Lagrangian scheme for scalar advection

(1)

HAL Id: hal-01821098

https://hal.archives-ouvertes.fr/hal-01821098

Submitted on 22 Jun 2018

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Lagrangian scheme for scalar advection

Benoît Trouette, Georges Halim Atallah, Stéphane Vincent

To cite this version:

Benoît Trouette, Georges Halim Atallah, Stéphane Vincent. Lagrangian scheme for scalar advection.

Turbulence and Interactions TI2018, Jun 2018, Trois Ilets, France. �hal-01821098�

(2)

Lagrangian scheme for scalar advection

B. Trouette, G. Halim Atallah and S. Vincent benoit.trouette@u-pem.fr

P ^ROBLEM

Solve advection–diffusion equation, for low diffusivity ( Γ ) or high Péclet num- bers values. Applications: pollutant transport, two phase-flow, . . .

S ^PLITTING A ^PPROACH

Φ ^? − Φ ⁿ

∆t + ∇ · (u ⁿ Φ ⁿ ) = 0 ⇒ QUICK, MUSCL, WENO and VSM [1] schemes Φ ⁿ⁺¹ − Φ ^?

∆t = ∇·(Γ∇Φ ⁿ⁺¹ ) ⇒ centered scheme, implicit, direct solver (MUMPS)

C ^ODE D ^ESCRPITION

• Finite-Volumes on staggered grids.

• Augmented Lagrangian or KSP [2] for Pressure / Velocity coupling.

• VOF, Level Set, Front Traking.

• Lagrangian particle tracking.

• DNS and LES turbulence modelling.

• Penalty methods.

16 32 64 128 256 512 1024 2048 4096

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

A(n procs) E(n procs = A(n procs)/n procs

n

_procs

Ideal speed up MPI Speed up on Occigen (CINES), N

³

=128

³

N

³

=256

³

N

³

=512

³

L AGRANGIAN SCHEME

1. M markers (Lagrangian particles) of positions X _m and volumes δV _m carry the local information φ _m of the Eulerian field Φ . At initial time, φ ⁰ _m = Φ ⁰ (X _m ) , m = 1, . . . , M . 2. The markers are advected with the fluid velocity: ^dX _dt ⁿ⁺¹ ^m = u ⁿ , m = 1, . . . , M .

3. Post advection value of Φ is evaluated with averages on each Eulerian cell Ω _i,j .

Φ ^? _i,j = X

m:X ⁿ⁺¹ _m ∈Ω _i,j

φ ⁿ _m δV _m ⁰

,

X

m:X ⁿ⁺¹ _m ∈Ω _i,j

δV _m ⁰ with δV _m ⁰ = δV _m ∩ Ω _i,j

4. Φ ⁿ⁺¹ is then obtained solving the unsteady diffusion equation.

5. The local (Lagrangian) information is updated according the variation of Φ at the particle position: ^∂φ _∂t ^m

X ⁿ⁺¹ _m

= ^∂ _∂t ^Φ

X ⁿ⁺¹ _m . For a first order integration scheme, φ ⁿ⁺¹ _m = φ ⁿ _m + Φ ⁿ⁺¹ (X ⁿ⁺¹ _m ) − Φ ^? (X ⁿ⁺¹ _m ), m = 1, . . . , M.

R ^EFERENCES

[1] S. Vincent et al. Eulerian-Lagrangian multiscale methods for solving scalar equations.

Application to incompressible two-phase flows. In Journal of Comptutational Physics (2010) [2] J-.P. Caltagirone & S. Vincent. A Kinematic Scalar Projection method (KSP) for incom-

pressible flows with variable density. In Open Journal of Fluid Dynamics (2015)

A CKNOWLEDGMENT

The authors are grateful for the compu- tational facilities of GENCI under project n ^o A0032B06115 and to M. El Ouafa, M.

Mbaye and to E. Belut & S. Lechêne (INRS).

