Numerical simulations of multiphase and turbulent...

Numerical simulations of multiphase andturbulent flows

Daniel Fuster , Cansu Ozhan

CNRS. D’Alembert Institute. UPMC. Paris VI.

C. Ozhan et al () Efficient AMR criteria for vortical structures October 26, 2014 1 / 44

Overview

Numerical development

Gerris . In particular I develop a new branch of it and Paris Simulator


Overview


Gerris . In particular I develop a new branch of it and Paris SimulatorTools for turbulence simulations


Overview



Turbulence models (LES).


Overview



Turbulence models (LES).Special numerical schemes (skew-symmetric formulation)


Overview



Turbulence models (LES).Special numerical schemes (skew-symmetric formulation)Numerical tools for the analysis of turbulent structures (FFTW for flow fieldvariables and interfaces)


Overview




Special Adaptive Mesh Refinement features.


Overview




Special Adaptive Mesh Refinement features.Subgrid models (particle models)


ApplicationsInvestigation of atomization processes: Fuster et al (JFM, 2013)



Wave turbulence: Deike et al (PRL, 2014)



Wave turbulence: Deike et al (PRL, 2014)

Turbulent and reactive flows in catalytic convertersOzhan et al (CES, 2014)


Overview

Numerical developmentGerris (in particular I develop a new branch of it) and Paris Simulator

Tools for turbulence simulationsTurbulence models (LES).Special numerical schemes (skew-symmetric formulation)Numerical tools for the analysis of turbulent structures (FFTW for flow fieldvariables and interfaces)

Special Adaptive Mesh Refinement features and subgrid models.


Overview




Special Adaptive Mesh Refinement features and subgrid models.


Numerical schemes for turbulent flow simulations

How to handle multiphase flows and turbulence in Gerris?


Gerris allows for the DNS of multiphase flows

Proper numerical schemes?DNS are expensive, how to reduce the computational effort?




Adaptive Mesh Refinement (AMR)




Adaptive Mesh Refinement (AMR)Subgrid models (Particle module)


Skew-symmetric formulation [J. Comp. Phys, 2013]

Isotropic turbulence

E(k) = αǫ2/3k−5/3fL(kLint)fν(kLintRe−3/4L )

Lint = 0.5Lbox

ReL =

√

∫

∞

0E(k)dkLint

ν= 375

k =

∫

∞

0

E(k)dk = 0.5,

Skew-Symmetric vs. Bell-Cotella-Glaz

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

k/k 0

tu’/Lint

323

643

1283

2563

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

k/k 0

tu’/Lint

323

643

1283

2563





Lint = 0.5Lbox

ReL =

√

∫

∞

0E(k)dkLint

ν= 375

k =

∫

∞

0

E(k)dk = 0.5,

Skew-Symmetric vs. Bell-Cotella-Glaz

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0 0.2 0.4 0.6 0.8 1

Tay

lor

scal

e

tu’/Lint

323

643

1283

2563

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0 0.2 0.4 0.6 0.8 1

Tay

lor

scal

e

tu’/Lint

323

643

1283

2563





Lint = 0.5Lbox

ReL =

√

∫

∞

0E(k)dkLint

ν= 375

k =

∫

∞

0

E(k)dk = 0.5,

We significantly reduce the error of high order statistics


Effect of multiple phases (same Reynolds in both phases)


Can other numerical schemes came to rescue?

We are currently working with S. Zaleski on the implentation of amomentum conservative scheme (Paris Simulator)




The idea is to advect momentum ”geometrically” (as in VOF): Rudman’smethod (IJNMF, 1998) but without the need of working with a subgrid.





It avoids instabilities at large momentum ratios






but does it also improve the quality of the solution in turbulentsimulations?






but does it also improve the quality of the solution in turbulentsimulations?Work in progress


Motivation

We are interested in developing numerical tools for the simulation of turbulentflows.

In particular this work is motivated by the flow simulation inside a catalyticconverter system:


Motivation

We are interested in developing numerical tools for the simulation of turbulentflows.

In particular this work is motivated by the flow simulation inside a catalyticconverter system:

Adaptive Mesh Refinement (AMR) help us to save a significant amount ofcomputational time!


Problematic:

DNS is not always possible because the smallest scale on the problemcan be really small


Problematic:

DNS is not always possible because the smallest scale on the problemcan be really small

What we want is to get as close as possible to the real solution trying tooptimize the grid distribution


Optimizing the grid distribution

In order to dynamically adapt the grid one needs to adapt the grid




How to estimate the error?





and maybe more important... to estimate the error of what?






