numerical simulations of basic interfacial instabilities with the improved two-fluid model

IntroductionMathematical models

Numerical modelsSimulationsConclusions

Numerical Simulations Of Basic InterfacialInstabilities With the Improved Two-Fluid Model

Luka trubelj, Iztok Tiselj

Joef Stefan Institute, Reactor Engineering Division R4

International Conference

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Luka trubelj, Iztok Tiselj Joef Stefan Institute

Outlook of presentation

1 Introduction

2 Mathematical models

3 Numerical models

4 SimulationsKelvin-Helmholtz instabilityRayleigh-Taylor instability

5 ConclusionsOpen questions

Introduction

Two-phase flowsStratified flows (flows with large interfaces)Dispersed flows (flows with many smalldroplets/bubbles/particles)Mixed or transitional flows

Introduction

Introduction Motivation

Cold leg

Cold water ECC injection

Downcomer

Steam

Hot water

Mixing of hot and cold water

DCC on the jetJet instabilities

Turbulence in the liquid phase

Turbulence and momentum transfer at the interface

Heat and mass transfer at the interface

Bubble entrainment Bubble migration

Mixed water

Heat transfer to the wallsTurbulence production by the jet and entrained bubbels

Figure: Most important flow phenomena during a pressurized thermalshock situation with partially filled cold leg of primary system in nuclearpower plant.

Mathematical models Two-fluid model

Mass balance equation

kt + ~vk k = 0 (1)

Momentum balance equation

(kk~vk)t + (kk~vk )~vk = kp + (kk~vk) + kk~g + ~FD,k + ~FS,k (2)

Mathematical models Two-fluid model

Mass balance equation

kt + ~vk k = 0 (1)

Momentum balance equation

(kk~vk)t + (kk~vk )~vk = kp + (kk~vk) + kk~g + ~FD,k + ~FS,k (2)

Mathematical models Interfacial forces

General drag force for stratified flow

~FD,1 =12 (~v2 ~v1) mix

r(3)

r =t100

Splitted surface tension force

~FS,k = k~FS = k (4)

Mathematical models Interfacial forces

General drag force for stratified flow

~FD,1 =12 (~v2 ~v1) mix

r(3)

r =t100

Splitted surface tension force

~FS,k = k~FS = k (4)

Mathematical models Interface sharpening

Why to use a special interface tracking model?

RT-short1.aviMedia File (video/avi)

Conservative level set (Olsson, 2005)

1

+ [1 (1 1)~n] = 1 (5)

=x2

=x32

The interface is still smeared over 3 cells

Figure: Void fraction smeared over several cells using conservativelevel set method.

The smearing is not increasing during time

The interface is still smeared over 3 cells

Figure: Void fraction smeared over several cells using conservativelevel set method.

The smearing is not increasing during time

Numerical models

Spatial discretization: finite differenceConservative calculation of the fluxes with the high resolutionscheme for advectionStaggered grid

Time scheme: Euler explicitPressure correction: SIMPLE

Solver: CGSTABOperator splitting for the drag force

Numerical models

Kelvin-Helmholtz instabilityRayleigh-Taylor instability

Simulations

Several test cases were used to validate the improved two-fluidmodel with interface sharpening for flows with large interfaces:

Rayleigh-Taylor instability, dam break (the interfacesharpening, the drag force)Pressure jump over a droplet, droplet oscillations, risingbubble, wetting angle, Kelvin-Helmholtz and Rayleigh-Taylorinstability (the surface tension force implementation)

Simulations

Kelvin-Helmholtz instability

Fluid properties1 = 780 kg/m32 = 1000 kg/m31 = 0.0015 Pas2 = 0.001 Pas = 0.04 N/mg = 9.81 m/s2

Proposed by Tiselj,2004 Figure: Tilted tube: initial conditions.

KH.aviMedia File (video/avi)

c = 2pi

g(21)

Table: Onset of instability tonset and the most unstable wavelength c :experiment, theory and simulations on various grids.

nx ny tonset [s] c [mm]Theoretical 1.7 27Experimental 1.88 0.07 25 45

Numerical183030 1.93 42.5244040 2.04 40.4366060 2.12 39.1

Figure: Kelvin-Helmholtz instability amplitude growth: theoretical, inexperiment and in numerical simulations on different grids.

Rayleigh-Taylor instability

Fluid properties1 = 3 kg/m32 = 1 kg/m31 = 0.03 Pas2 = 0.01 Pas = 0.01; 0.04; 0.16 N/mg = 9.81 m/s2

DimensionsL = 4 mH = 1 m

Most unstable wavelength = c

3 = 2pi

3

g(21)

RT-wide.aviMedia File (video/avi)

Table: Most unstable wavelength of Rayleigh-Taylor instability:comparison between the theory and numerical simulation.

= 0.01 N/m = 0.04 N/m = 0.16 N/mnx ny [mm] [mm] [mm]

Th. 243 487 973

Num.12832 358-437 492-787 787-131325664 331-496 496-793 992-1323512128 362-498 498-797 996-1328

Open questions

Conclusions

Two-fluid model for flows with large interfaces was improvedThe interface was sharpened with the conservative level setmodel and the numerical diffusion was eliminatedThe surface tension force was implementedThe interfacial drag force was modified to more universalformulationThe improved two-fluid model was validated on test cases:

Rayleigh-Taylor instabilityKelvin-Helmholtz instability

It was shown that the developed two-fluid model with interfacesharpening can be used as an interface tracking model

Open questions

Conclusions

Open questions

Conclusions

Open questions

Conclusions

Open questions

Conclusions

Open questions

Development of the coupled modelImplementation of the energy equation with energy and masstransfer between phasesImplementation of the turbulence model

Open questions

thanks-up.aviMedia File (video/avi)

IntroductionMathematical modelsNumerical modelsSimulationsConclusionsAppendix

numerical simulations of basic interfacial instabilities with the improved two-fluid model

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