R ^ESULTS

Advection/Diffusion of a pollutant peak

max( ) = 0.985 Exact solution

L=1 m, R=0.1 m, =10^-6 m²/s u(y)=- (y-L/2)/2 m/s

v(x)=+ (x-L/2)/2 m/s

0(r)=(R-r)/R, if r<R =0, if r>R

time = 0 s

max( ) = 0.979

time = 1 s

max( ) = 0.973

time = 2 s

max( ) = 0.969

time = 3 s

max( ) = 0.964

time = 4 s

Quick scheme

max( ) = 0.620 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Muscl scheme

max( ) = 0.638 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Weno 5 scheme

max( ) = 0.857 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Lagrangian scheme (2 ppdpc) max( ) = 0.922 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Weno 5 scheme

max( ) = 0.917 | max(

_exact

) = 0.964

Mesh 256

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Lagrangian scheme (4 ppdpc) max( ) = 0.925 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Weno 5 scheme

max( ) = 0.948 | max(

_exact

) = 0.964

Mesh 512

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Lagrangian scheme (8 ppdpc) max( ) = 0.957 | max(

_exact

) = 0.964

Mesh 128

²

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

X

Y

0.3 0.4 0.6 0.7

0.6 0.7 0.8 0.9

Computational cost in % of the Ref. (Weno, 512 ² (13h)) Weno, 128 ² 1.2% Lag., 128 ² , 2–4 ppdpc ∼ 2 %

Weno, 256 ² 19.2% Lag., 128 ² , 8 ppdpc 6.1%

Weno, 512 ² Ref. Lag., 128 ² , 16 ppdpc 48%

Phase inversion problem with KSP method [2]

Phase 1

1=900 kg/m³

1=0.1 Pa.s C=0

Phase 2

2=1000 kg/m³

2=0.1 Pa.s C=1

H=0.1 m g=9.81 m/s²

=0.045 N/m

time=0.0 time=0.4 time=1.2 time=2.0 time=3.2

Snapshots of the color function over time. Mesh 128 ² . Blue is KSP, VOF-PLIC, black line is C = 0.5 for the Augmented Lagrangian, VOF-PLIC method and orange line stands for the Lagrangian advection scheme with 4 ppdpc and KSP.

Ventilated cavity (collaboration with INRS)

X Y

Z

Lag.

Weno

C=0.25

X Y

Z

Lag.

Weno

C=0.5

X Y

Z

Lag.

Weno

C=0.75

Simulation of air flow at Re = 1500 (injection) in a cuboid cavity. Air flow carries a tracer gas with air properties ( Γ ≈ 10 ⁻⁵ m ² /s). Iso-surfaces 0.25, 0.5 and 0.75 of con- centration C for Lagrangian scheme with 4 ppdpc (left) and Weno 5 scheme (right) are plotted for t = 1.5 second. The non diffusive character of the Lagrangian scheme is ob- served. (Mesh: 64 × 32 × 32 , ∆t = 5 × 10 ⁻³ s.)

-5 -4 -3 -2 -1 0

-2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 log

₁₀

( ∆ x)

Weno Lag. 2 ppdpc 4 ppdpc 8 ppdpc 16 ppdpc

slope 2

slope 1

log

10

(err

L2

) log

10

(err

∞

Lagrangian scheme for scalar advection

HAL Id: hal-01821098

https://hal.archives-ouvertes.fr/hal-01821098

Lagrangian scheme for scalar advection

Benoît Trouette, Georges Halim Atallah, Stéphane Vincent

To cite this version:

Lagrangian scheme for scalar advection

B. Trouette, G. Halim Atallah and S. Vincent benoit.trouette@u-pem.fr

P ROBLEM

Solve advection–diffusion equation, for low diffusivity ( Γ ) or high Péclet num- bers values. Applications: pollutant transport, two phase-flow, . . .

S PLITTING A PPROACH

Φ ? − Φ n

∆t + ∇ · (u n Φ n ) = 0 ⇒ QUICK, MUSCL, WENO and VSM [1] schemes Φ n+1 − Φ ?

∆t = ∇·(Γ∇Φ n+1 ) ⇒ centered scheme, implicit, direct solver (MUMPS)

C ODE D ESCRPITION

• Finite-Volumes on staggered grids.

• Augmented Lagrangian or KSP [2] for Pressure / Velocity coupling.

• VOF, Level Set, Front Traking.

• Lagrangian particle tracking.

• DNS and LES turbulence modelling.

• Penalty methods.

16 32 64 128 256 512 1024 2048 4096

n

Ideal speed up MPI Speed up on Occigen (CINES), N

=128

N

=256

N

=512

L AGRANGIAN SCHEME

1. M markers (Lagrangian particles) of positions X m and volumes δV m carry the local information φ m of the Eulerian field Φ . At initial time, φ 0 m = Φ 0 (X m ) , m = 1, . . . , M . 2. The markers are advected with the fluid velocity: dX dt n+1 m = u n , m = 1, . . . , M .

