Kai Schneider Mam Cdp 09

8/11/2019 Kai Schneider Mam Cdp 09

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Fully adaptive multiresolution methods

for evolutionary PDEs

Kai Schneider

M2P2-CNRS & CMI, Universit de Provence, Marseille, France

Joint work with : Margarete Domingues, INPE, BrazilSonia Gomes, Campinas, Brazil

Workshop on Multiresolution and Adaptive Methods for Convection-Dominated Problems

January 22-23, 2009Laboratoire Jacques Louis Lions, Paris, France

Olivier Roussel, TCP, Unversitt Karlsruhe, Germany


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Outline

Motivation

Introduction

Adaptivity in space and time

Adaptive multiresolution method

Local time stepping / Controlled time stepping

Applications to reaction-diffusion equations

Applications to compressible Euler and Navier-Stokes

Conclusions and perspectives


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Motivation

Context: Systems of nonlinear partial differential equations(PDEs) of hyperbolic or parabolic type.

Turbulent reactive or non-reactive flows exhibit a multitude of active spatial and temporal scales.

Scales are mostly not uniformly distributedin the space-time domain,

Efficient numerical discretizations could take advantage of this property -> adaptivity in space and time

Reduction of the computational complexitywith respect to uniform discretizations

while controlling the accuracyof the adaptive discretization.

Here: adaptive multiresolution techniques


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Introduction

- Multiresolution schemes(Harten 1995)o Solution on fine grid -> solution on coarse grid + detailso Details small -> interpolation, no computation (CPU time reduced)o 2d non-linear hyperbolic problems (Bihari-Harten 1996, Abgrall-Harten 1996, Chiavassa-Donat 2001,

Dahmen et al. 2001, )

- Adaptive Multiresolution schemes(Mller 2001, Cohen et al. 2002, Roussel et al. 2003, Brger et al. 2007, )

o Details small -> interpolation and remove from memory (CPU time and memory reduction)

- Aim of this talko fully adaptive schemes (space + time) for 2d and 3d problemso Compare with Adaptive Mesh Refinement (preliminary results)


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Adaptivity: space and time

Numerical method:finite volume schemes

Space adaptivity (MR):Hartens multiresolution (MR) for cell averages.

Decay of the wavelet coeffcients to obtain information on local regularity of the solution.

coarser grids in regions where coeffcients are small and the solution is smooth,

while fine grids where coeffcients are significant and the solution has strong variations.

Controlled Time Stepping (CTS):The time integration with variable time steps,

time step size selection is based on estimated local truncation errors.

When the estimated local error is smaller than a given tolerance, the time step is increased to

make the integration more effcient.

Local time stepping (LTS):Scale-dependent time steps. Different time steps, according to each cell scale: if t is

used for the cells in the finest level, then a double time step 2t is used in coarser level with double spacing.

Required missing values in ghost cells are interpolated in intermediate time levels.


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= (l,i)0i


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Ul1= Pll1Ul

Ul+1= Pll+1 Ul

Pll+1 Pll1

Pll1 Pll+1 =

Dl,i = Ul,i Ul,i P

U

N

U

N1

Ul (Ul1, Dl)

M : UL (U0, D1, . . . , DL)


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Dl,i |Dl,i| < l

Ul = (ul,i)0lL, il


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tU= D(U)

U= (, v, e)t

D(U) = (f(U) + (U,U)) + S(U)

2nd

(l, i)

tUl,i = Dl,i

Ul,i = 1

|l,i|l,i

U dV

Dl,i := 1

|l,i|

l,i

D dV= 1

|l,i|

l,i

(f(U) + (U,U)) nl,i ds +Si

Ref. Roussel, Schneider, Tsigulin, Bockhorn. JCP 188(2003)


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Un+1 = M1 T() M E(t) Un

T()

E(t)

O(N log N)

N



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Conservative flux computation

Ingoing and outgoing flux computation in 2D for two different levels



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Dl,i < U, l,i > l,i

| < U, l,i > | < |Dl,i| < l

||l,i||L1 = 1 l = 2

d(lL)L

||l,i||L2 = 1 l = 2d

2(lL)L

||l,i||H1 = 1

l = 2(

d

21)(lL)

