2 D Bwd-Facing Step - Zentraler Informatikdienstinfo.tuwien.ac.at/ViennaOpenFOAMUserGroup/NumOpenFOAM_P05.pdf · 2 D Bwd-Facing Step Incompressible Navier Stokes ... Univ. Assist

[302.044] Numerical Methodsin Fluid Dynamics

2�D Bwd-Facing StepIncompressible Navier�Stokes (simpleFoam),

SIMPLE Algorithm and Turbulence Models

Univ. Assist. MSc. Francesco Romanò

[email protected]

January 8th, 2015

mailto:[email protected]

IncompressibleNavier�Stokes

Backward�FacingStep

simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm

Outline

1 Incompressible Navier�Stokes

2 Backward�Facing Step

3 simpleFoam

Turbulence Model

Meshing

Solver Set-up

Mesh Re�nement

SIMPLE Algorithm

[302.044] � Univ. Assist. MSc. Francesco Romanò 2/14



simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm

2�D Incompressible Navier�StokesEquations

Incompressible Navier�Stokes Equations

∇ · u = 0

∇ · (uu) +∇ · (νeff∇u) = −∇p+BC & IC

+Turbulence parameters

u = velocity �eld;p = relative pressure �eld divided by ρ;νeff = e�ective kinematic viscosity.




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


Incompressible Navier�Stokes Equations∇ · u = 0

∇ · (uu) +∇ · (νeff∇u) = −∇p+BC & IC






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


Incompressible Navier�Stokes Equations∇ · u = 0

∇ · (uu) +∇ · (νeff∇u) = −∇p+BC & IC






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm

2�D Backward�Facing StepMathematical Problem

Backward�Facing Step Problem

∇ · u = 0

∇ · (uu) +∇ · (νeff∇u) = −∇p

BC :

left edge: Inlet → u = 10 m/s i;

step: No�Slip → u = 0;

right edge: Out�ow → p = 0;

bottom edge: No�Slip → u = 0;

top edge: No�Slip → u = 0;

IC : u = 0 , (p = 0);

Turbulence parameters





simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm

2�D Backward�Facing StepMathematical Problem

Backward�Facing Step Problem

∇ · u = 0

∇ · (uu) +∇ · (νeff∇u) = −∇p

BC :

left edge: Inlet → u = 10 m/s i;

step: No�Slip → u = 0;

right edge: Out�ow → p = 0;

bottom edge: No�Slip → u = 0;

top edge: No�Slip → u = 0;

IC : u = 0 , (p = 0);

Turbulence parameters





simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm

2�D Backward�Facing StepOpenFOAM simpleFoam Solver

OpenFOAM simpleFoam Solver

. simpleFoam assumes an incompressible, steady, viscous�ow;

. simpleFoam has full access to all the turbulence models inthe incompressibleTurbulenceModels library and thenon-Newtonian ones in incompressibleTransportModels

library implemented in OpenFOAM:




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm





library implemented in OpenFOAM;

. Solving the NSEs using a RANS (Reynolds AveragedNavier-Stokes) model implies that the simulation cannotrepresent the turbulence phenomenon in all its scales, butit just mimes its macroscopical e�ects;

. Considering turbulence models, in transportProperties isnecessary to set the properties of the chosen model1.

1In this case, the k − ε turbulence model will be used[302.044] � Univ. Assist. MSc. Francesco Romanò 6/14



simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm











simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm











simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


OpenFOAM simpleFoam Solver: k − ε Model

(ρk) ,t+(ρkui) ,xi =[(µ′ + µt

σk

)k,xj

]+

Pk + Pb − ρε− YM + Sk

(ρε) ,t+(ρεui) ,xi =[(µ′ + µt

σε

)ε,xj

]+

C1εεk (Pk + C3εPb)− C2ερ

ε2

k + Sε

where: k = Turbulent kinetic energy , ε = Dissipation

µt = ρCµk2

ε, Pk = −ρu′iu′j

∂uj∂xi

= µtS2 = µt2SijSij

S = Modulus of the mean rate-of-strain tensor

Pb = βgiµtPrt

T,xi , β = −1

ρρ,T |p

C1ε = 1.44 , C2ε = 1.92 , Cµ = 0.09 , σk = 1.0 , σε = 1.3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



