Phenomenological and Numerical Studies of Superfluid Helium Dynamics in the Two-Fluid Model...

Phenomenological and Numerical Studies of Superfluid Helium

Dynamics in the Two-Fluid Model

Christine Darve

Mini-Workshop on Thermal Modeling and Thermal Experiments for Accelerator Magnets

September, 30th – October, 1st 2009

Northwestern University / Mechanical Engineering Department CD-30/09/09

Overview

a) Two-fluid model for Helium II

b) Motivations for numerical modelization

c) Existing 1-D, 2-D and 3-D numerical simulations

d) Governing equations using P, vn, vs and T variables

e) Computing stage

f) Conclusion

Reference papers:ICEC: Phenomenological approach: a 3-D model of superfluid helium suitable for numerical analysisby C. Darve, N. A. Patankar, S. W.Van Sciver

LT25: Numerical approach: A method for the three-dimensional numerical simulation of He II

by L. Bottura, C. Darve, N.A. Patankar, S.W. Van SciverCERN, Accelerator Technology Department, Geneva, SwitzerlandFermi National Accelerator Laboratory, Accelerator Division, Batavia, IL, USA

Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA National High Magnetic Laboratory, Florida State University Tallahassee, FL, USA


Two-fluid model for Helium IIThe Superfluid fraction:

Atoms that have undergone BE condensation

Finite density, but NO viscosity, carry NO entropy

irrotational behavior for an inviscid fluid

vortices can be generated

0 v s

334322

) (

q

Ts

Aq

TsdT

s

nGMn

High flux regime

dT/dx ~ q3

Low flux regime dT/dx ~ q

Laminar

Turbulent

Mut

ual-

frict

ion

regi

me

Order of magnitude (for q = 5,000 W/m2, T= 1.8 K)

sec

m035.0

v

TS

qn

sec

m017.0v s

36

m

m10L μm32~if then

Spacing between vortices, where L, is the total length of vortex line per unit volume

L

1~

2

wLo n

Line density in steady state

conditions

vL

vs

vn

Differently from classical hydrodynamics, the dissipation in He II vortex motion is NOT due to the viscosity term ν∇2v in the NS equation

v

pv)v( 2

v

t v


Motivations for numerical modelization

The knowledge of cooling characteristics of He II is indispensable to design superconducting magnets !

Few examples of applications (see introduction talks):

Thermal counter-flow / TatsumotoFundamental understanding of 2-fluid flow

Determination of the Critical Heat Flux / Yoshikawa, ShiraiSupraconductor cooling

Particle Image Velocimetry technique / Zhang, Fuzier, van SciverEffect of Normal and superfluid component

2nd Sound / NHMFL, Fuzier, van Sciver


Thermal counter-flow / Tatsumoto

“Numerical analysis for steady-state two-dimensional heat transfer from a flat plate at one side of a duct containing pressurized He II”By H. Tatsumoto, K. Fukuda b, M. Shiotsu

Continuity and momentum balances conservation > PEnergy balance conservation Lax algorithm >T


Determination of Critical Heat Flux / Yoshikawa, Shirai

“Experiments and 3-D numerical analyses for heat transfer from a flat plate in a duct with contractions filled with liquid He II”By Yoshikawa K., Shirai Y., Shiotsu M., Hama K.


Static and Forced-flow in He II

“Experimental measurement and modeling of transient heat transfer in forced flow of He II at high velocities” by S. Fuzier, S.W Van Sciver

Second sound + Modelization of forced convection + counter-flow + pressure effect Forced flow up to 22 m/s Use of high non-linear effective thermal conductivity : keff and Fanning friction factor

343

Ts

Af(T)

s

nGM

Validation of model in range : Low flow ( ) +High flow ( )T P

3

1

1

31

1

}1

{

dx

dT

dx

dP

sTf

dx

dTTfq

s


Probing the microscopic scale of He II

Particle Image Velocimetry technique:

- permit 2-D and 3-D flow visualization- capable to follow the normal velocity, vn

NHMFL measured internal convection phenomenaChallenges: Particle choices (density, size)

Need numerical simulation to better understand what happen

Ts

qn

vLarge discrepancies were

observed even with slip velocity correction

Interaction with superfluid component?