L


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Local Time Stepping (LTS) : main aspects

On the finest scale L, t is imposed by the stability conditionof the explicit scheme

On larger scales < L, t = 2Lt

One LTS cycle: tn tn+2L

At intermediate steps of the evolution of fine cells, requiredinformation of coarser neighbours are interpolated in time.
http://find/


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Scheme of local scale-dependent time-stepping

Ref. Domingues,Gomes, Roussel and Schneider. JCP 227 (2008)

interpolation (cheap)

evolution (expensive)

update

update

xx

x x

t t

tt

1st. stage

1st. stage2nd. stage

2nd. stage

1st. time stepRK2

2nd. time step

return to the stored value (no cost)


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Controlled Time Stepping (CTS) : main aspects
http://find/


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MR/CTS/LTS scheme

Combination of MR, CTS and LTS strategies:

1. MR/CTS is applied to determine the time step trequired to

attain a specified accuracy with a global time stepping;2. the MR/LTS cycle is computed using the obtained step size

tfor the evolution of the cell averages on the finest scale;

3. another MR/CTS time step is then done to adjust the nexttime step, and so on.
http://find/http://goback/


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Numerical validation

Error analysis

StabilityConvection-diffusion equation: tu + xu=

1Pe

2xxu, TVD if (Bihari 1996)

t x2

4P e1 + x, x 2L (7)

Accuracy

||uLex uLMR|| ||u

Lex u

LFV|| + ||u

LFV u

LMR|| (8)

Discretization error: ||uLex uL

FV|| 2L

Perturbation error: ||uLFV uL

MR|| n= T

t (Cohen et al 2002)

We want theperturbation error to be of the same order as the discretizationerror. Therefore we choose

= C 2(+1)L

P e + 2L+2 , C >0 (9)



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% CPU time compression % Memory compression

MR

L

13121110987

100

8060

40

20

0

MR

L

13121110987

100

8060

40

20

0

L-error L1-error

O(x2)MRFV

L13121110987

1e+01

1e+00

1e-01

1e-02

1e-03

1e-04

O(x2)MRFV

L13121110987

1e+00

1e-01

1e-02

1e-03

1e-04

1e-05

Convection-diffusion: P e= 10000, t= 0.2, C= 5.108



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Viscous Burgers equation: tu + x u2

2 = 1Re2xxuAnalogously, we set

= C 2(+1)L

Re + 2L+2 , C >0 (10)

MR

L

13121110987

100

80

60

40

20

0

MR

L

13121110987

100

80

60

40

20

0

O(x2

)

MRFV

L

13121110987

1e+00

1e-01

1e-02

1e-03

1e-04

1e-05

1e-06

% CPU time compression % Memory compression L1-error

Re = 1000, t= 0.2, C= 5.108



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Coherent Vortex Simulation

Compressible Navier-Stokes equations

Turbulent weakly compressible 3d mixing layer


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max||||max 0

=+

total coherent incoherent

Coherent Vortex Simul ation

(M. Farge and K. Schneider. Flow, Turbulence and Combustion, 66, 2001.).

P i i l f CVS (I)


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Principle of CVS (I)

CVS of incompressible turbulent flows: decomposition of the vorticity

= u into coherent and incoherent parts using thresholding of

the wavelet coefficients.

Evolution of the coherent flow is then computed deterministically in a

dynamically adapted wavelet basis and the influence of the incoherent

components is statistically modelled (Farge & Schneider 2001).

Here: compressible flows.

Decompose the conservative variables U = (, u1, u2, u3, e) into

a biorthogonal wavelet series.

A decomposition of the conservative variables into coherent and inco-herent components is then obtained by decomposing the conservative

variables into wavelet coefficients, applying a thresholding and recon-

structing the coherent and incoherent contributions from the strong

and weak coefficients, respectively.

