fro30131 0 advanced modeling in porous media: use of dns

FRO30131 0

Advanced Modeling in Porous Media: Use of DNS

F. Duval , F. Fichot 1 and M. Quintard

Institut de Protection et de Sfiret6 Nucl6aire (IPSN)D6partement de Recherches en Skurit - CEA Cadarache

B.P. - 13108 St Paul lez Durance, nance

Institut de Wcanique des Fluides de Toulouse JMFT)

All6e du Professeur Camille Soula, 31400 Toulouse, France

Two-phase flows in porous media appear in a large number of engineering applications including heat-

exchangers, drying process and nuclear reactors. More particularly, in the framework of its research pro-

gramme on severe nuclear reactor accidents, IPSN investigates the water flooding of an overheated porous

bed, where such complex flows are likely to exist. The goal is to describe the flow with a general model,

covering rods and debris beds regions in the vessel.

Although the Direct Numerical Simulation (DNS) of two-phase flows is possible with several methods,

applications axe now limited to small computational domains, typically of the order of a few centimeters.

Due to the large size of the reactor vessel, a calculation of the flow from first principles (DNS with exact

conservation equations for each phase) would require an infinite computation time. Numerical solutions at

the reactor scale can only be obtained by using averaged models. Volume averaging is the most traditio-

nal way of deriving such models. Averaging volumes include all phases (solid, liquid, vapor) and several

interfaces. For nuclear safety, a control volume must include a few rods or a few debris particles, with acharacteristic dimension of a few centimeters. The difficulty usually met with averaged models is the closure

of several transport or source terms which appear in the averaged conservation equations (for example the

interfacial drag or the heat transfers between phases). Such terms result from averages over the interfaceslocated inside the control volume and are often written by introducing effective transport properties. In

the past, the closure of these terms was obtained, when possible, from one-dimensional experiments that

allowed measurements of heat flux or pressure drops. For more complex flows e.g. in a debris bed), the

experimental measurement of local velocities, pressure or temperatures is often impossible and the effective

properties cannot be determined easily. An alternative way is to perform "numerical experiments" with

numerical simulations of the local flow. As mentionned. above, the domain of application of DNS corres-

ponds to computation sizes of the same order as the control volumes necessary to derive averaged models.

Therefore DNS appears as a powerful tool to investigate the local features of a two-phase flow in complex

geometries. Moreover, computational results can easily be averaged to obtain effective properties.

There are several efficient methods dedicated to direct numerical simulation of two-phase flows. In thisstudy, Level Set and Diffuse interface methods are analyzed. Both methods represent interfaces on afixed grid by transition zones across which fluid properties vary sharply but continuously. However, thistransition layer does not have the same meaning in both methods. Whithin the framework of the Second

Gradient theory, Diffuse interface models can be derived for both single-component and binary fluids.The latter model is often referred to as the Cahn-Hilliard or Phase-Field model. From a numerical point

of view, the pore-scale characteristic length is of a few orders of interface thickness, this indicates thatinterface can easily be captured by the computational grid. In this sense, Diffuse interface methods are

particularly well suited for DNS at the pore level. Direct numerical simulation of immiscible two-phase

flows in a representative region of a porous medium involves complex geometries and requires unstructuredgrids. A finite element numerical scheme is introduced for solving this model. wo-dimensional results

including contact line motion, stationary bubbles and two-phase flow at the pore-scale are presented.

Future applications and consequences for averaged models are discussed.

Progress and Future Trends in Core Degradation Modeling, Cadarache France), November 21-23, 2001

advanced Modeling in Porous Media Use of DNS

Advanced Modeling in Porous Media

Use of DNS

Fabien Duval', Florian Fichot' and Michel Quintard'

Institut de Protection et de Sfiret6 Nucl6aire (IPSN), D6partement de Recherches en SfturM (DRS)

CEA Cadarache, 13108 St Paul Lez Durance, France.

2 Institut de Wcanique des Fluides de Toulouse (IMFT)

All6e du Professeur Camille Soula, 31400 Toulouse, France.

ww ri vbuyltNIW� XV(.101a

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar November 21-23, 2001 Cadarache - France


Outline of the presentation

• Context of the study

Macroscopic modeling of two-phase heat transfer in porous media

Scaling-up theory The volume averaging method

Use of Direct Numerical Simulation (DNS) at the pore-scale

• Direct Numerical Simulation

Level Set and Second Gradient methods

The Second Gradient theory single component and binary fluid case

• Cahn-Hilliard model incompressible mixture of two second gradient fluids

Finite element numerical scheme

Computational results

• Conclusion and perspective

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar November 21-23, 2001 Cadarache - France 2


Context of the study

Macroscopic modeling of two-phase heat transfer in porous media

• Motivation Debris bed cooling, drying process, heat-exchangers ...

