Spatial Discretisation

Transcription

Spatial Discretisation
Markus Widhalm
Folie > Vortrag > Autor
Dokumentname > 23.11.2004
Content
spatial discretisation
upwind scheme
central scheme
upwind flux
basic
gradient computation
central flux
basic
artificial dissipation
turbulence flux
full viscous discretization
thin shear layer approximation
Road Map
Spatial discretisation
for flux computation
Inviscid
1st
order
2nd
order
viscous
1st or
2nd
order
Upwind Verfahren
Central scheme +
artificial dissipation
Full viscous flux
Gradient
computation
Thin layer
approximation
Central scheme
Central scheme
Basics
e.g. the continuity equation
ρt + uρx = 0
discretization of the continuity equation
n
"
ρn+1
−
ρ
u ! n
i
n
i
=−
ρi+1 − ρi−1
∆t
2∆x
second - order central scheme
n ... time step, i ... spatial step
Central scheme
Basics
Flux computation
+&*(



F! = 


density
x-momentum
y-momentum
z-momentum
energy
F!f ace






012,
/.
$& $'
&
$
!
$
($
&
%
$
!"#
)(
'
($$
&
$
!
$
&$
&
%
$
!"# -.,
,
)&
" 1
1 !!
Fl + F!r − α̃ (w
=
!r − w
! l)
2
2
Central scheme
Basics
What is α?
α decides about the dissipation scheme
scalar dissipation - α becomes the maximum eigenvalue
matrix dissipation - α becomes a matrix with 3 eigenvalues
Central scheme
Basics
What is α?
α decides about the dissipation scheme
scalar dissipation - α becomes the maximum eigenvalue
matrix dissipation - α becomes a matrix with 3 eigenvalues
How do we compute the difference?
(w
!r − w
! l)
Pj5
Pj4
Pj6
Pj2
Pj0
!
D
= (w
!r − w
! l)
Pj3
Pj7
Fi
Pj8
Pj1
Pj9
dual
control
volume
= "2 (!ur − !ul ) − "4 (L(!ur ) − L(!ul ))
Scalar dissipation
Parameter
Parameter input
Inviscid flux discretization type: Central
Scalar dissipation
Parameter
Parameter input
Central dissipation scheme: Scalar_dissipation
2nd order dissipation coefficient: 0.5
Inverse 4th order dissipation coefficient: 64
Scalar dissipation
Parameter
Why do we need two parameters?
handles discontinuities
Scalar dissipation
Parameter
-1.5
Z
k2 = 0.25
k2 = 0.5
k2 = 0.8
-1
shock
capturing
low
-0.5
cp
M = 0.74
α = 2.0°
0
0.5
high
1
1.5
0
0.2
0.4
X
0.6
0.8
2nd order
coefficient
1
= 0.25
4
1
= 0.5
2
1
= 0.66
1.5
1
= 0.8
1.25
1
Scalar dissipation
Parameter
M = 0.74
α = 2.0°
k2 = 0.25
k2 = 0.5
k2 = 0.8
shock
capturing
low
high
2nd order
coefficient
1
= 0.25
4
1
= 0.5
2
1
= 0.66
1.5
1
= 0.8
1.25
Scalar dissipation
Parameter
dissipation
high
0.1
z
k4
k4
k4
k4
0.05
pl
M = 0.74
α = 2.0°
Inverse 4th order dissipation coefficient
dissipation in smooth regions
64 →
low
0.2
0.4
x
0.6
0.8
16 →
32 →
= 16
= 32
= 64
= 128
0
0
Inverse 4th
order coefficient
128 →
1
16
1
32
1
64
1
128
1
Matrix dissipation
Parameter
Parameter input
Central dissipation scheme: Matrix_dissipation
Inverse 4th order dissipation coefficient: 64
Matrix dissipation terms coefficient: 0.5
Minimum artificial dissipation for acoustic waves: 0.2
Minimum artificial dissipation for velocity: 0.2
Matrix dissipation
Parameter
For robustness reasons we need extra parameters:
Matrix dissipation
Parameter
For robustness reasons we need extra parameters:
α̃("ul , "ur , λ) = kCM · T |λ|T −1
|λ| = diag(|λ1 |, |λ2 |, |λ3 |)
|λ1 | = max(|vn + Af |), δa (|vn | + Af )
|λ2 | = max(|vn − Af |), δa (|vn | + Af )
|λ3 | = max(|vn |), δv (|vn | + Af )
Matrix dissipation
Parameter
Parameter input
With the matrix dissipation - change from matrix to scalar !!!
Acoustic
Velocity
Scalar
Matrix
0.0
0.0
0%
100%
0.2
0.2
20%
80%
0.7
0.3
70%
30%
1.0
1.0
100%
0%
Scalar & Matrix dissipation
What is the main difference between both dissipation schemes ?
scalar dissipation:
very stable
very less parameters needed
matrix dissipation:
scales with all three eigenvalues of the flux jacobian
less dissipative than scalar dissipation
mainly seen in the boundary layer region
needs experience for the extra parameters to run a stable
computation
Upwind scheme
Upwind scheme
Basics
e.g. the continuity equation
ρt + uρx = 0
discretization of the continuity equation
n
"
ρn+1
−
ρ
u ! n
i
n
i
=−
ρi − ρi−1
∆t
∆x
first - order upwind scheme
n ... time step, i ... spatial step
Upwind scheme
Basics
Flux computation