P i i l f CVS (II)


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Principle of CVS (II)

Dimensionless density and pressure are decomposed into

= C+ I , (5)

p = pC+pI .

where C and pC respectively denote the coherent part of the den-sity and pressure fields, while I and pI denote the correspondingincoherent parts.

Velocity u1, u

2, u

3, temperature T and energy e, are decomposed

using the Favre averaging technique, i.e. density weighted.

For a quantity we obtain,

=C+ I , where C=()C()C

(6)

Finally, retaining only the coherent contributions of the conservativevariables we obtain the filtere d compressible Navier-Stokes equations

which describe the flow evolution of the coherent flow UC. Theinfluence of the incoherent contributions UI is in the current approachcompletely negleted.


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NavierStokes equations for compressible flows (I)

Three-dimensional compressible flow of a Newtonian fluid in the Stokes

hypothesis in a domain R3.

t=

xj

uj

t( ui) =

xj

uiuj + p i,j i,j

t( e) =

xj

( e + p) uj ui i,j

T

xj

, p,T

and e

denote the dimensionless density, pressure, temperatureand specific total energy per unit of mass, respectively; (u1, u2, u3)

T

is the dimensionless velocity vector.

Navier Stokes equations for compressible flows (II)


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NavierStokes equations for compressible flows (II)

The components of the dimensionless viscous strain tensor i,j are

i,j = Re

uixj+

uj

xi2

3 uk

xki,j

,

where denotes the dimensionless molecular viscosity and Re the

Reynolds number. The dimensionless conduc tiv ity is defined by

=

( 1)M a2 Re P r,

where , M a and P r respectively denote the specific heat ratio and

the Mach and Prandtl numbers.

The system is completed by an equation of state for a calorically ideal

gas

p = T

M a2.

and suitable initial and boundary conditions.


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NavierStokes equations for compressible flows (III)

Assuming the temperature to be larger than 120 K, the molecular

viscosity varies with the temperature according to the dimensionlessSutherland law

=T32

1 + Ts

T + Ts

where Ts 0.404.

Denoting by (x,y,z) the three Cartesian directions, this system of

equations can be written in the following compact form

U

t=

F

x

G

y

H

z

where U= (, u1, u2, u3, e)T

denotes the vector of the conser-vative quantities, and F, G, H are the flux vectors in the directions

x, y, and z, respectively.


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Time evolution (I)

Explicit 2-4 Mac Cormack scheme, which is second-order accurate in

time, fourth-order in space for the convective terms, and second-order

in space for the diffusive terms

Ul,i,j,k = Unl,i,j,k + t7 Fnl,i,j,k+ 8 F

n

l,i+1,j,k

Fn

l,i+2,j,k6x

+ t

7 Gnl,i,j,k +8 Gnl,1,j+1,k Gnl,i,j+2,k

6y

+ t7 H

nl,i,j,k +8 H

nl,1,j,k+1 H

nl,i,j,k+2

6z


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Time evolution (II)

Un+1l,i,j,k =U

nl,i,j,k + U

l,i,j,k

2+ t

2

7 Fnl,i,j,k +8 Fnl,i1,j,k Fnl,i2,j,k6x

+t

2

7 Gnl,i,j,k +8 Gnl,1,j1,k Gnl,i,j2,k

6y

+t

27 Hnl,i,j,k + 8 Hnl,1,j,k1 Hnl,i,j,k2

6z

Note that, for the computation of the diffusive terms, we do not

use a decentered scheme. Here the diffusive terms are approximated

the same way as if we were using a second-order Runge-Kutta-Heun

method in time, together with a second-order centered scheme in

space.


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= [30, 30]3

M a= 0.3

P r = 0.7

Re= 50

t= 80

N= 1283

Flow configuration of the mixing layer


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We initialize the test-case by setting two layers of a fluid stacked one upon theother one, each of them with the same velocity norm but opposed directions.

Lx

Ly

Lz

Uo

Fig. 2. Flow configuration: domain and initial basic flowu

0 of the three-dimensionalmixing layer.