• Methodology Heuristic (intuitive) or Theoretical (scaling-up) approach

• Heuristic approach

- Starts directly at the macroscopic scale

- Leads to intuitive average models debris bed cooling problem

- Effective properties are very difficult to determine -----------------

(ex heat exchange coefficients and thermal dispersion tensors)

• Scaling-up theory (Homogenization or Volume Averaging) macroscopic scale

- Starts at the microscopic scale (i.e. pore scale) ------------------ --

- Leads to comprehensive average models

- Effective properties estimated from the scaling-up theory

(clear relation between pore-scale structure and macroscopic description) pore scale


advanced Modeling i Porous Media Use of DNS

Scaling-up The Volume Averaging Method

(Carbonell Whitaker 1984), Quintard & Whitaker 2000)...

I Averaging volume V large compared with the pore size

L - but small compared to the macroscopic length L (Darcy's scale)

2. Macroscopic equations average of the pore-scalea Itransport equations over V, (T13)'= VO To dV

YO

3� Spatial decomposition T = (TOY, T3 : n order to

eliminate the point temperatures from macroscopic equations

y -phase 4, Closure based on the governing equations for

the spatial deviations T,3V 0�0 +

To = boo. T + So, ((T �T3),3 +

boo, so, closure (or mapping) variablesThree-phase system hquid-vaporflow

through a porous medium Closed form of the macroscopic transport equations

a (TO)0 + V,3),3Ef3 (PCP)O �TO)I' - K - V(TO)O + .. + ho, ((TI3) - M)01 +

at



Effective properties Experimental issue

0 (TO) 11 0EO (pC,),, � � + O (pCp),, vo)" - V(TO) = (K* V(T3)11) +... + 110 (TO)

atDarcy's scale convection thermal dispersion heat exchange flux

0 Estimating from experiments the effective coefficients K*, ho, ... is a very difficult taskO

• Experimental challenge Darcy's scale velocities in the sample

and void fraction measurement

• Inherent difficulty experimental measurements provide the

point temperatures in the solid phase and the bulk

temperature in the fluid phases

How to estimate average temperatures

from point measurements

• Lateral heat losses, contact points, wall effects (porosity) ...

=t, How to represent their impact

on the effective thermal properties

Data to derive effective properties correlations are very difficult to obtain


Modeling in Porous Media Use of DNS

Effective properties Use of Direct Numerical Simulation (DNS)

0 Effective properties and closure problems

K = okol k)3 noboo dA + k,3 no-, boo dA , ho = k,6 no, Vso, dA13 V f f

.45'Y A � 01

The method of volume averaging provides closure problemsfor mapping variables bo and s,

Closure problems can be solved for spatially periodic

9 model of a porous medium

=> Determination of effective properties fromthe solution of closure problems

- Analytical solutions for simple ID unit cells

- Knowledge of the two-phase flowSpatially periodic model at the pore scale for 2D-3D unit cells DNS

(the period stands for the aeraging volume)

Direct Numerical Simulation at the pore level

to assess or to modify closure relations between average values and deviations

to analyze and to predict the local transfers and their effects on effective propertieshat cannot be obtained from ex-er'

. iments


advanced Modeling i Porous Media Use of DNS

Direct Numerical Simulation

0 An overview of existing methods

- Level Set methods Chang et al. 1996), Osher et l. 2001)

- Front-Tracking methods (Juric (1996). Tryg-vason et at'. (2001) ...

- Multiphase models (Ga-\7rilvuk et al. 1997), Saurel & Lenietayer 2001) ...

- Diffuse interface or Second Gradient methods (Jacqmin 1999). Jamet et al. 2001)...

0 Level Set and Diffuse interface methods

Track implicitly interfaces on fixed grids (interfaces -_ transition zones)

Surface tension represented as a continuous body force

• Level Set approach

- Classical thermodynamic description of interfaces surface of zero thickness

- Phase indicator "color" function

- Transition zone spreading of fluid properties over a numerical diffusion zone

• Diffuse interface approach

- Thermodynamic modeling of capillary layers surface of non-zero thickness (i.e. diffuse)

- Phase indicator density or mass fraction

- Transition zone resulting from the thermodynamic modeling


Avanced Modeling i Porous Media Use of DNS

The second gradient theory single component case

• Specific internal energy of a fluid of second grade

Ae (p, s, VP) eo (p, s) + - (VP),

2pclassical part internal capillarity

Intrinsic properties

Interface of non-zero thickness (analytical density profile for plane interfaces)X+

Surface tension at equilibrium : o f A (VP)2 dx

X

Equations of motion derived from Hamilton's principle (Gouin 1987), Jamet 2001))

dv 0 2 A VP)21P -V(P ApV V AVp V - (stress form)

dt 2regularization

surface tension effects

• Numerical difficulties arising from the stress form

- Unstructured grids (Finite Volume alerkin) : high order derivatives

- Preconditioning for low Mach number flows (Guillard & Viozat 1999)) : complex Eq Of State

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar Noveinber 21-23, 2001 Cadarache - France 8

Avanced Modeling in Porons Media Use of DNS

The second gradient theory two components case

• Specific internal energy of a mixture of two fluids of second grade

= (PI, P2, SI, S2, PI, P2)