Figure 17 Optical cryostat with PIV setup

Beam dump

Optical portLaser sheet

CCD camera

Synchronizer

Beam expandingoptics

Dual laser head

Computer

NHMFL, T. Zhang & S. van Sciver, 2001 - 2004


Rao et al. : Forced convection – Steady state and transient (vertical micron-wide GM duct heated at the bottom)

• Method:Finite difference algorithm; 4th order Runge-Kutta, explicit in time

• Variables: Pressure, temperature, normal velocity at BC

• Assumptions: Two-fluid model and the simplified model [Kashani]

• Result: Good agreement of both methods with experimental results by Ramada

Existing 1-D Numerical Simulations

3/1

)(

1

x

T

TKxx

TCpu

t

TCpq

Bottura et al. : THEA - Simulation of quench propagation

• Method: Finite element algorithm, Taylor-Galerkin, explicit in time

• Variables: Pressure, temperature, velocity

• Assumptions: Use a single-fluid model; add couterflow heat exchange in the energy conservation balance to benchmark

• Result: Good agreement with experimental results by Srinivasan and Hofmann, Kashani et al., Lottin and van Sciver


Ramadan and Witt: Compared single-fluid and two-fluid models (natural conv. in large He II baths)

• Variables: Pressure, temperature, velocity

• Assumptions: Ignore the thermomechanical effect term and the Gorter-Mellink mutual friction term in the momentum equations for both components

• Result: Illustrate the weakness of the single-fluid model

Tatsumoto: SUPER-2D–Steady state and transient (rectangular duct with varying ratio of heated surface)

• Method: Finite difference, First order upwind scheme, explicit in time

• Variables: Pressure, temperature, heat flux

• Assumptions: Two-fluid model and the energy dissipation based on the mutual friction between the superfluid and normal-fluid components

• Result: Predict the steady state critical heat flux to a precision of about 9 %



Doi, Shirai, Shiotsu, Yoshikawa – Kyoto : SUPER-3-D, Steady state

[1] “3-D numerical analyses for heat transfer from a flat plate in a duct with contractions filled with pressurized He II”.[2] “Experiments and 3-D numerical analyses for HT from a flat plate in a duct with contractions filled with liquid He II”.(duct w/ 1 and 2 contractions calculation of Critical Heat Flux)

• Method: Finite difference, First order upwind scheme, explicit in time

Energy balance -> s ->T

Adams-Bashforth method -> vs -> v ->vn

Variables: Pressure, temperature, heat flux , dt=0.5 sec [1] and dt=2 sec [2]

• Assumptions: Two-fluid model and the energy dissipation based on the mutual friction between the superfluid and normal-fluid components

• Result: Predict (wrt experiment) the steady state CHF to a precision of about 14 %

• Large memory and time (Parallelized computation using Message passing Interface - MPI)

We proposed a different set of equation to ease the calculation of 1-D, 2-D and 3-D structures.



1. Formulate new and complete Helium II approximations based on the two-fluid model and the theory of GM mutual friction using p, T, vn and vs as variables

Mass, momentum and energy balances conservations permit to derive a partial differential equation (PDE) system of the form:

2. Construct a numerical 1-D then 3-D solver for Helium II based on existing PDE solver

Calculate shape functions, associated local and global derivatives, jacobian matrix and determinant of 3-D FE (cubic, tetrahedral and wedge)

Implement the new formulations in 1-D PDE solver for space and time discretization

Add and modify library for 3-D matrix and vector operations

Implement a protocol to identify nodes where algebraic and boundary conditions can be imposed

Proposal - PDE (p,vn,vs,T)

m ut a u g u s uq


Governing equations for the two-fluid model

Momentum density of He II :

vnvn svs

Density of He II:

n s

wvn vsRelative velocity:

Thermodynamic potential, :

2vn 1

3 vn

Stress tensor :

i p

sT

n

2

w

2

only depends on the normal fluid

Using thermodynamic properties: p

ih dTCd

TCp

di vv

d 1 c 2 dp

c 2 dh

where is the Gruneisen parameter, Cv is the specific heat at constant density, c is the speed of (first) sound and h is the specific enthalpy.

