DNS of the Turbulence Decay in Liquid Metal Flow Under an...

Introduction Case & Background Formulation & Methodology Results Concluding remarks

DNS of the Turbulence Decay in LiquidMetal Flow Under an Increasing Magnetic

Field

X.Albets-Chico , D. Grigoriadis, S. Kassinos

Department of Mechanical and Manufacturing EngineeringUniversity of Cyprus

International conference on MHD turbulenceDecember 21st-23rd, 2009. IIT Kanpur, India.


Outline

1 Introduction

2 Case & Background

3 Formulation & MethodologyFormulationNumerics

4 ResultsVerification & ValidationFinal Results

5 Concluding remarks


Motivation: Spatially varying magnetic fields

The nuclear fusion reaction takes placein the plasma confined in the tokamak




The energy (and other particles) isabsorbed by the blanket surroundingthe toroidal tokamak





The blanket concept is based onliquid-metal flow






The plasma is confined using a helicalmagnetic field






The plasma is confined using a helicalmagnetic field

The section of the pipes is expected toby circular


Circular Pipe Turbulent Flow

Turbulence

Generallythree-dimensional



Turbulence


Generated bywall instabilities



Turbulence


Generated bywall instabilities

Mixing of heat ,momentum andmass is enhanced.


Circular Pipe Magnetohydrodynamic flow

Effect of the magnetic field on electrically conducting fluid

Induced currentsj → −∇φ + u × B





Lorentz Force →j × B ∼ u × B × B






The current loopsdefine the localeffect of theLorentz force






The current loopsdefine the localeffect of theLorentz force

Turbulence issystematically suppressedby magnetic fields(Ha/Re > 0.025)


Objective: Magnetohydrodynamics & Turbulence ?

The action of strong magnetic fields suppressesturbulence




Fringing magnetic fields are related to three-dimensionaleffects





Three-dimensional effects increase the complexity andinstabilities of the flow





Three-dimensional effects increase the complexity andinstabilities of the flow

Coexistence of the two phenomena is here investigated


Case

y

z

x

R

Magnet

14R

20R

B

C

F

D

Magnetic field = B yfit(x)

Flow

Magnet

A

E

30

34R

R

ExperimentalConfiguration.ALEX facility,Argonne NationalLab., Illinois, USA.


Case

y

z

x

R

Magnet

14R

20R

B

C

F

D


Flow

Magnet

A

E

30

34R

R

Case reportedExperimentally[Picologlou(1986),Reed(1987)] andNumerically(Asymptoticmethods [forinstance,Reed(1987),Molokov(2003),Bülher(2006)], andrecently by DNS[Ni (2007),Albets(2009)])


Case

y

z

x

R

Magnet

14R

20R

B

C

F

D


Flow

Magnet

A

E

30

34R

R

Case addressedby current work.Analyzed for thevery first time.Role of inertiadisqualifies theuse of asymptoticmethods. Thereare also importantexperimentalaccuracyproblems. Is DNSan option?


Background (i)Experimental work, decreasing magnetic field. Upstreamconditions Ha = 6640, N = 10700 (ALEX Results: Picologlouet al.. 1986, Reed et al.. 1987 )


Background (i)Experimental work, decreasing magnetic field. Upstreamconditions Ha = 6640, N = 10700 (ALEX Results: Picologlouet al.. 1986, Reed et al.. 1987 )Attention to the accuracy in Experiments!!



Approximately 18R are needed to decrease from Ha = 6640 toHD conditions?



Approximately 18R are needed to decrease from Ha = 6640 toHD conditions?The ERROR signal output of the local magnetic field becomesvery important (negative value!)... The last ACCEPTABLE valueis at Ha = 60! (17R dowstream of the high mag. field area)



Approximately 18R are needed to decrease from Ha = 6640 toHD conditions?The ERROR signal output of the local magnetic field becomesvery important (negative value!)... The last ACCEPTABLE valueis at Ha = 60! (17R dowstream of the high mag. field area)For a Increasing Magnetic field: Experimental analysis presentenormous difficulties to provide accurate information about thetransition from (turbulent) HD flow to strong MHD flow(Ha/Re > 0.025, Ha ≈ 100 turbulence decayment)


Background (ii). Size of the problem?

How to characterize this area? What is the most reliabledescription of the evolution of the magnetic field from HDconditions to Ha ≈ 50 and beyond?




tanh− based function can be applied By = Bd1+tanh[0.45(x/R)]

2(IEA Liquid Breeder Blanket SubTask, 2005) (mean error of 1%)






Consequences






Consequences

Required distance to evolve from HD conditions(Ha = 0.005, Ha ≈ 0) to strong MHD conditions(Ha = 6640) is 23R






Consequences

Required distance to evolve from HD conditions(Ha = 0.005, Ha ≈ 0) to strong MHD conditions(Ha = 6640) is 23RRequired distance to reach accurate fully-developedturbulent HD flow at experimental conditions (Reτ ≈ 520) isbetween 5 − 10D.






