Energy conserving schemes in OpenFOAM

Davide Modesti Matteo Bernardini Sergio Pirozzoli

Dipartimento di Ingegneria Meccanica ed AerospazialeUniversita’ di Roma ”La Sapienza”

HPC methods for Engineering - Milan, ItalyJune 17-19,2015

contact: davide.modesti@uniroma1.it

Index

1 Introduction2 Numerical solvers3 Conservation principles4 Is OpenFOAM energy conserving?5 Energy-Conserving Schemes6 Implementation into OpenFOAM:rhoEnergyFoam7 Decaying homogeneous isotropic turbulence8 Taylor-Green flow9 Taylor-Green flow: Kinetic energy spectra10 Circular Cylinder at low Reynolds number11 Compressible channel flow: DNS12 Computational Efficiency13 Scalability14 Conclusions15 Tutorial:Time reversibility on unstructured meshes16 Tutorial:Time reversibility, extremely coarse mesh

Introduction

Objectives

I Give the idea of ”discrete energy consistency”I Capability of different schemes/solver to satisfy the kinetic

energy conservation property of Navier-Stokes equationsI Comparative study: OpenFOAM vs. other CFD solversI Test Cases:

1. Decaying isotropic turbulence2. Taylor-Green flow3. Laminar circular cylinder4. DNS of supersonic channel flows

Numerical solvers

OpenFOAM

I Open Source(GPL)I Unstructured,

Finite Volume(FV)I dnsFoam,

incompressible DNSPISO algorithm

I rhoCentralFoam,compressible,Kurganov andTadmor (TVD).

Numerical solvers

OpenFOAM

I Open Source(GPL)I Unstructured,

Finite Volume(FV)I dnsFoam,

incompressible DNSPISO algorithm

I rhoCentralFoam,compressible,Kurganov andTadmor (TVD).

Commercial Solver

I ProprietaryI Unstructured(FV)I Upwinding, TVD

Numerical solvers

OpenFOAM

I Open Source(GPL)I Unstructured,

Finite Volume(FV)I dnsFoam,

incompressible DNSPISO algorithm

I rhoCentralFoam,compressible,Kurganov andTadmor (TVD).

Commercial Solver

I ProprietaryI Unstructured(FV)I Upwinding, TVD

Finite Differencesin-house solver

I Compressibleenergy-conserving

I 3D, CartesianI Arbitrary order of

accuracy

Conservation principles

Kinetic energy

Conservation principles

Kinetic Energy

It is important to satisfy the conservation of the kinetic energy in theinviscid and incompressible limit, in an unbounded domain

Conservation principles

Kinetic Energy

It is important to satisfy the conservation of the kinetic energy in theinviscid and incompressible limit, in an unbounded domain

Discrete energy conservation:

the nonlinear terms in the Navier-Stokes equations:I do not contribute to the net variation of kinetic energyI do not dissipate or inject spurious energy

Is OpenFOAM energy conserving?cfd-online Forum

I May 27, 2011 longamon asked:”Hey Foamers...........has anyone used kinetic energyconservative schemes in OF?”

Low-dissipative schemes

I Vuorinen, V., et al. ”A low-dissipative, scale-selectivediscretization scheme for the Navier-Stokes equations.”Computers & Fluids 70 (2012): 195-205.

I Vuorinen, V., et al. ”On the implementation oflow-dissipative Runge-Kutta projection methods for timedependent flows using OpenFOAM.” Computers & Fluids 93(2014): 153-163.

Energy-Conserving SchemesNumerical Flux:

∂(ρ uϕ)∂x = 1

h

(f̂j+1/2 − f̂j−1/2

), f̂j+1/2 = 1

2 (ρj uj ϕj + ρj+1 uj+1 ϕj+1)

Energy-Conserving SchemesNumerical Flux:

∂(ρ uϕ)∂x = 1

h

(f̂j+1/2 − f̂j−1/2

), f̂j+1/2 = 1

2 (ρj uj ϕj + ρj+1 uj+1 ϕj+1)

A conservative and energy-consistent scheme has been proposed byDucros et al. (2000) and generalized by Pirozzoli (2010):

FD : f̂j+1/2 = 18 (ρj + ρj+1) (ϕj + ϕj+1) (uj + uj+1)

j j + 1

j + 1/2

Implementation into OpenFOAM:rhoEnergyFoam

Space discretization

Energy consistent numerical fluxes implementedin the OpenFOAM library:

I stability without numerical dissipation

Time integration

Explicit fourth-order Runge-Kutta time integration:I low-storage implementation, suitable for LES and DNS.

