field.

U
d
U
F
d
U
U
F
U
F
U
F
U
F
U
F
U









/
)
(
ˆ
max

),
)
(
ˆ
(
)
(
ˆ

),
(
ˆ
)
(
ˆ
)
(
ˆ










2
/
1
ˆ

j
F
2
/
1
2
/
1
2
/
1
2
/
1
)
(

)
/
)
(
ˆ
(
)
(






j
j
C
j
j
C
R
U
d
U
F
d
L


)
(
ˆ
)
(
)
(
ˆ
2
/
1
j
j
C
c
j
U
F
L
U
F




(3) Calculate numerical flux in characteristic field

k
j
k
k
c
j
F
F
2
/
1
3
1
2
/
1
ˆ
)
ˆ
(







6
/
)
(
ˆ
6
/
)
(
ˆ
5
3
/
)
(
ˆ
ˆ
3
/
)
(
ˆ
6
/
)
(
ˆ
5
6
/
)
(
ˆ
ˆ
6
/
)
(
ˆ
11
6
/
)
(
ˆ
7
3
/
)
(
ˆ
ˆ
2
1
2
2
/
1
1
1
2
2
/
1
1
1
2
1
2
/
1
c
j
c
j
c
j
j
c
j
c
j
c
j
j
c
j
c
j
c
j
j
U
F
U
F
U
F
F
U
F
U
F
U
F
F
U
F
U
F
U
F
F





























3
.
0

,
6
.
0

,
1
.
0

where
)
(
~

,
~
~
3
2
1
3
1


















k
k
k
i
i
k
k
4
/
)
)
(
ˆ
)
(
ˆ
4
)
(
ˆ
3
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
4
/
)
)
(
ˆ
)
(
ˆ
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
4
/
)
)
(
ˆ
3
)
(
ˆ
4
)
(
ˆ
(
12
/
)
)
(
ˆ
)
(
ˆ
2
)
(
ˆ
(
13
2
2
1
2
2
1
3
2
1
1
2
1
1
2
2
1
2
2
1
2
1
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F
U
F


















































(4) Calculate numerical flux in physical space

c
j
j
C
j
F
R
F
)
ˆ
(
)
(
ˆ
2
/
1
2
/
1
2
/
1







(*) a mirror imagine (with respect to j+1/2)


Procedure to that in step(3) is used to calculate



2
/
1
ˆ
j
F

(*) only difference between WENO and MP only in step (3) is described in the
following


(a) Calculate

original

interface value

60
/
)
)
(
ˆ
3
)
(
ˆ
27
)
(
ˆ
47
)
(
ˆ
13
)
(
ˆ
2
(
)
ˆ
(
2
1
1
2
2
/
1
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
F
















420
/
)
)
(
ˆ
4
)
(
ˆ
38
)
(
ˆ
214
)
(
ˆ
319
)
(
ˆ
101
)
(
ˆ
25
)
(
ˆ
3
(
)
ˆ
(
3
2
1
1
2
3
2
/
1
c
j
c
j
c
j
c
j
c
j
c
j
c
j
c
j
U
F
U
F
U
F
U
F
U
F
U
F
U
F
F























7
tt

order scheme

5
tt

order scheme


(b) Determine
discontinuity









)
)
ˆ
(
)
ˆ
)((
)
ˆ
(
)
ˆ
((
2
/
1
,
,
2
/
1
,
c
MP
m
c
j
m
c
j
m
c
j
m
F
F
F
F

)
(
ˆ

of
component

one

is

)
5
1
(

ˆ
,
j
j
m
U
F
m
F




(*) limiter
needed



)
)
ˆ
(
)
ˆ
((
,
)
ˆ
(
)
ˆ
(
minmod
)
ˆ
(
)
ˆ
(
1
,
,
,
1
,
,
c
j
m
c
j
m
c
j
m
c
j
m
c
j
m
c
MP
m
F
F
F
F
F
F








Wh
ere


(c) Limiting
procedure

)]
ˆ
,
ˆ
,
)
ˆ
max((
),
ˆ
,
)
ˆ
(
,
)
ˆ
min[max((
ˆ
)]
ˆ
,
ˆ
,
)
ˆ
min((
),
ˆ
,
)
ˆ
(
,
)
ˆ
max[min((
ˆ
3
/
4
)
)
ˆ
(
)
ˆ
((
5
.
0
)
ˆ
(
ˆ

,
5
.
0
ˆ
ˆ

),
)
ˆ
(
)
ˆ
((
5
.
0
ˆ

),
)
ˆ
(
)
ˆ
((
)
ˆ
(
ˆ
)

,

,
4

,
4
(
minmod

,
)
ˆ
(
2
)
ˆ
(
)
ˆ
(
,
1
,
,
max
,
1
,
,
min
4
2
/
1
1
,
,
,
4
2
/
1
1
,
,
1
,
,
,
1
1
1
4
2
/
1
,
1
,
1
,
LC
m
UL
m
c
j
m
MD
m
c
j
m
c
j
m
m
LC
m
UL
m
c
j
m
MD
m
c
j
m
c
j
m
m
M
j
c
j
m
c
j
m
c
j
m
LC
m
M
j
AV
m
MD
m
c
j
m
c
j
m
AV
m
c
j
m
c
j
m
c
j
m
UL
m
j
j
j
j
j
j
M
j
c
j
m
c
j
m
c
j
m
j
F
F
F
F
F
F
F
F
F
F
F
F
F
F
d
F
F
F
F
d
F
F
F
F
F
F
F
F
F
d
d
d
d
d
d
d
F
F
F
d




































)
ˆ
,
ˆ
,
)
ˆ
((
median
)
ˆ
(
max
min
2
/
1
,
2
/
1
,
m
m
c
j
m
c
j
m
F
F
F
F