Euler-characteristic boundary conditions for lattice ...fdubois/...LODI { Local One-Dimensional Inviscid equations Objective oT nd the or u i to set the Dirichlet boundary condition

Euler-characteristic boundary conditions

for lattice Boltzmann methods

Salvador Izquierdo, Norberto [email protected]

Fluid Mechanics Group (University of Zaragoza - Spain)

CEMAGREF - Antony (France)December 2008

Fluid Mechanics Group (UZ) December 2008 1 / 23

Outline

1 Introduction

2 Euler-characteristic boundary conditions for LB

3 Results

4 Conclusions


Introduction

Table of contents

1 Introduction


3 Results

4 Conclusions


Introduction

Motivation: open boundaries

Universidad_de_Zaragoza::Área_de_Mecánica_de_Fluidos::lb_preconditioning_urv

Appendix B :: ‘problems’ in turbulent simulations

LANDSAT 7 on 9/15/1999

Alejandro Selkirk IslandRe 21400

LB LES

Open boundary conditions (constant pressure, far-�eld environment)

Non-reecting open boundaries


Introduction

Motivation: examples

Microcantilever

Force

Pout

10 μm

gfn::uz

Classical CFD Navier-Stokes equations

by Antonio Gómez

Grupo de fluidodinámica numérica

Microcantilever: uid-structure interaction where small pressureperturbations modify the behavior.

Industrial boiler: ames generate pressure waves which have inuence onthe combustion.


Introduction

Review of solutions

Zero-gradient boundary conditions at outlet (Incompressible!?)

[Yu et al. (2005)][Yang's Thesis (2007)]

Non-reecting boundary conditions (thermodynamical consistency?)

Absorbing layers [Ricot et al. (2008), Tekitek et al. (2008), da Silva (2007)]Characteristic BC [Kam et al. (2007), Dehee (2008)] (without details of theimplementation )


Euler-characteristic boundary conditions for LB

Table of contents

1 Introduction


3 Results

4 Conclusions



Lattice Boltzmann method

LBM with MRT collision operator

f(xi + ei�t; t+ �t)� f(xi; t) = �M�1S[m(xi; t)�m

eq(xi; t)]

Transformation matrix D2Q9

0BBB@

�

e

�

jxqxjyqypxxpxy

1CCCA =

0BBB@

1 1 1 1 1 1 1 1 1

�4 �1 �1 �1 �1 2 2 2 2

4 �2 �2 �2 �2 1 1 1 1

0 1 0 �1 0 1 �1 �1 1

0 �2 0 2 0 1 �1 �1 1

0 0 1 0 �1 1 1 �1 �1

0 0 �2 0 2 1 1 �1 �1

0 1 �1 1 �1 0 0 0 0

0 0 0 0 0 1 �1 1 �1

1CCCA

0BBB@

f0f1f2f3f4f5f6f7f8

1CCCA

Relaxation matrix

MRT: S = diag(0; se; s�; 0; sq; 0; sq; s� ; s�)! related to transport properties

SRT: � = 1=s� ! viscosity (and stability)

TRT: s� = se = s�, sq



LB boundary conditions: velocity

INLET

xb xf

dq dx

OUTLET

xbxf

dqdxUBB { velocity bounce-back

APPROACH: reection rule + Dirichlet BC (+ correction)

f��(xf ; t+ 1) = ~f�(xf ; t)� 2feq�� (xb; t̂)

where:feq�� = !��0c

�2s (e� � u)

is the anti-symmetric part of the feq�



LB boundary conditions: pressure

INLET

xb xf

dq dx

OUTLET

xbxf

dqdx

PAB { pressure anti-bounce-back

APPROACH: reection rule + Dirichlet BC + correction

f��(xf ; t+ 1) = � ~f�(xf ; t)

+ 2feq+� (xb; t̂)

+ (2� s�)�f+� (xf ; t)� f

eq+� (xf ; t)

�

where:

feq+� = !��+

1

2!��0c

�4s

�(e� � u)

2� c

2s(u � u)

�

f+� =

1

2(f� + f��)

are the symmetric part of feq� and f�Fluid Mechanics Group (UZ) December 2008 10 / 23


LODI { Local One-Dimensional Inviscid equations

Objective

To �nd the � or ui to set the Dirichlet boundary condition in the open boundary,extracting the pressure-wave reection component. Based on [Poinsot and Lelle(1992)]

Solving in the boundary the 2D Euler equations in x-direction:

@�

@t+ u

@�

@x+ �

@u

@x= 0

@u

@t+ u

@u

@x+

1

�

@p

@x= 0

@v

@t+ u

@v

@x= 0

@�E

@t+

@ [u(�E + p)]

@x= 0

Eigenvalues!