Implementation into OpenFOAM:rhoEnergyFoam

rhoEnergyFoam:

I compressible unsteady solver (subsonic and supersonic shock-freeflows)

I exact kinetic energy conservation in the inviscid andincompressible limit

I 2nd order accurate in space, 4th order in timeI Integration with the OpenFOAM thermodynamic and

turbulence librariesI Suitable for DNS and LES of compressible flows.

Decaying homogeneous isotropic turbulence

Test case proposed by Honein and Moin (2004)

E(k) = A

(k

k0

)4e−2 (k/k0)2

Viscous

Reλ ≈ 100 Mt0 = urms〈a〉

= 0.2

Inviscid

Decaying homogeneous isotropic turbulence

Mesh 323 t/τ = 5

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

FDFD

Viscous

1 10 100k

E(k)

1 10 100k

E(k)

FDk2FDk2

Inviscid

Decaying homogeneous isotropic turbulence

Mesh 323 t/τ = 5

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

FDrEFoam

FDrEFoam

Viscous

1 10 100k

E(k)

1 10 100k

E(k)

FDrEFoam

k2

FDrEFoam

k2

Inviscid

Decaying homogeneous isotropic turbulence

Mesh 323 t/τ = 5

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

FDrEFoam

dnsFoam

FDrEFoam

dnsFoam

Viscous

1 10 100k

E(k)

1 10 100k

E(k)

FDrEFoam

dnsFoamk2

FDrEFoam

dnsFoamk2

Inviscid

Decaying homogeneous isotropic turbulence

Mesh 323 t/τ = 5

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

FDrEFoam

dnsFoamCommercial

FDrEFoam

dnsFoamCommercial

Viscous

1 10 100k

E(k)

1 10 100k

E(k)

FDrEFoam

dnsFoamCommercial

k2

FDrEFoam

dnsFoamCommercial

k2

Inviscid

Decaying homogeneous isotropic turbulence

Mesh 643 t/τ = 5

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

1e-07

1e-06

1e-05

0.0001

0.001

1 10 100k

E(k)

FDrEFoam

dnsFoamCommercial

FDrEFoam

dnsFoamCommercial

Viscous

1 10 100k

E(k)

1 10 100k

E(k)

FDrEFoam

dnsFoamCommercial

k2

FDrEFoam

dnsFoamCommercial

k2

Inviscid

Decaying homogeneous isotropic turbulenceMesh 323

00.10.20.30.40.50.60.70.80.9

11.1

0 5 10 15 20 25 30 35 40

KTKT 0

t/τ

FDrhoEnergyFoam

dnsFoamCommercial

Inviscid

Taylor-Green flow

Test case proposed by Duponcheel et al. (2008)

Initial Conditions

u = u0 sin (k0 x) cos (k0 y) cos (k0 z)v = −u0 cos (k0 x) sin (k0 y) cos (k0 z)w = 0

Time reversibility

Euler equations are time reversible,that is:u(t, x)⇒ −u(−t, x)

Taylor-Green flow

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

FDFD

Taylor-Green flow

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

FDrhoEnergyFoam

FDrhoEnergyFoam

Taylor-Green flow

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

FDrhoEnergyFoam

dnsFoam

FDrhoEnergyFoam

dnsFoam

Taylor-Green flow

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

FDrhoEnergyFoam

dnsFoamrhoCentralFoam

FDrhoEnergyFoam

dnsFoamrhoCentralFoam

Taylor-Green flow

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

FDrhoEnergyFoam

dnsFoamrhoCentralFoam

Commercial

FDrhoEnergyFoam

dnsFoamrhoCentralFoam

Commercial

Taylor-Green flow

dnsFoam rhoEnergyFoam

Taylor-Green flow: Kinetic energy spectra

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10k

E(k) k−2

1e-14

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1 10k

E(k) k−2

t = 0t = 16t = 0t = 16

Circular Cylinder at low Reynolds number

Regimes

Williamson (1996)I Re < 49 stationary laminarI 49 < Re < 190 laminar,

vortex sheddingI 190 < Re < 260 transitional

wake

Re=40, Mesh=64x64

l/d CD

rhoEnergyFoam 2.40 1.54icoFOAM 2.36 1.53rhoCentralFoam 1.19 1.90Commercial 1.58 1.63Taira et al. (2007) 2.30 1.56Linnick et al. (2005) 2.28 1.54Coutanceau et al. (1977)* 2.13 1.59

Circular Cylinder at low Reynolds number

rhoEnergyFoam

Vorticity

Re=200, Mesh=128x128

CD CL

rhoEnergyFoam 1.33± 0.050 ±0.66icoFoam 1.33± 0.045 ±0.69Commercial 1.26±? ±0.57rhoCentralFoam 1.30±? ±0.40Taira et al. 1.35± 0.044 ±0.69Linnick et al.* 1.34± 0.044 ±0.69

Compressible channel flow: DNSrhoEnergyFoam

Streamwise velocity in the YZ plane

Computational set-up

I Parameters:Reτ (= uτh/ν) = 220,Mb(= ub/aw) = 1.5

I Forcing: momentum andtotal energy equation, tomaintain constant massflow rate.