3. Post advection value of Φ is evaluated with averages on each Eulerian cell Ω i,j .

Φ ? i,j = X

m:X n+1 m ∈Ω i,j

φ n m δV m 0

,

X

m:X n+1 m ∈Ω i,j

δV m 0 with δV m 0 = δV m ∩ Ω i,j

4. Φ n+1 is then obtained solving the unsteady diffusion equation.

5. The local (Lagrangian) information is updated according the variation of Φ at the particle position: ∂φ ∂t m

X n+1 m

= ∂ ∂t Φ

X n+1 m . For a first order integration scheme, φ n+1 m = φ n m + Φ n+1 (X n+1 m ) − Φ ? (X n+1 m ), m = 1, . . . , M.

R EFERENCES

[1] S. Vincent et al. Eulerian-Lagrangian multiscale methods for solving scalar equations.

Application to incompressible two-phase flows. In Journal of Comptutational Physics (2010) [2] J-.P. Caltagirone & S. Vincent. A Kinematic Scalar Projection method (KSP) for incom-

pressible flows with variable density. In Open Journal of Fluid Dynamics (2015)

A CKNOWLEDGMENT

The authors are grateful for the compu- tational facilities of GENCI under project n o A0032B06115 and to M. El Ouafa, M.

Mbaye and to E. Belut & S. Lechêne (INRS).

R ESULTS

Advection/Diffusion of a pollutant peak

Quick scheme

max( ) = 0.620 | max(

) = 0.964

Mesh 128

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

Muscl scheme

max( ) = 0.638 | max(

) = 0.964

Mesh 128

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

Weno 5 scheme

max( ) = 0.857 | max(

) = 0.964

Mesh 128

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

Lagrangian scheme (2 ppdpc) max( ) = 0.922 | max(

) = 0.964

Mesh 128

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

Weno 5 scheme

max( ) = 0.917 | max(

) = 0.964

Mesh 256

, CFL = 0.5 Isolines: 0.9, 0.5, 0.1, 0.01

Lagrangian scheme (4 ppdpc) max( ) = 0.925 | max(

) = 0.964

Mesh 128

P ^ROBLEM

S ^PLITTING A ^PPROACH

Φ ^? − Φ ⁿ

∆t + ∇ · (u ⁿ Φ ⁿ ) = 0 ⇒ QUICK, MUSCL, WENO and VSM [1] schemes Φ ⁿ⁺¹ − Φ ^?

∆t = ∇·(Γ∇Φ ⁿ⁺¹ ) ⇒ centered scheme, implicit, direct solver (MUMPS)

C ^ODE D ^ESCRPITION

3. Post advection value of Φ is evaluated with averages on each Eulerian cell Ω _i,j .

Φ ^? _i,j = X

m:X ⁿ⁺¹ _m ∈Ω _i,j

φ ⁿ _m δV _m ⁰

m:X ⁿ⁺¹ _m ∈Ω _i,j

δV _m ⁰ with δV _m ⁰ = δV _m ∩ Ω _i,j

4. Φ ⁿ⁺¹ is then obtained solving the unsteady diffusion equation.

5. The local (Lagrangian) information is updated according the variation of Φ at the particle position: ^∂φ _∂t ^m

X ⁿ⁺¹ _m

= ^∂ _∂t ^Φ

X ⁿ⁺¹ _m . For a first order integration scheme, φ ⁿ⁺¹ _m = φ ⁿ _m + Φ ⁿ⁺¹ (X ⁿ⁺¹ _m ) − Φ ^? (X ⁿ⁺¹ _m ), m = 1, . . . , M.

R ^EFERENCES

The authors are grateful for the compu- tational facilities of GENCI under project n ^o A0032B06115 and to M. El Ouafa, M.

R ^ESULTS

Computational cost in % of the Ref. (Weno, 512 ² (13h)) Weno, 128 ² 1.2% Lag., 128 ² , 2–4 ppdpc ∼ 2 %

Weno, 256 ² 19.2% Lag., 128 ² , 8 ppdpc 6.1%

Weno, 512 ² Ref. Lag., 128 ² , 16 ppdpc 48%

Snapshots of the color function over time. Mesh 128 ² . Blue is KSP, VOF-PLIC, black line is C = 0.5 for the Augmented Lagrangian, VOF-PLIC method and orange line stands for the Lagrangian advection scheme with 4 ppdpc and KSP.