!
F =


density
x-momentum
y-momentum
z-momentum
energy
F!f ace






+&*(
012,
/.
$& $'
&
$
!
$
($
&
%
$
#
"
!
)(
'
($$
&
$
!
$
$
&
&
$%
#
"
!
,
.
,
)&
" 1#
#
1 !!
!
#
Fl + Fr − Ā (w
=
!r, w
! l , !nl,r )# (w
!r − w
! l)
2
2
Upwind scheme
Basics
2nd ( higher order) discretization explanatory for ρ:
ρf ace
!
"
1 #
= ρnode + ψ ∇ρnode #li − #li−1
2
+&*(
012,
/.
&$& $'
$
!
&$($
%
$
!"#
)(
'
$&($$
!
$
%&$&
$
#
!"
,
,-.
)&
Upwind scheme
Basics
2nd ( higher order) discretization explanatory for ρ:
A
ρf ace
!
"
1 #
= ρnode + ψ ∇ρnode #li − #li−1
2
B
A) We need the gradients from the state variables!
+&*(
B) We need a limiter for discontinuities!
012,
/.
&$& $'
$
!
&$($
%
$
!"#
)(
'
second - order upwind scheme
$&($$
!
$
%&$&
$
#
!"
,
,-.
)&
Upwind scheme
Parameter
How do we control this features through the parameterfile?
Considerations have to be done for:
First order upwind
Second order upwind
Gradient computation
Limiters used
Upwind scheme
Parameter
Parameter input
Inviscid flux discretization type: Upwind
Upwind scheme
Parameter
Parameter input
Upwind flux:
Roe
Van_Leer
AUSMDV
AUSMP
AUSM_Van_Leer
EFM
MAPS+
Upwind scheme
Parameter
Roe: Roe, P.L., Approximate Riemann Solvers Parametric Vectors and
Differences Schemes. JCP, Vol. 43, 1981
AUSM (Advective upstream splitting mehtod) : Liou, M.S.; Steffen, C.J.,
A new flux splitting scheme. JCP, Vol. 107, 1993
AUSMP: Wada, Y.; Liou, M.S., A flux splitting scheme with highresolution and robustness for discontinuities. AIAA 94-0083, 1994
EFM (Equilibrium flux method) : Pullin, D.I., Direct simulation methods
for compressible inviscid ideal gas flow, JCP, Vol. 34, 1980
MAPS+ (Mach number based Advection Pressure Splitting): Rossow,
C.C., A flux splitting scheme for compressible and incompressible flows.
JCP, Vol. 164, 2000
Upwind scheme
Parameter
Parameter input
Upwind flux:
Order of upwind flux (1-2): 2
Order of additional euqations (1-2): 2
Roe
Van_Leer
AUSMDV
AUSMP
AUSM_Van_Leer
EFM
MAPS+
Upwind scheme
Parameter
Parameter input
Upwind flux:
Order of additional equations (1-2): 2
Reconstruction of gradients:
Green_Gauss
Least_square
Roe
Van_Leer
AUSMDV
AUSMP
AUSM_Van_Leer
EFM
MAPS+
Upwind scheme
Parameter
Green_Gauss
n
!
1
1
!
∇φ =
(φi + φ0 ) !ni Si
V i 2