Test cases

In order to measure the efficiency of AMR techniques we need:

Simplified test cases with analytical solution






Test cases

In order to measure the efficiency of AMR techniques we need:

Simplified test cases with analytical solution

We want to measure the global and local efficiency of the method


We need to chose:

A method to estimate the error

A norm to adapt the grid:

AMR criteria → Error×∆xn


Because of their efficiency, we compare different a-posteriori error estimationmethods:

Error indicatorsWe trust in our intuition to define variables that we expect to beproportional to the error




Gradient (velocity, tracer)




Gradient (velocity, tracer)Vorticity norm




Gradient (velocity, tracer)Vorticity normHelicity norm (3D)



A Hessian based h-refinement algorithmWe measure the error of a given quantity based on the differencebetween two consecutive levels



A Hessian based h-refinement algorithmWe measure the error of a given quantity based on the differencebetween two consecutive levelsIt is an estimation of the discretization error on a given variable




Conservative variables (momentum, energy...)




Conservative variables (momentum, energy...)Primitive variables (velocity, pressure...)




Conservative variables (momentum, energy...)Primitive variables (velocity, pressure...)Variables related to turbulent flows (vorticity, helicity...)



A residual based methodWe try to obtain a measure of the discretization error when solving agiven equation

MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (1)




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (1)

For each cell, we identify the regime to estimate the error:




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (1)

For each cell, we identify the regime to estimate the error:In advection dominated flows (steady-state)

F conv(Y ) = S (2)

We can solve for the exact problem in 1D and therefore to obtain anestimation of the discretization error




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (1)


F conv(Y ) = S (2)

We can solve for the exact problem in 1D and therefore to obtain anestimation of the discretization errorSame applies for the diffusion limit




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (1)


F conv(Y ) = S (2)

We can solve for the exact problem in 1D and therefore to obtain anestimation of the discretization errorSame applies for the diffusion limitand reaction limit


A residual based method

MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)



MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)

We can show that a measure of the discretization error introduced ineach term by difference between two consecutive levels.



MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)


Errorc = Cconv||FLconv − F

L−1conv||Lp

(4)



MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)



L−1conv||Lp

(4)

For diffusion dominated flows

Errord = Cdiff ||FLdiff − F

L−1

diff ||Lp(5)



MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)



L−1conv||Lp

(4)



L−1

diff ||Lp(5)

For reaction dominated flows

Errord = Cdiff ||SL − S

L−1||Lp(6)



MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (3)



L−1conv||Lp

(4)



L−1

diff ||Lp(5)

For reaction dominated flows

Errord = Cdiff ||SL − S

L−1||Lp(6)

For each cell the mechanism controlling the error can be differentdepending on the local Reynolds number, etc....This criteria also reduces the error propagation (we reduce the error ofthe RHS of the equation)




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (7)

Navier-Stokes (with velocity as primitive variable)




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (7)

Navier-Stokes (with velocity as primitive variable)Vorticity equation




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (7)

Navier-Stokes (with velocity as primitive variable)Vorticity equationHelicity equations




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (7)

Navier-Stokes (with velocity as primitive variable)Vorticity equationHelicity equationsKinetic energy equation




MY

n+1 − Yn

∆t= F conv(Y

n+1/2) + F diff (Yn+1/2) + S (7)

Navier-Stokes (with velocity as primitive variable)Vorticity equationHelicity equationsKinetic energy equation....



Error indicatorWhich quantity?




A Hessian based h-refinement algorithmWhich quantity?





A residual based methodWhich equation?





A residual based methodWhich equation?

The optimal choice is problem dependent


Error estimation for the transport equation: Propagation error only controls inregions where the error is smallSteady problem


Error estimation for the transport equation: Propagation error only controls inregions where the error is smallSteady problem

Transient T = Tsteady + T (t)eiωt


Three test cases related to our problem of interest:



Test cases

Dissipation of the Lamb-Oseen vortex



Test cases


Linear growth of random noise in a shear layer



Test cases



Isotropic turbulence test case


We choose three test cases related to our problem of interest:

Test cases





Lamb-Oseen vortex

0.05

0.10

0.15

2 4 6 8 10

uθ

r

lc

νt=0.0νt=0.5νt=1.0νt=2.0

uθ =Γ

2π r

[

1− exp

(

−r2

4 ν t

)]

.


Lamb-Oseen vortex

0.05

0.10

0.15

2 4 6 8 10

uθ

r

lc

νt=0.0νt=0.5νt=1.0νt=2.0

uθ =Γ

2π r

[

1− exp

(

−r2

4 ν t

)]

.

10-7

10-6

10-5

10-4

10-3

10-2

101 102 103 104

ε L1(u

)

lc/∆x

FIXAMR 1.0

101 102 103 104

(Ek) n

um

/(E

k) t

heo

lc/∆x

FIXAMR


Lamb-Oseen vortex

0.05

0.10

0.15

2 4 6 8 10

uθ

r

lc

νt=0.0νt=0.5νt=1.0νt=2.0

uθ =Γ

2π r

[

1− exp

(

−r2

4 ν t

)]

.