LODI with wave amplitudes

Wave amplitudes: (isothermal! ! p = c2s�)

L =

8>><>>:

L1L2L3L4

9>>=>>;

=

8>>>>><>>>>>:

(u� cs)�@p@x� �cs @u@x

�

u @v@x

u�c2s

@�@x� @p

@x

�

(u+ cs)�@p@x

+ �cs@u@x

�

9>>>>>=>>>>>;

LODI equation using L (without the energy equation):

@�

@t+

1

2c2s(L4 + L1) +

1

c2sL3 = 0

@u

@t+

1

2�cs(L4 � L1) = 0

@v

@t+ L2 = 0

x

y

L1 (ux-cs)L1 (ux-cs)

L4 (ux+cs)L4 (ux+cs)

L2 (ux)L2 (ux)L3 (ux)L3 (ux)



Equilibrium distribution functions

Modi�ed meq� at the boundary: (�! heat capacity ratio)

eeq = �2(2� �)�+ �0(u2 + v2)�eq = �+ �0(u

2 + v2)

qeqx = ��0uqeqy = ��0v

peqxx = �0(u2 � v2)

peqxy = �0uv

In the continuum limit:

Speed of sound ! cs =p�RT =

q�p�

Viscosity ! � = 13

�1

s�� 1

2

�

Bulk viscosity ! � = 2��6

�1

se� 1

2

�



Implementation

x

y

L1 (ux-cs)L1 (ux-cs)

L4 (ux+cs)L4 (ux+cs)

L2 (ux)L2 (ux)L3 (ux)L3 (ux)

Approach: UBB or PAB with computed � and ui from LODI

LODI discretization (OUTLET{PAB)

�(t̂) � �(t̂� 1)� �t2c2s

�L4(t̂� 1) + L1(t̂� 1)

�� 1

c2sL3(t̂� 1)

u(t̂) � u(t̂� 1)� �t2�cs

�L4(t̂� 1)� L1(t̂� 1)

�

v(t̂) � v(t̂� 1)� �tL2(t̂� 1)

Models for Lin

L1(xb; t̂� 1) = k1(p(xb; t̂� 1)� pb)where:

k1 = �1(1�Ma2)cs

L


Results

Table of contents

1 Introduction


3 Results

4 Conclusions


Results

Results I: 1D wave

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

T

X

(a)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

X

(b)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

X

(c)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

X

(d)

(a) equilibrium distribution functions

(b) inlet: UBB; outlet:PAB

(c) characteristic boundary conditions (CBC)

(d) CBC with corrections


Results

Results II: reection ratio

se

|r|

0.8 1.2 1.60

0.1

0.2InletOutlet

u00.06 0.12

ρmax0.08 0.16

sν

s e(|r

inle

t| min)

1.2 1.60.5

0.6

0.7

0.8

0.9

1κ

|r|

1 2 3 4 50

0.1

0.2InletOutlet

sνκ

(|rou

tlet| m

in)

1.2 1.61

1.1

1.2

1.3

u00.04 0.08

ρmax0.08 0.16

Evaluate performance of two parameters:

bulk viscosity in the uid domain (se)heat capacity ratio in the boundary (�)


Results

Results III: mass balance

Laminar channel

CBC: (i) avoids pressure reection + (ii) allows mass conservation(well-posedness of BC)


Results

Results IV: unsteady simulation

Flow around a square cylinder

UBB-PAB: resonance has e�ects on the solution

CBC is the solution


Results

Results IV: unsteady simulation

Time steps

Cd

0 25000 50000 75000 100000

-6

-4

-2

0

2 UBB-PAB

Time steps

Cd

0 25000 50000 75000 100000

-6

-4

-2

0

2 CBC

93000 96000 990001.39

1.395

1.4

1.405

1.41

1.415

93000 96000 990001.39

1.395

1.4

1.405

1.41

1.415

Frequency

FF

T(C

d)

0.001 0.002 0.003 0.004 0.0050

0.001

0.002

0.003

0.004

0.005

0.006

0.007 CBCUBB-PAB


Conclusions

Table of contents

1 Introduction


3 Results

4 Conclusions


Conclusions

Conclusions

No previous implementation description of CBC for LB (isothermal)

Characteristic boundary conditions:

Presented for 2D open boundaries with Dirichlet conditionsDirect application for: 3D, walls and Neumann conditionsReduction of the interaction up to 99%

Key points:1 NSCBC by Poinsot and Lele (1992)2 Multireection boundary conditions by Ginzburg et al.(2008)


Conclusions

Bibliography

T. Colonius, Annu. Rev. Fluid Mech. 36, 315 (2004)

T. Poinsot and S. K. Lele, J. Comput. Phys. 101, 104 (1992)

I. Ginzburg, F. Verhaeghe, and D. d'Humi�eres, Commun. Comput. Phys.,3(2),427478, (2008).

S. Izquierdo and N. Fueyo, Phys. Rev. E. 78, 046707 (2008)


IntroductionEuler-characteristic boundary conditions for LBResultsConclusions

Documents

Euler-characteristic boundary conditions for lattice ...fdubois/...LODI { Local One-Dimensional Inviscid equations Objective oT nd the or u i to set the Dirichlet boundary condition