Least_square

φx
! =  φy  = x ∼
∇φ
= R−1 QT b
φz
Upwind scheme
Error of the reconstruction on hybrid meshes
φ=x+y
Upwind scheme
Error of the reconstruction on hybrid meshes
φ=x+y
Upwind scheme
Parameter
Reconstruction of gradients inside structured part of a hybrid mesh Automatically with a MUSCL scheme
lines found during the preprocessing step
hl
h
pll
hr
pr
pl
prr
face
h
pr
pl
face
hl
hr
prr
pll
h
pl
pr
face
Upwind scheme
Parameter
Parameter input
Upwind flux:
Order of additional equations (1-2): 2
Green_Gauss
Least_square
Limiter freezing convergence: 0
Mach number limit for limiter: 0
Venkatakrishnan limiter constant: 1
Lowest pressure for 2nd order: 0
Roe
Van_Leer
AUSMDV
AUSMP
AUSM_Van_Leer
EFM
MAPS+
Upwind scheme
Limiters
Limiter freezing convergence:
Upwind scheme
Limiters
Mach number limit for limiter:
Detail !
Upwind scheme
Limiters
Venkatakrishnan limiter constant
Upwind scheme
Limiters
Venkatakrishnan limiter constant
-1.5
-1.5
M = 0.74
α = 2.0°
-1
-1
0
-0.5 Z
0.5
cp
cp
-0.5
Venkata 0
Venkata 0.5
Venkata 1
0 Venkata 2
1
0
0.2
0.4
0.5
X
0.6
0.8
Z
Venkata 0
1 Venkata 0.5
Venkata 1
Upwind scheme
Stability
Lowest pressure for 2nd order: 0.001
Viscous flux computation
Basics
Consider friction of the fluid



F! = 


density
x-momentum
y-momentum
z-momentum
energy



 F
!inviscid + F!viscid
 ! =F

+
n × T ransportequations
(n Equation T urbulencemodel)
Basics
Consider friction of the fluid



F! = 


density
x-momentum
y-momentum
z-momentum
energy



 F
!inviscid + F!viscid
 ! =F

+
n × T ransportequations
(n Equation T urbulencemodel)
Basics
Viscous part effective for momentum and energy equation
momentum
D"v
= ∇P + ρf"
Dt
 
0
σxx - p
0  +  τyx
p
τzx
ρ

p
P = − 0
0
0
p
0
ρf" ... volumef orces
τxy
σyy - p
τyz

τxz

τyz
σzz - p
σ ... normal stress
How do we model P ?
τ ... shear stress
!
"
Newtonian fluid
∂vi
∂vj
2
pij = −pδij + µ
+
− δij ∇
∂xj
∂xi
3
δi,j ... Kronecker tensor
Parameter
Parameter input
Viscous flux type TSL/Full (0/1): 1
Full viscous
Face
Pi
Pj
!
"
1
! j + ∇w
! i
! =
∇w
∇w
2
Parameter
Parameter input
Viscous flux type TSL/Full (0/1): 0
(Full) Thin Shear Layer
Face
Pi
Pj
wj − wi
!
∇w ≈
∆x
Parameter
some comments on Thin Shear Layer approximation:
more stable
almost same solution
maybe less accurate
High-Lift computation (SAE, SA)
error was negligible
viscous flux jacobian depends on p0 and p1 only
some comments on Full Viscous approximation
exact formulation
complex limiting of gradients
Parameter
Parameter input
Turbulent convection scheme for central RANS scheme:
Central
AUSMDV
Roe
ConsVarAveragedRoe
default settings:
laminar and one equation turbulence models: Central
probably less stable but more accurate
all other turbulence models: Roe
more stable but less accurate
Thank you for your attention.
Have fun with TAU

Spatial Discretisation

Transcription

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