December, 2007VKI Lecture1 Boundary conditions. December, 2007VKI Lecture2 BC essential for...

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