- Equations of motion Hamilton's principle (Gouin 1990), Chevalier 1998))

- Dissipative model drag force and viscous constraints

• A simplified form of the specific internal energy P = PC, C mass fraction

e (P, C V = e (P, C +A VC)2 (isothermal case)2p

Cahn-Hilliard model A particular case for two incompressible fluids

OC + v - c (p), = F(c - A 2C generalized chemical potentialOt I,- I.., -1

diffusion

P dv -Vp + pVc + V o + &)g (potential form), v 0dt 'llar'cap ity

Thermodynamic closure Volumetric free energy of the mixture FW

Immiscibility modeling double well form/- 2 1Pe C 'Jacqrnin 1996)

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar November 21-23.) 2001 Cadarache - Fance 9


Cahn-Hilliard model Finite element numerical scheme - SUPG

• Cahn-Hilliard mass fraction equation Newton method (� P Galerkin basis functions)

Cn+1 Cn

+ n Vcn+' d + V;5 KVP'+ d = 0, � + Tpvn . V�OAt

�O Pn+1 - F (C)n+1 ) d V�o AVc'+ d = 0

• Cahn-Hilliard Navier-Stokes equations Pro'ection method (�b P2 Galerkin basis functions

First step Transport-Diffusion �b + TRV' V�b

,�n+l - Vn n �n+l V . �rn+P + (V 1 dQ (-vpn + /,n+1vCn+1 + �n+lg dQ

At

Second step Pressure Poisson Equation V�O VP'+' d V� _ VPn dQ P �OV . n+ 1 dQAt

,v1+1 - At V (Pn+i - nThird step Velocity Projection �b v` d ) dQP

Numerical scheme implemented from the PELICANS object-oriented platform developed at IPSN/DRS

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar November 21-23, 2001 Cadarach - France 10

Avaiwe(] Mo6ling i Poi-mis Media L-':s(, f DNS- ----- ---------- - - -- -- - - -- ------- ------------- - - ------------

- -----------

Stationary bubble Laplace relation

Two-dimensional case AP aR R� R

0.05

0.04

0.03 Cahn-HilliardLaplace

0.02

0.0

00 0.2 4 0.6 0.8 1

coohttioi f the re-5811-re &renc AP

------- ----------------------- - - ------------------- -----------------Duval, F. Fichot and M. Quintard WARE CATHARE Seminar Nmvinber 21--23, 2001 (adarach - 1_--i-ance

Avaiwed Modcffiip, last Porous Medii : se of )NS

- - - --------------

Impact of a rising bubble on a solid wall under microgravity

pf Pe p 11gil0. 0 6 x 0 0 12 m rectangle 19, 1 I:g!l= 2 0=90 Bo M-2

P'q py (T

(a) (b)

(C)------- --------------------------------- - ----- ----------

Duval, F. Fichot and M. Quintard ICARE/CATHARE Seminar Novemher 21-23 201 Ci(],irache Fraiic(I 12

Avaiwed Modellug iij Porotis Media Us(,, of DNS- -------- - -- ---------------- - ---------------

Coalescence of two bubbles

10-30.015 x 0.0 12 rn rectangle, 2.5; (T 2.6 N. n

(a) (b)

------------------

(d) IJ------- --- - ------- --------- - --------------

---------------------------------------- - ------------------------------ --- --------------- -------Duval, F. Fichot and M. Quintard ICAREX ATHARE Seminar Novetiiboi- 21-23, 2001 Cadaracli - Fnnicc 13

AWUMI(i MoclelUig iii Poroiis Me(fia Uo of MS

Examples of a bubble moving through a porous medium

Constricted capillary

Weber number We Re.Ca P(u)'db inertial force01 static force

.. .... .. .... ... .

.... ...... . .

--------------------

.. ........ . .-------------------------

case high Weber umber case 2 IwAIr NVcber nmber

----------------------- ---------- - ---- ------------ - ---------------- ------------------------------------------------------------------ - ----------- - -------------------- --Duval, F. Fichot and M. Quintard WARE CATHARE Seminar Nowtiibcr 21-23. 2001 Cadaracli - Uralicc


Conclusion and perspective

• The Volume Averaging Method provides comprehensive models

and exhibits closure problems for the determination of the effective properties

• Two-dimensional numerical experiments have shown the interest and the

potentialities of Diffuse interface methods for two-phase flows at small length scales

Scaling-up theory and DNS provide a powerful tool for determining

the effective properties of a porous medium subjected to a two-phase flow

• Two and three-dimensional DNS at the pore level (unit cells) Influence of important

parameters (void fraction, velocity field ... on the effective properties thermal

dispersion tensors, heat exchange coefficients ...

• From these numerical experiments Derivation of correlations for various

pore-scale configurations

=:�. Correlations will be introduced into averaged models (ICARE/CATHARE)


fro30131 0 advanced modeling in porous media: use of dns

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