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0.5

0.25

y = 0

Coherent Vortex SimulationCoherent Vortex Simulation


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Slices of vorticity at y = 0


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Coherent Vortex SimulationCoherent Vortex Simulation

Adaptive gr id


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1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e+02

1e+03

1 10

E

k

DNSCVS Epsilon = 0.2

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e+02

1e+03

1 10

E

k


1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e+02

1e+03

1 10

E

k


t = 80

L1

L2

H1

L1

L2

H1



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= [

60,60]3

Ma= 0.3

P r = 0.7

Re= 200

t= 80

N= 2563 1283


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Fig. 17. Time evolution of a weakly compressible mixing layer at resolution N= 2563

in the quasi-2D regime. CVS computation with = 0.03 and norm #3. First row:

Two-dimensional cut of vorticity at y = 0, 10 isolines of vorticity between 0.1 and

1. Second row: Corresponding isosurfaces of vorticity |||| = 0.5 (black) and ||||= 0.25 (gray). Third Row: Corresponding adaptive mesh of the CVS computation.The corresponding time instants are t = 19 (left), t = 37(center) and t = 78(right).

1e-05

1e-04

1e-03

1e-02

1e-01

1e+00

1e+01

1e+02

1e+03

10 100

E

k

CVS

830000

840000

850000

860000

870000

0 10 20 30 40 50 60 70 80

E

t

CVS

6000

7000

8000

9000

10000

0 10 20 30 40 50 60 70 80

Z

t

CVS

Fig. 18. Energy spectra in the streamwise direction at t= 80(left). Time evolutionof the kinetic energy (center) and enstrophy (right) for the CVS computations at

Re= 200, N= 2563.

Conclusions (CVS I)


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( )

Adaptive multiresolution method to solve the three-dimensional com-

pressible NavierStokes equations in a Cartesian geometry.

Extension of the Coherent Vortex Simulation approach to compress-

ible flows.

Time evolution of the coherent flow contributions computed effi-

ciently using the adaptive multiresolution method.

Generic test case: weakly compressible turbulent mixing layers.

Different thresholding rules, i.e. L1, L2 and H1 norms.

H1 based threshold yields the best results in terms of accuracy and

efficiency.

Conclusions (CVS II)


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CVS required only about 1/3 of the CPU time needed for DNS and

allows furthermore a memory reduction by almost a factor 5. Never-

theless all dynamically active scales of the flow are well resolved.

Drawbacks:

Explicit time discretization, imposes a time step limitation due to

stability reasons, i.e. the smallest spatial scale dictates the actual

size of the time step ( local time stepping strategies).

Using local time stepping the time step on larger scales can be in-

creased without violating the stability criterion of the explicit time

integration (further speed up).

Generalisation to complex geometries: volume penalization approach

(cf. Angot et al. 1999, Schneider & Farge 2005).

http://www.cmi.univ-mrs.fr/~kschneid http://wavelets.ens.fr


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Compressible Euler equations

Multiresolution or Adaptive Mesh Refinement ?

2D Riemann problem: Lax-Liou test case 5

3D expanding circular shock wave

(joint work with Ralf Deiterding, Oak Ridge, USA)


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2D/3D Euler equations

The compressible Euler equations:

Q

t+

F

r = 0, with Q=

ve

and F =

v

u2 +p(e+p)v

where t is time,ris 2D position vector with |r|=

(x2 +y2),

=(r, t) density,v

=v

(r,t

) velocity with components (v1

,v2

),e=e(r, t) energy per unit of mass andp=p(r, t) pressure.
http://find/


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The equation of state for an ideal gas

p=RT = ( 1) e

|v|2

2

,

completes the system, whereT =T(r, t) is temperature,

specific heat ratio andRuniversal gas constant.

In dimensionless form, we obtain the same system of equations,but the equation of state becomes p= T

Ma2 , where Ma denotes

the Mach number.