Consequences

Required distance to evolve from HD conditions(Ha = 0.005, Ha ≈ 0) to strong MHD conditions(Ha = 6640) is 23RRequired distance to reach accurate fully-developedturbulent HD flow at experimental conditions (Reτ ≈ 520) isbetween 5 − 10D.

Computational domain of at least 50R for conducting cases,even more for insulating cases! (development length!,three-dimensional effects of R ×O(Ha1/2) R. Holroyd and J.S. Walker, Journal of Fluid Mechanics 84, 79 (1978)


Governing EquationsConservation of mass, momentum and charge:

∇ · u = 0 (1)∂u∂t

+ u · ∇u = −∇p +1

Re∇2u + N(j × B) (2)

∇ · j = 0 (3)

Ohm’s Law:

j = σ(−∇φ + u × B) (4)

∇2φ = ∇ · (u × B) (5)

ν viscosityσ conductivityρ densityu velocityj current densityp pressuret timeφ electric potentialB magnetic field

Re = UbR/νN = RσB2/ρUb

Ha = RB√

σ/ρν


Boundary Conditions

Wall:~uwall = 0, (No slip)∂p∂n |wall = 0,∂φ∂n |wall = −[∇τ · (c∇τφw )], J. S. Walker, Journal de Mécanique 20, 79 (1981).

Inlet:~uinlet = definedpinlet → inlet penalty b.c.,∂φ∂n |inlet = 0

Outlet:∂~uoutlet

∂t + Uconv∂~uoutlet

∂n = 0, (Convective)poutlet → outlet penalty b.c.,∂φ∂n |outlet = 0


Numerical Method, Grid Generation

Unstructured Nodal-basedFinite Volume Code

Collocated scheme

Fractional-step method(Chorin, 1968)

Time discretization:Crank-Nicholson /Semi-implicit /Explicitintegration of the LorentzForce

Conservative computation ofthe Lorentz force (Ni, 2007)

Direct Numerical Simulation(no model used)

z/R

y/R

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

θ

r

Hybrid quads/triangles b.layer/core

Very strong near-wall stretching toresolve Ha layers

Core designed to resolveturbulence scales



z/R

y/R

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

θ

r

Very strong near-wallstretching to resolve Halayers (up to Ha ≈ 10000)→ ∆r+

min = 0.013→ ∆rmin/R = 0.00005

Core → turbulence scalesat Reτ ≈ 520 →∆x+ = 10.4,∆θ+

max=7.2,∆r+max ∼7



SUMMARY

Explicit Lorentz Force...TIME-STEPING REQUIREMENTS!∆t ≤ 2/σB2→ Ha = 7000 → 3 · 105 time-steps are neededto compute one turn-over time

(development length = 50R)ESTIMATED MESH REQUIREMENTS! → 17 M6

elements

Something needs to be done... impossible to address...


Numerical Strategy

D

A

B

C’Ha=500

Ha=0, N=0, Reb≈4000

Magnet

Magnet

A

20 R

Ha=500, N=62.7

Ha=50, N=0.62

B

C’’

Fully-developed hydrodynamicturbulent flow

C’

C’’

D

C’

Transition to low mag. field

Ha=50, N=0.62

Ha=50Ha=0.005 Ha=3

Ha=500, N=62.7

Ha=3

Flow

B

≈50 R

20 R

7 R

20 R


Numerical Strategy

Magnets’ magnetic field generates Ha = 50 conditionsBy = 501+tanh[0.45(x/R)]

2Magnets’ magnetic field generates Ha = 500 conditionsBy = 5001+tanh[0.45(x/R)]

2

−10 −5 0 5 100

100

200

300

400

500

600

X/R

Ha

−10 −5 0 50

20

40

60

80

100

X/R

Ha

−10 −5 0 50

20

40

60

80

100

X/R

Ha

it is possible to find a shift to match both values withoutassuming too much... why?


Numerical Strategy∂(tanh(x))

∂x = 1 − tanh2(x)...so...∂(tanh(x))

∂x → 0 faster than tanh(x)for x → −∞ we chooseappropriate x/R which, in combination with the shift (while matchingboth magnetic field values), produces a difference in the slope that issmall ...

−3.5 −3 −2.51

1.5

2

2.5

3

3.5

4

4.5

5

X/R

Ha

6%relativedifferencein slope

−10 −5 0 5 10 150

100

200

300

400

500

600

X/R

Ha

The investigateddomain is extended withoutincreasingcomputationalrequirements!

This feature allows to reuse computational information and extendfurther downstream our domain, very useful to assess inertia effects!!!