Local efficiency: Estimated error vs. Real error

10-7

10-6

10-5

10-4

10-3

10-2

10-7 10-6 10-5 10-4 10-3 10-2

ε tru

e

εestimated



Test cases





Noise growth in a shear layer

U = ∆Uerf (y/δc)

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-1.0 0.0 1.0

y/δ

c

U/∆U


Noise growth in a shear layer

U = ∆Uerf (y/δc)

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-1.0 0.0 1.0

y/δ

c

U/∆U

-28-26-24-22-20-18-16-14-12-10

0 10 20 30 40 50 60

Am

pli

tud

e

t(∆U/δc)

SimulationTheory

10-4

10-3

10-2

10-1

100

10-2 10-1 100 101

ε(αc

i)

(δc/∆x)

FIXAMR

0.1

1.0

10-2 10-1 100 101

(αc i

) nu

m/(

αci)

theo

(δc/∆x)

FIXAMR



Test cases







Lint = 0.5Lbox

ReL =

√

∫

∞

0E(k)dkLint

ν= 375

k =

∫

∞

0

E(k)dk = 0.5,

The characteristic lengthscale changes in time

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

k/k 0

tu’/Lint

323

643

1283

2563

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0 0.2 0.4 0.6 0.8 1

Tay

lor

scal

e

tu’/Lint

323

643

1283

2563


Isotropic turbulence : L1 norm, Hessian based error estimator

0.1

1

1 10 100

I k

∆x/η

FIXVel

Helicity

VortDisp



0.1

1

1 10 100

I k

∆x/η

FIXVel

Helicity

VortDisp

100

101

102

1 10 100

I M2

∆x/η

FIXVel

Helicity

VortDisp



0.1

1

1 10 100

I k

∆x/η

FIXVel

Helicity

VortDisp

100

101

102

1 10 100

I M2

∆x/η

FIXVel

Helicity

VortDisp

10-1

100

101

102

103

1 10 100

I M3

∆x/η

FIXVel

Helicity

VortDisp



0.1

1

1 10 100

I k

∆x/η

FIXVel

Helicity

VortDisp

100

101

102

1 10 100

I M2

∆x/η

FIXVel

Helicity

VortDisp

10-1

100

101

102

103

1 10 100

I M3

∆x/η

FIXVel

Helicity

VortDisp

101

102

103

104

105

1 10 100

I M4

∆x/η

FIXVel

Helicity

VortDisp


Isotropic turbulence : L∞ norm, Hessian based error estimator

0.1

1

1 10 100

I k

∆x/η

FIXHelicity

Disp

100

101

102

1 10 100

I M2

∆x/η

FIXHelicity

Disp

100

101

102

103

1 10 100

I M3

∆x/η

FIXHelicity

Disp

101

102

103

104

105

1 10 100

I M4

∆x/η

FIXHelicity

Disp


Isotropic turbulence : Residual based error estimator: L1 vs. L∞ norms

0.1

1

1 10 100

I k

∆x/η

FIXNS (L1)

NS (Linf)

100

101

102

1 10 100

I M2

∆x/η

FIXNS (L1)

NS (Linf)

100

101

102

103

1 10 100

I M3

∆x/η

FIXNS (L1)

NS (Linf)

101

102

103

104

105

1 10 100

I M4

∆x/η

FIXNS (L1)

NS (Linf)


Perspectives: To investigate the process of bubble breakup in forced isotropicturbulence


Experimental validation : Benjamin [2002, Experiments in Fluids]We look at the distribution of flow through a catalytic converter:



σV =1

m

∫

A

|Ui − Ue| δm

Re = 20000



σV =1

m

∫

A

|Ui − Ue| δm

Re = 20000

1

10

103 104 105 106

Non-u

nif

orm

ity i

ndex

Degree of freedom

ExperimentalSimulation, AMR

Simulation, FIX



σV =1

m

∫

A

|Ui − Ue| δm

Re = 60000

1

10

103 104 105 106

Non-u

nif

orm

ity i

ndex

Degree of freedom


Simulation, FIX



σV =1

m

∫

A

|Ui − Ue| δm

Re = 80000

1

10

103 104 105 106

Non-u

nif

orm

ity i

ndex

Degree of freedom


Simulation, FIX


Conclusions

AMR is shown to provide more accurate solutions for three different testcases


Conclusions


Simple test cases do not allow to see significant differents among thevarious methods


Conclusions



For the isotropic turbulence test case we do see large differences for highorder statistics


Conclusions




L∞ performs betters than L1 norm for low resolved simulations whenlooking at high order statistics... but it is not always easy to control!


Conclusions




L∞ performs betters than L1 norm for low resolved simulations whenlooking at high order statistics... but it is not always easy to control!

AMR allow us to reproduce the flow distribution inside a catalyticconverter at much lower resolutions


Numerical simulations of multiphase and turbulent...

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