Inviscid implosion phenomenon (2d)
http://find/


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Inviscid implosion phenomenon (2d)

The initial conditions are

(r, 0) =

1 if r r00.125 if r>r0,

e(r, 0) =

2.5 if r r00.25 if r>r0,

v1 =v2 = 0 and r0 denotes the initial radius.This initial condition is stretched in one direction and a rotation in the axes isapplied.

r=

rX2

a2 +

Y2

b2,

X = xcos ysin ,Y = xsin +ycos

The parameters of the ellipse: a= 1/3, b= 1, the rotation angle is = /3

with an initial radius r0 = 1, computational domain is = [2, 2]2,= 102.

Ref.: Domingues et al., ANM, 2009, in press

M lti l ti C t ti lli ti l i l i
http://find/


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Multiresolution Computation : elliptical implosion

Density Grid

Ref.: Domingues et al., ANM, 2009, in press

Comparison for the numerical solutions of the 2D Euler equations for t=0.5 with L=10 and = 2 103.

Method Error CPU

E Ti M
http://elliptic-implosion/mesh.mpeghttp://elliptic-implosion/density.mpeg


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E Time Memory

(%) (103 sec) (%) (%)

FV-RK2,CFL(0)=0.18(Ref.) 0.60 45 100 100

MR-RK2, CFL(0) =0.18 0.67 10 23 18MR/LTS-RK2, CFL(0) =0.18 1.09 9 19 16

MR/CTS/LTS-RK2(3), CFL(0) =0.24 0.66 8 18 18

FV-RK3,CFL(0)=0.18(Ref.) 0.59 65 100 100

MR-RK3, CFL(0) =0.18 0.66 12 18 18

MR/CTS-RK2(3), CFL(0) =0.24 0.63 9 14 18


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2d Riemann problem: Lax-Liu test case 5

Computational domain is = [0, 1] [0, 1],4 free-slip boundary conditions andPhysical parameters Ma= 1 and = 1.4.

Initial conditions:Parameters Domain position

1 2 3 4

Density() 1.00 2.00 1.00 3.00Presure (p) 1.00 1.00 1.00 1.00

Velocity Component (v1) -0.75 -0.75 0.75 0.75Velocity Component (v2) -0.50 0.50 0.50 -0.50 x

y

3 4

12
http://find/


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MR and AMR computations

MR method: 2nd order MUSCL with AUSM+-up Scheme fluxvector splitting Liou(JCP, 2006) with van Albada limiter is used.RK2. Wavelet threshold = 0.01.

AMR method: 2nd order unsplit shock-capturing MUSCL scheme

with AUSMDV flux vector splitting Wada& Liou (SIAM J.Comput.Sci., 1997) . Limiting and reconstruction in primitive variables withMinmod limiter. Modified RK2. Adaptive parameters=p= 0.05 and p== 0.05, with coarser level 128 128.

Computations at final time 0.3.Target CFL number is 0.45.

In collaboration with Ralf Deiterding, Oak Ridge, USA

Adaptive Multiresolution Computation : Lax-Liu test case 5
http://find/


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Adaptive Multiresolution Computation : Lax-Liu test case 5

Density Grid

AMR simulation


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AMR simulation

Uniform r1,2,3= 2, 2, 2 Reference solutionx= 1/1024 x= 1/1024 x= 1/4096

Reference solution computed with Wave Propagation Method.


Summary of the results for MR and AMR/LT
http://find/


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MR

Level Le1() Overhead Grid Compression Overhead[102] per it. cell (%) per it. (%)

L=8 4.13 0.58 24.98 14.6L=9 2.79 0.52 13.23 6.8

L=10 1.84 0.63 6.58 4.2

AMRLevel Le

1() Overhead Grid Compression Overhead

[102] per it. cell (%) per it. (%)

L=8 4.00 0.13 68.2 8.7L=9 2.66 0.03 44.4 1.3

L=10 1.57 0.12 26.2 3.1

Preliminary resultsIn collaboration with Ralf Deiterding, Oak Ridge, USA
http://find/


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MR computations


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MR computations

Numerical Parameters: L= 7, = 0.001, RK2 scheme, MUSCLAUSM+up flux, CFL= 0.8.