Numerical Strategy

20R

C

D

A

R

C

B

D

AB

BC

y

z

x

Flow

Constant magnetic field,fully-develped MHD flow?

Hydrodynamic flow Reb≈4000

Rapid transition to MHD flow, decayment of turbulence


Numerical Strategy

A

B

AB

20R

y

z

x

Flow

Time-history...t, u,v,w

Periodic Boundary-conditions

Flow

Hydrodynamic flow Reb≈4000


Numerical Strategy

D

R

B

20Ry

z

x

Flow: inlet defined in time and position


Numerical Strategy

7R

C’

D

R

C’’

B

D

BC’

C

C’C’’

20Ry

z

x

Flow: inlet defined in time and position


Rapid transition to MHD flow (I) (low N, low Ha)

Rapid transition to MHD flow (II)


Numerical Strategy

20R

C’’

D

R

D

C’

C’’

C’

C’’

y

z

x

Flow: inlet defined (u,v,w)in time and position


Rapid transition to MHD flow (II)


Results


Verification. Connecting magnetic fields

The information is accurately transferred to the newdomain, identical magnetic fields

x/R

P/R

σUbB

d2 ,u x/U

b

By/B

d

-15 -10 -5 0 5

0.2

0.4

0.6

0.8

1

1.2

0.2

0.4

0.6

0.8

1

Quasi-fully developed MHD flow(turbulent interaction)

Fully developed turbulent HD flow

Computation 1Computation 2

ux at z/R=0

P

ux at z/R=0.75

x/R-15 -10 -5 0 5 10

Computation 1Computation 2

N=0.62Ha=50

N ≈0Ha≈0

N=2.5e-3Ha=3.15

Increasing Magnetic field

Evaluation of the injection into higher magnetic field underprocess...


Validation &Verification . Hydrodynamic flow.Accurate resolution of the Fully-developed turbulent flow atReτ ≈ 520 →Resolution of the turbulence scales, near-wall viscoussublayer

100

1020

5

10

15

20

y+

u+ x

Present periodicPresentDurst et al.Wagner et al.Log law

100

1020

0.5

1

1.5

2

2.5

3

y+

(ux,u

r,u

θ)+ r

ms,

p+ rm

s

ux

uθ

ur

p

Present periodicPresentDurst et al.Wagner et al.

The injection profile is also assessed...while evaluatinginlet/outlet effects...perfect agreement regardingmomentum!!


Validation. MagnetoHydrodynamic flow.Accurate resolution of the fully-developed MHD flow up toHa = 7000Resolution of the Ha layers and side layersHa = 10, Ha = 50, Ha = 2000, Ha = 7000

−1 −0.5 0 0.5 10

0.02

0.04

0.06

0.08

0.1

y/R, z/R

ux

ν∂

p/∂

xR

2

Analytic (Gold, 1962)Computed

−1 −0.5 0 0.5 10

0.005

0.01

0.015

0.02

0.025

y/R, z/R

ux

ν∂

p/∂

xR

2

Analytic (Gold, 1962)Computed

−1 −0.5 0 0.5 10

1

2

3

4

5

x 10−4

y/R, z/R

ux

ν∂

p/∂

xR

2

Shercliff, 1962Computed

−1 −0.5 0 0.5 10

0.5

1

1.5x 10

−4

y/R, z/R

ux

ν∂

p/∂

xR

2

Shercliff, 1962Computed


Transition to low MHD conditions ( Ha = 50) (1/3).


Transition to low MHD conditions ( Ha = 50) (2/3).

ux at z/R=0.75=y/R=0.75

ux at z/R=0

y/R=0.75, z/R=0

z/R=0.75

x/R

u x/U

b

By/

Bd

-30 -20 -10 0 10

0.6

0.8

1

1.2

1.4

0.2

0.4

0.6

0.8

1

Conducting c=0.027

Fully-developed turbulent HD flow

Insulating


Conducting c=0.027


Insulating

x/R

P/R

σUbB

d2

By/B

d

-30 -20 -10 0 10

0

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1



(ux’ ux

’)1/2 at z/R=0.95

(ux’ ux

’)1/2 at z/R=0

x/R

(ux’ u x’ )1/

2 /uτ

By/B

d

-30 -20 -10 0 100

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

Conducting c=0.027


Insulating

The mean velocity evolves to a "flatter"profile, Ha and Roberts layers effects

The MHD pressure drop is higher than theturbulent one. Conductivity effectsincrease the pressured-drop requirementsdue to the net braking Lorentz force.

The turbulent fluctuating levels decreasenon-homogeneously! Overall agreementwith experimental observation for theturbulence decay (Ha/Re > 0.025), as theturbulence indicators are not completelydampened out.


