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December, 2007 VKI Lecture 1
Boundary conditions
December, 2007 VKI Lecture 2
BC essential for thermo-acoustics
u’=0p’=0
Acoustic analysis of a Turbomeca combustor includingthe swirler, the casing and the combustion chamber
C. Sensiau (CERFACS/UM2) – AVSP code
December, 2007 VKI Lecture 3
BC essential for thermo-acoustics
C. Martin (CERFACS) – AVBP code
December, 2007 VKI Lecture 4
Numerical test• 1D convection equation (D=0)
• Initial and boundary conditions:
m/s1,m8m8,0 00
Uxx
fU
t
f
0),8( tf Zero order extrapolation
m2.0,4/exp)0,( 22 aaxxf
December, 2007 VKI Lecture 5
Numerical test 0
x
f
t
f
t=3
t=6
t=9
t=12
t=15
t=18
t=21
t=24
t=27
December, 2007 VKI Lecture 6
Basic EquationsPrimitive form:
Simpler for analytical work
Not included in wave decomposition
December, 2007 VKI Lecture 7
Decomposition in waves in 1D
AoTx
VA
t
V
uP
u
u
A
.
/1.
.
cu
cu
u
..
..
..
c
c
c
L
/110
/110
/101 2
2/2/0
2/12/10
2/2/11
cc
cc
L
A can be diagonalized:
A = L-1L
- 1D Eqs:
3/mkguspeed
smcuspeed /
smcuspeed /
- Introducing the characteristic variables:
Pc
u
Pc
u
cP
W
W
W
W
1
1/ 2
3
2
1
VLW
AoTLx
W
t
W.
- Multiplying the state Eq. by L:
December, 2007 VKI Lecture 8
Remarks
• Wi with positive (resp. negative) speed of propagation may enter or leave the domain, depending on the boundary
• in 3D, the matrices A, B and C can be diagonalized BUT they have different eigenvectors, meaning that the definition of the characteristic variables is not unique.
M<1uU + cU - c
uU + cU - c
December, 2007 VKI Lecture 9
Decomposition in waves: 3D• Define a local orthonormal basis with the
inward vector normal to the boundary
21,, ttn zyx nnnn ,,
CnBnAnE zyxn • Introduce the normal matrix :
• Define the characteristic variables by: nnnnnn LLEVLW ..,. 1
nuucucuuuudiag nnnnnnn
.,,,,,
nu
nu
nucun cun
December, 2007 VKI Lecture 10
Which wave is doing what ?
WAVE SPEED INLET (un >0) OUTLET (un <0)
Wn1 entropy un in out
Wn2 shear un in out
Wn3 shear un in out
Wn4 acoustic un + c in in
Wn5 acoustic un - c out out
December, 2007 VKI Lecture 11
General implementation• Compute the predicted variation of V as given by the scheme of
integration with all physical terms without boundary conditions.
Note this predicted variation.PV
CininPinoutC WLWLVVVV ,11 ..
• Compute the corrected variation of the solution during the iteration as:
• Assess the corrected ingoing wave(s) depending on the physical
condition at the boundary. Note its (their) contribution.Cinin WLV ,1.
CinW ,
inPout WLVV 1
• Estimate the ingoing wave(s) and remove its (their) contribution(s).
Note the remaining variation.
December, 2007 VKI Lecture 12
Pressure imposed outlet• Compute the predicted value of P, viz. PP, and decompose it into
waves:
• Wn4 is entering the domain; the contribution of the outgoing wave
reads:
• The corrected value of Wn4 is computed through the relation:
54
2 nnP WW
cP
tn
Cn P
cWW
25,4
5
2 nout W
cP
OK !
Desired pressure variation at the boundary
tCnn
CinoutC PWWc
PPP ,45,
2
• The final (corrected) update of P is then:
December, 2007 VKI Lecture 13
Defining waves: non-reflecting BC
• Very simple in principle: Wn4 =0
• « Normal derivative » approach:
• « Full residual » approach:
• No theory to guide our choice … Numerical tests required
04
tn
Wcu n
n
04
tt
Wn
December, 2007 VKI Lecture 14
1D entropy wave
Same result with boththe “normal derivative” and the “full residual”approaches
December, 2007 VKI Lecture 15
2D test case• A simple case: 2D inviscid shear layer with zero velocity and constant pressure at t=0
Full residual Normal residual
December, 2007 VKI Lecture 16
Outlet with relaxation on PP
cWW nn
254 • Start from
• Cut the link between ingoing and outgoing waves to makethe condition non-reflecting
• Set to relax the pressure at the boundary towards the target value Pt
• To avoid over-relaxation, Pt should be less than unity.
Pt = 0 means ‘perfectly non-reflecting’ (ill posed)
Pc
Wn
24
tPPP BtP
December, 2007 VKI Lecture 17
Inlet with relaxation on velocity and Temperature
• Cut the link between ingoing and outgoing waves
• Set to drive VB towards Vt
• Use either the normal or the full residual approach
to compute the waves and correct the ingoing ones via:
tVVV Bt
December, 2007 VKI Lecture 18
Integral boundary condition• in some situations, the target value is not known pointwise. E.g.: the outlet pressure of a swirled flow
• use the relaxation BC framework
• rely on integral values to generate the relaxation term to avoid disturbing the natural solution at the boundary
Boundary
Boundarybulk
1dSV
SVtV Bt
December, 2007 VKI Lecture 19
Integral boundary condition• periodic pulsated channel flow (laminar)
U(y
,t) /
Ubu
lk
)sin(10bulk tuuu t
Integral BCs to impose the flow rate
?
December, 2007 VKI Lecture 20
Everything is in the details
Lodato, Domingo and Vervish – CORIA Rouen