Density initial condition.
http://find/


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Evolution of density att= 0.042, 0.084, 0.126, 0.210, 0.252, 0.294, 0.336, 0.378, 0.420(from left to right and top to bottom).
http://find/


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Adaptive grid: xyprojection grid att= 0.042, 0.084, 0.126, 0.210, 0.252, 0.294, 0.336, 0.378, 0.420(from left to right and top to bottom).

AMR computations


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Numerical Parameters: 2 levels with refinement factor 2 are used,

finest level: 120 120 120 grid (1.73 M cells), coarse grid of30 30 30 cells, minmod-limiter, CFL= 0.8, until physical timet= 0.84, 58 time steps.

Evolution of density at t= 0,t= 0.21 and t= 0.84(from left to right).Source: amroc.sourceforge.net/examples/euler/3d/html.

In collaboration with Ralf Deiterding, OakRidge, USA


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Evolution of the density solution, adaptive computations with 3levels and 2 buffer cells.Cut at z= 0 and z= 0.5 at time t= 0.84 (from left to right) .

Source: amroc.sourceforge.net/examples/euler/3d/html.

http://find/


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Reaction-diffusion equations

2D thermo-diffusive flames

3D flame balls

Governing equations


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Non-dimensional thermodiffusive equations

tT+ v T 2

T = s (1)

tY + v Y 1

Le

2Y = (2)

(T, Y) = Ze2

2 LeY exp

Ze(T 1)

1 + (T 1)

(reaction rate)

s(T) =

T+ 1 14

1 1

4 (heat loss due to radiation)

+ initial and boundary conditions

Y =Y1, T =

T Tu

Tb Tu , Le=

D (Lewis), =

Tb Tu

Tb , Ze= Ea

RTb (Zeldovich)

v given by the incompressible NS equations. When the fluid is at rest, v = 0.

Governing equations


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Planar flames

Flame propagation at the velocity vf

When the fresh mixture is advected at v = vf steady planar flame

YT

EDCBA

1

0

AB: fresh mixture, BC: preheat zone, CD: reaction zone d= O(Ze1), DE: burnt mixture

Governing equations


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Thermodiffusive instability

x x

Burnt gas Fresh gas

< 0

Fresh gas Burnt gas

yy

mass

heat

Stable: for Le = 1, Z e= 10 (animation) - Unstable: for Le= 0.3, Ze = 10 (animation)

Asymptotic theory for Ze >>1 (Sivashinsky 1977, Joulin-Clavin 1979)

1) Ze(Le 1) < 2 : cellular flames 2) Ze(Le 1) > 16 : pulsating flames

2D Flame front


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Temperature Reaction rate Adaptive grid

stableLe = 1.0

Unstable

Le = 0.3

Application to TD flames

Th fl b ll fi ti


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The flame ball configuration

Simplest experiment to study the interaction of chemistry and transport ofgases (experimental: Ronney 1984, theory: Buckmaster-Joulin-Ronney 1990-91)

Enables to study the flammability limit of lean gaseous mixtures

HOT BURNT GAS

COLD PREMIXED GAS

radiation

heat conduction

reactant diffusion

Problem: the combustion chamber is finite Interaction with wall



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pp

Interaction flame front-adiabatic wall: the 1D case

Lean mixture H2-air, Ze= 10, = 0.64, = [0, 30]

Radiation neglected

Adiabatic walls Neuman boundary conditions

Objective: study the inflence of Le

Profiles of T and for Le= 0.3 (animation 1)

Profiles of T and for Le= 1 (animation 2)

Profiles of T and for Le= 1.4 (animation 3)

Ref. Roussel, Schneider, CTM10(2006)

http://movies/flamewall1d_Le0.3.gifhttp://movies/flamewall1d_Le1.gifhttp://movies/flamewall1d_Le1.4.gifhttp://movies/flamewall1d_Le1.4.gifhttp://movies/flamewall1d_Le1.gifhttp://movies/flamewall1d_Le0.3.gif


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Interaction flame front-adiabatic wall: the 1D case

Le= 0.9Le= 0.8Le= 0.7Le= 0.3

121086420

2

1.5

1

0.5

0

Le = 1.4Le= 1

Le= 0.95

121086420

7

6

5

4

3

2

1

0

t t

Flame velocity vf for Le


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Interaction flame ball-adiabatic wall

Radiation neglected, Ze= 10, = 0.64, = [50, 50]d

Adiabatic walls Neuman boundary conditions

At t= 0, the radius of the flame ball is r0 = 2.