Transition to low MHD conditions ( Ha = 50) (3/3)

y/R-R, z/R-R

u+ xrm

s/u

τ0

10-3 10-2 10-10

0.5

1

1.5

2

2.5

3

z=0 c=0y=0 c=0Fully-Developed HDz=0 c=0.027y=0 c=0.027

x/R=-8 N ≈ 0

x/R=8 N = 0.62

(y-R)/R, (z-R)/R

u+ xrm

s/u

τ0

10-5 10-4 10-3 10-2 10-1 1000

0.1

0.2

0.3

0.4

0.5

z=0 c=0y=0 c=0z=0 c=0.027y=0 c=0.027

x/R=8 N = 0.62

Roberts layer effect

Ha layer effect

f,Hz

Su

100 101

10-1

100

101

102

103

104

105

HDMHD (Ha=0.55, N≈0))MHD (Ha=25, N=0.31)MHD (Ha=50, N=0.62)

Power Spectrum (4 turn-over time) z/R=0.95

2.95 ≈ 3

1.68 ≈ 5/3

The reduction of the fluctuating statistics isobserved for Ha > 25, N > 0.3...rapidslope of the magnetic field?

The reduction of the fluctuating statisticstakes places in a in-homogenous way,being the conducting Ha layer moreeffective to organize the flow and reducethe turbulence levels.

The inertial range of the power spectra isreduced by the magnetic field(−5/3 → −3) (agreement with spectralanalysis...)


Transition to moderate MHD conditions (1/4)Wall-conductivity effects ( c = 0.027, c = 0)



c = 0.0

c = 0.027


Transition to moderate MHD conditions (3/4)

x/R

u x/U

b

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

Fully-Developed Turbulent flow

z/R=0.75

z/R=0.75 , y/R=0.75

y/R=0.75

center-line

x/R

P/R

σUbB

d2

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

-0.06

-0.04

-0.02

0

0

0.2

0.4

0.6

0.8

1


z/R=0.75

y/R=0.75

center-line

x/R

(ux’u

x’)

1/2/u

τ0

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1


z/R=0.75z/R=0.75 , y/R=0.75

y/R=0.75

center-line

(Ha = 500, c = 0.0)

Overspeed-areas are generated due to thebraking Lorentz force. A fully-developedprofile is not completely achieved (inertialterms...)

Important three-dimensional effects takeplace. The MHD pressure drop is oneorder of magnitude larger than the HD one.

The (turbulent) fluctuations are completelydampened out for values Ha >≈ 300N >≈ 25...However, important fluctuatinglevels take place... LAMINAR magneticflow instabilities!!!


Transition to moderate MHD conditions (4/4)

x/R

u x/U

b

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

0.8

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1z/R=0.75

Fully-developed MHD flow

z/R=0.75, y/R=0.75

y/R=0.75

Fully-developed Turbulent flow

center-line

x/R

P/R

σUbB

d2

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0

0.2

0.4

0.6

0.8

1

z/R=0.75


center-line, z/R=0.75, y/R=0.75 center-line, y/R=0.75


x/R

(u x’

u x’)

1/2

/uτ0

By/

Bd

-30 -25 -20 -15 -10 -5 0 5 10

0.5

1

1.5

0

0.2

0.4

0.6

0.8

1

z/R=0.75


z/R=0.75, y/R=0.75

y/R=0.75


center-line

(Ha = 500, c = 0.027)

A rapid transition to the fully-developed flatprofile is achieved by the moderatemagnetic field. Important jets take place,but they are smaller respect the insulatingones (wall-conductivity effect)!

The HD pressure drop is negligible respectto the MHD one (2 orders!×HD) due to thestrong braking Lorentz force in thedownstream area. Inertial effects aremasked.

The (turbulent) fluctuations are completelydampened out for values Ha >≈ 300N >≈ 25, the fluctuating effect ofinsulating configurations does not takeplace... (wall-conductivity effect)...


Concluding Remarks

Present results agree with experiments (Ha/Re > 0.025)when predicting the decayment of the turbulence producedby the magnetic field.


Concluding Remarks


Turbulence is suppressed in a non-homegenous way,Conductivity and Ha layers are the major actors...


Concluding Remarks


Turbulence is suppressed in a non-homegenous way,Conductivity and Ha layers are the major actors...Important over-speed areas are generated by themoderate and strong increasing magnetic fields.Conductivity reduces the intensity of the jets and maskinertial terms role. Three-dimensional effects are quicklydissipated


Concluding Remarks


Turbulence is suppressed in a non-homegenous way,Conductivity and Ha layers are the major actors...Important over-speed areas are generated by themoderate and strong increasing magnetic fields.Conductivity reduces the intensity of the jets and maskinertial terms role. Three-dimensional effects are quicklydissipated

Pulsating phenomena are generated close to theedge-pole face of the magnet for the insulating case.Turbulence is completely suppressed forN >≈ 25, Ha >≈ 300. Additional moderate mag. fieldsshould be analyzed...

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