nd

wTdsd

pdi

2

2

2

d1

dp sdT

n

2

dw

2

n n p,T

ii p,T Tpss ,

k k p,T State variables : p, T

p,T


Governing equations – Eq. I and Eq. II

t v 0

n

t nvn m

s

t svs m

where m is the rate at which normal fluid is created from superfluid

Equation I- Continuity equation & mass balance conservation for normal fluid and superfluid:

where g is the acceleration of the gravity field

gvvvvvv

pt sssnnn

ssnn

Momentum equation for superfluid [Donnelly]

s

vs

t svsvs Ft sg

Equation II and Equation III- Momentum equations

Force associated with turbulence that appears only when the relative velocity between the superfluid and normal fluid components is larger than a critical value

Force of mutual friction is given by GM for counterflow situationwhere AGM is a function of T and, possibly, of w

wF 2wA nsGMt

Ft Lof n

2

s2 nsnw

2w


s

vs

t svsvs

s

p ssT

sn

2w2 Ft sg

n

vn

t nvnvn

n

p ssT

sn

2w2 Ft ng mw

Momentum balance conservation – Eq. II & Eq. III

Momentum equation for the normal fluid becomes:

Momentum equation for the superfluid becomes:mass exchange

force due to pressure gradient

acceleration terms

thermo-mechanical effect


Energy balance conservation – Eq. IV

it vi pvTssw w

2 sn

2w

m

2

Ftw kT vn q

internal heat convection through entropy transport

represents the internal energy dissipation associated with turbulence, see later…

originates from the transformation of superfluid into normal fluid and vice versa

Equation IV - Internal energy conservation :

qTkwsTpiit n

nsss

ssn

nnssnn

gvvwwvv

vv

vv

vv

2

2222

22222

st svn 0

Irreversible motion the entropy is conserved

222222

222222w

mwTsp

t ntns

ssss

nnnssnn

gvvwFwwvv

vv

vvv

We can express the kinetic energy as

(A)-(B)

i t

iv pv Tssw w2sn

2w Ftw kT vn q

(B)

(A)


- Energy dissipated by viscous dissipation is small compared to other sources of heat transport (e.g. mutual friction) -> treated as a source perturbation

Substitutions and assumptions

- Friction force is given by GM for counterflow situation: wF 2wA nsGMt

- The divergence of the total velocity is computed through the chain relation:

Note that the normal and superfluid velocities appear explicitly

vn

vn

s

vs

n

vn

s

vs vn

n

vs

s

- Contributions related explicitly to the mass exchange m are small when compared to other terms -> drop them from the balances

kT kT

where AGM is a function of T and, possibly, of w

- Terms containing differentials of quantities other than variables (p, vn, vs, T) perturbations with respect to the leading terms of the equations

- Variations of the Gruneisen parameter are small


PDE (p,vn,vs,T) form – Continuity & Energy (Eq. I & IV)

2

2

2

222

222

222

nss

ss

nnn

snsnnssssns

ssnsnns

nnssnn

wsTccq

wAwATksTw

csTwcpt

p

wwvvv

wvwvvv

vvvvvv

2

22

2

22

22

nss

ssv

nnv

snsnnsssnn

vss

sns

svsnsnns

nvnv

wsTTCTCq

wAwATkTCsT

wTCsTwTCt

TC

wwvvv

wvwvvv

v

vvvvv

thermal capacity

effect of exp/comp

energy dissipation due to GMconduction

external heat viscous dissipation

convection

effect of energy exchange n<-> s

entropy change

1st sound- decompression of

mass exchange n<->s

generated by viscosity

convection of the mass

associated to S exchange

due to mutual friction conduction


PDE (p,vn,vs,T) form – Momentum (Eq. II & III)

n

vn

tn

p nvnvn

sn

wvn

sn

wvs ssT Asnw

2vn Asnw2vs ng

s

vs

ts

p

sn

wvn svsvs

sn

wvs ssT Asnw

2vn Asnw2vs sg

viscous effect

gravity effect

mutual friction

thermomechanical effect

mass exchange momentum

normal<-> superfluid

transport of momentum

force due to variation of pressure

acc. mass


THEA: commercial code by CryoSoft, 1-D Thermal, Hydraulic and Electric Analysis of superconducting cables

use parts of THEA capable of solving generic partial differential equations in a 1-D system of the form

Numerical Formulations and THEA

qusuguau

m t

mass matrix advection matrixdiffusion matrix source matrix

forcing vector

dNaNA JIJ

J

IIJ

dNsNS JIJ

J

IIJ

QUSGAt

UM )(

dNmNM JIJ

J

IIJ

dNgNG JIJ

J

IIJ

Write the PDE system as a weighted residual at the nodes with identical weight and shape functions to obtain the system of ODE with discretized matrices

dqNQ J

J

JJ

where


Computing phase

Fortran 77

Submit jobs to Fermilab Farm using CONDOR

Submit jobs to the GRID

Advantages:

User defined environment

Submit jobs in parallel

Inconvenient:

Difficult process to investigate the code instabilities !