2D: Evolution of T and mesh for Le= 0.3 (animations 1-2)

2D: Evolution of T for Le= 1 (animation 3)

2D: Evolution of T for Le= 1.4 (animation 4)

3D: Evolution of T and mesh for Le= 1 (animations 5-6)

Analogy with capillarity for a fluid droplet


http://movies/flamewall2d_Le0.3_temp.avihttp://movies/flamewall2d_Le0.3_temp.avi


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pp

Interaction flame ball-adiabatic wall

Le= 1.4Le = 1

Le= 0.3

t

706050403020100

140

120

100

80

6040

20

0

Le = 1.4Le= 1

Le = 0.3

t

1086420

8000

7000

6000

5000

4000

3000

2000

1000

0

R = d in 2D R = d in 3D




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pplication to TD flames

Interaction flame ball-adiabatic wall: Performances

d Le N max % CPU % Mem

2 0.3 2562 25.50% 14.10%

2 1 2562 21.50% 11.75%2 1.4 2562 21.00% 11.10%

3 1 1283 12.98% 4.38%



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Interaction flame ball-vortex

v

X

Phenomenon which happens e.g. in furnaces

Thermodiffusive model, v analytic solution of Navier-Stokes

Evolution of T and mesh for Ze= 10, Le= 0.3, no radiation (animations)



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Interaction flame ball-vortex

F V 5122F V 2562

M R

t

10.80.60.40.20

40

35

30

25

20

15

10

5

0

= 0 = 0

t

10.80.60.40.20

40

35

30

25

20

15

10

5

0

R =

d: for MR and FV methods with and without vortex

Ref. Roussel, Schneider Comp. Fluids34(2005)

3D flame ball, Le = 1


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Temperature Adaptive grid

Splitting flame ball computed with the MR/LTS method


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Iso-surfaces and isolines on the cut-plane for temperature (top) and concentration (bottom) withL=8 scales, Le=0.3, Ze=10,k=0.1.

Ref. Domingues Gomes, Roussel and Schneider. JCP 227 (2008)

Temperature Concentration grid xy grid yz

Splitting flame ball: projections of the cell centers used onthe adaptive mesh
http://movies/movie_jcp2008_temp.avihttp://movies/conc_jcp2008.avihttp://movies/movie_jcp2008_xymesh.avihttp://movies/movie_jcp2008_yzmesh.avihttp://movies/movie_jcp2008_yzmesh.avihttp://movies/movie_jcp2008_xymesh.avihttp://movies/movie_jcp2008_temp.avihttp://movies/conc_jcp2008.avi


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Splitting flame ball: CPU and memory compressions for thedifferent methods with L=8 scales


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Method % CPU time % Memory Integral reaction

rate

MR 2.7 1.05 669.09

MR/LTS 2.3 1.05 669.11



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Conclusions

Finite volume discretization with explicit time integration (both of second-order) tosolve evolutionary PDEs in Cartesian geometry.

Efficient space-adaptive multiresolution method (MR) with local time stepping (LTS).CPU speed-up and memory reduction, while controling the accuracy.

Further speed-up due to an improved time advancement using larger time steps onlarge scales without violating the stability condition of the explicit scheme.

However, synchronization of the tree data structure necessary.

Time-step control (CTS) for space adaptive schemes (embedded Runge-Kutta schemes)and combination with LTS.

Applications to reaction-diffusion equations, compressible Euler and Navier-Stokes equations.

Next: develop level dependent time step control which allows to adapt the time step within acycle of the level dependent time stepping MR/LTS.

http://www.cmi.univ-mrs.fr/~kschneid http://wavelets.ens.fr

Documents

Kai Schneider Mam Cdp 09