(ρk) ,t+(ρkui) ,xi =[(µ′ + µt

σk

)k,xj

]+

Pk + Pb − ρε− YM + Sk

(ρε) ,t+(ρεui) ,xi =[(µ′ + µt

σε

)ε,xj

]+

C1εεk (Pk + C3εPb)− C2ερ

ε2

k + Sε

where: k = Turbulent kinetic energy , ε = Dissipation

µt = ρCµk2

ε, Pk = −ρu′iu′j

∂uj∂xi

= µtS2 = µt2SijSij

S = Modulus of the mean rate-of-strain tensor

Pb = βgiµtPrt

T,xi , β = −1

ρρ,T |p

C1ε = 1.44 , C2ε = 1.92 , Cµ = 0.09 , σk = 1.0 , σε = 1.3[302.044] � Univ. Assist. MSc. Francesco Romanò 7/14



simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. The k − ε model needs initial conditions to be set;

. The following hypothesis will be used for the simulation ofthe Backward-Facing Step case:

◦ isotropic initial turbulence;◦ initial turbulence �uctuation intensity: 5%;◦ turbulent reference length scale: 10%;

. The correspondent parameters set-up is then resulting:

u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm







u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm







u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm





◦ isotropic initial turbulence;

◦ initial turbulence �uctuation intensity: 5%;◦ turbulent reference length scale: 10%;


u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm





◦ isotropic initial turbulence;◦ initial turbulence �uctuation intensity: 5%;

◦ turbulent reference length scale: 10%;


u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm







u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm







u′ = v′ = w′ = 5%Uin = 0.5m/s

k =3

2(0.5m/s)

2= 0.375m2/s2

ε =C0.75µ k1.5

l=

0.090.75 · 0.3751.5

0.1 · 25.4 · 10−3m2/s3 = 14.855m2/s3




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


OpenFOAM simpleFoam Solver: Meshing

. As mesh, a very coarse non-orthogonal one is employed:

. Algorithm steps:

◦ create the geometry writing the blockMeshDict �le andlocating it in <problem_folder>/constant/polyMesh/;

◦ run the command <problem_folder>$ blockMesh.




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm




. Algorithm steps:






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm




. Algorithm steps:






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm




. Algorithm steps:






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm




. Algorithm steps:






simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


OpenFOAM simpleFoam Solver: Solver Set-up

. For an incompressible, viscous �ow, considering a turbulentregime, not only the the kinematic viscosity has to bede�ned in transportProperties. To be set are theturbulence parameters, too;

. In order to choose the turbulence model to use (k− ε in thiscase) a RASProperties dictionary has to be written;

. Because the employed grid is slightly Non-Orthogonal,the �ag nNonOrthogonalCorrectors in fvSolution is setto 0;




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:

◦ set the �uid and turbulent model properties intransportProperties and RASProperties and locatethem in <problem_folder>/constant/;

◦ set the initial conditions in the p, U, epsilon, k, nut andnuTilda �les and locate them in <problem_folder>/0/;

◦ set time step and data saving/storing options writingthe controlDict �le and locate it in<problem_folder>/system/;

◦ set discretization and splitting scheme details writingthe fvSchemes and fvSolution �les and locate themin <problem_folder>/system/;

◦ run the solver using <problem_folder>$ simpleFoam;◦ visualize with <problem_folder>$ paraFoam.




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:





◦ run the solver using <problem_folder>$ simpleFoam;

◦ visualize with <problem_folder>$ paraFoam.




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Algorithm steps:









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm



. Case folder structure:

→ <problem_folder>/

→ 0/

U p

epsilon k

nut nuTilda

→ constant/

transportProperties

RASProperties

→ polyMesh/

blockMeshDict

→ system/

controlDict

fvSchemes

fvSolution




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


OpenFOAM simpleFoam Solver: Mesh Re�nement

. It is expectable that, considering the complexity of the �ow,a graded mesh is needful;

. Generally speaking, the regions of highest shear areparticularly critical, so they require the �nest grid;

. The inlet uniform �ow passing over the step generates shearon the �uid below (through a vortex in the bottom part ofthe domain). Therefore, the region of high shear will benearby both: the centreline and the walls.




simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm









simpleFoam

TurbulenceModel

Meshing

Solver Set-up

MeshRe�nement

SIMPLEAlgorithm


OpenFOAM simpleFoam Solver: SIMPLE Algorithm


Documents

2 D Bwd-Facing Step - Zentraler Informatikdienstinfo.tuwien.ac.at/ViennaOpenFOAMUserGroup/NumOpenFOAM_P05.pdf · 2 D Bwd-Facing Step Incompressible Navier Stokes ... Univ. Assist