Need to use Linux (dual boot machine) or Cygwin

Interactive graphical data analysis programs: PAW

Visualization means are limited: Tecplot

Not User friendly at all !


We first consider a scalar problem with one degree of freedom (Temperature) in a 3-D space (see topologies)

This problem is a typical parabolic equation in time, in the 3-D space and can be tested against analytic results of a 1-D problem

Verification of the 3-D code on a scalar problem

QTkt

TCp

TU

Cp m 00

0

z

y

x

a

a

a kg

0s

qq

QUSGAt

UM )(

ODE to solve

0

20

40

60

80

100

120

140

0 1 2 3 4 5axis

Tem

pera

ture

(K

)

Analytic result

Numerical - unstructured mesh

Numerical - structured mesh

Scalar problem - conduction in steady state (Model A and B, Case II)

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

5.2

0 0.1 0.2 0.3 0.4 0.5Nodes

Tem

pera

ture

(K

)

5 Cube

10 Wedge

30 Tetra

Analytic

Influence of element for a steady state problem (Model A’, Case II)


Some results using THEA - PDESolver

“PIV Measurements of He II Counterflow Around a Cylinder”By S. Fuzier, S. W. Van Sciver, and T. Zhang

FIGURE 1. Turbulent structures for D = 6.35 mm. T = 2.03 K and q = 7.2 kW/m2 , n = 23 mm/s and

Preliminary results:

Preliminary results:


Computing Using COMSOL - example

2-D Simulation using asymmetric conditions - Application modes

•Weakly compressible NS vn, P

•Convection and conduction T

•Weakly compressible NS vs, P

Advantages:

Very user friendly !

Possibility to modify governing equation

Add coupling between variables

Use integrated numerical stabilization for normal fluid

Schemes are helpful to stabilize the solution without changing the solution too much Artificial diffusion (overdamping)

Inconvenient of Physical parameterization:

Adapt the governing equation for the superfluid behavior

Viscosity = 0 instability ; work with artificial diffusion


Using COMSOL

Based on Tatsumoto’s example


Using COMSOL - Based on NHMFL’s example

Normal component velocity


New materials: o A new set of equation to validate o COMSOL or others to validate this PDE

Remains opened questions in Superfluid Helium behavior Physics:• PIV: factor 2 between theoretical and numerical model• S. Fuzier’s model to understand: issue with medium range ~ 8 m/s

coefficientQuestion: how to better understand the phenomenology of 2-fluid flowOne answer: by simulation approach

Challenges: o Add coupling, which can introduces inherent physical

stabilisation.. o Non-linearity of superfluid component

Keys to successful numerical simulations:o limit the computing time and complexity: CPU usedo Use a user friendly visualization tool

Conclusion


Extra – slide : Using COMSOL


Extra – slide : Helium II – PDE simplification

v

s

n

C

000

000

000

0001

m

vssnnns

svsns

svn

sns

ssnss

snsns

nnn

nsss

nssn

ssnn

CwsTTCwsTTC

s

s

wsTcwsTc

vv

wvw

wwv

vv

a

220

022

22

2222

k

k

000

0000

0000

000

g

00

00

00

00

22

22

22

22

ww

ww

s

wAwA

wAwA

wAwA

wAwA

nsns

nsns

nsns

nsns

q

q vn c2vn

n

c 2vs

s

Twss w

2wsn

2 ng

sg

q v CvTvnn

CvTvs

s

Twss w

2wsn

2

m ut a u g u s uq

T

p

s

n

v

vu

PDE system to solve

Vector of unknowns:

and preliminary simplificationTo implement in the code

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Phenomenological and Numerical Studies of Superfluid Helium Dynamics in the Two-Fluid Model...

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