I Box dimensions:Lx × Ly × Lz =4πh× 2h× 4/3πh

I Mesh resolution: ∆x+ ≈10, ∆y+ < 4, ∆z+ ≈ 5About 9 milions cells

Compressible channel flow: DNSMean velocity

0

5

10

15

20

25

0.1 1 10 100

u+

y+

0

5

10

15

20

25

0.1 1 10 100

u+

y+

FDrhoEnergyFoam

FDrhoEnergyFoam

Compressible channel flow: DNSStreamwise velocity fluctuations

0

1

2

3

4

5

6

7

8

9

10

0 50 100 150 200 250

ρ/ρ

wu

′ 2+

y+

0

1

2

3

4

5

6

7

8

9

10

0 50 100 150 200 250

ρ/ρ

wu

′ 2+

y+

FDrhoEnergyFoam

FDrhoEnergyFoam

Compressible channel flow: DNSWall-normal velocity fluctuations

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 50 100 150 200 250

ρ/ρ

wv

′ 2+

y+

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 50 100 150 200 250

ρ/ρ

wv

′ 2+

y+

FDrhoEnergyFoam

FDrhoEnergyFoam

Compressible channel flow: DNSSpanwise velocity fluctuations

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

ρ/ρ

ww

′ 2+

y+

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

ρ/ρ

ww

′ 2+

y+

FDrhoEnergyFoam

FDrhoEnergyFoam

Compressible channel flow: DNSReynolds stress

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

ρ/ρ

wu′ v′+

y+

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250

ρ/ρ

wu′ v′+

y+

FDrhoEnergyFoam

FDrhoEnergyFoam

Computational Efficiency

rhoCentralFoam gives the same results as the in-house(FD) solver.

Computational Efficiency

rhoCentralFoam gives the same results as the in-house(FD) solver.

⇓

What about the computational efficiency ?

Computational Efficiency

rhoCentralFoam gives the same results as the in-house(FD) solver.

⇓

What about the computational efficiency ?

Computational times

FD rhoEnergyFoam rhoCentralFoam dnsFoam Commercial1.0 5.9 5.9 5.1 20

Scalability

Linear Scalability

I FD: up to 64K CPUs and moreI standard OpenFOAM solvers: up to 1K CPUs.

Bottlenecks: I/O and linear solversI rhoEnergyFoam: ?

Conclusions

Discrete kinetic energy conservation guarantees:I stability without addition of artificial dissipationI a greater fidelity to the physicsI OpenFOAM/Commercial are not energy-conservingI OpenFOAM can be modified, the proposed scheme is easy to be

implemented in an existing solver

Future work

I Provide rhoEnergyFoam with shock-capturing capabilitiesI Scalability analysis of rhoEnergyFoamI RANS and LES simulations of complex geometries

Tutorial:Time reversibility on unstructured meshes

X

Y

0 1 2 3 4 5 60

1

2

3

4

5

6

XY Plane

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16t∗

ωmax/ωmax0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10 12 14 16t

ωmax/ωmax0

rhoEnergyFoamrhoEnergyFoam

Tutorial:Time reversibility on unstructured meshes

Initial Time

X

Y

0 1 2 3 4 5 60

1

2

3

4

5

6

Ux: ­0.1 ­0.06 ­0.02 0.02 0.06 0.1

Reversal Time

X

Y

0 1 2 3 4 5 60

1

2

3

4

5

6

Ux: ­0.1 ­0.06 ­0.02 0.02 0.06 0.1

Final Time

X

Y

0 1 2 3 4 5 60

1

2

3

4

5

6

­Ux: ­0.1 ­0.04 0.02 0.08

Tutorial:Time reversibility, extremely coarse mesh

rhoEnergyFoam

I Time reversibility and energy consistency do no depend onthe mesh resolution !!!

I Let us consider a coarse mesh, 83.I Kinetic energy conservation is guaranteed ⇒ Time reversibility

Tutorial:Time reversibility, extremely coarse mesh

Tutorial:Time reversibility, extremely coarse mesh