Two-Phase Flow Modeling

Introduction 1

Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006

Seminar on Two-phase flow modelling

1) Introduction

by

Iztok Tiselj"Jožef Stefan“ Institute, Slovenia

Email: [email protected]

April 2006

Basic equations of two-phase flow 1



2) Basic equations of two-phase flow

by

Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia


Two-phase flow modelling, seminar at UPC, 2006

Table of contents

INTRODUCTION1) Introduction2) Basic equations of two-phase flows.

TWO-FLUID MODELS Lectures 3-6

INTERFACE TRACKING IN 3D TWO-PHASE FLOWS Lectures 7-10

ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS Lectures 11-14

DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18


Basic equations of two-phase flowsContents

- Introduction

- Navier-Stokes equations and constitutive (local instant formulation).

- Boundary conditions at the interface.

- Coalescence, break-up, single-to-two-phase flow transition.

- Averaging of the Navier-Stokes equations in two-phase flow.

- Recommended reference: M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer, 2006.


Basic equations of two-phase flowsIntroduction

Types of two-phase flows:

- gas-solid, liquid-solid - not considered in the present seminar.Interface between the phases is well defined, very accurate two-fluid models and Lagrangian models exist. Reference:C. T. Crowe, M. Sommerfeld, Y. Tsuji, Multiphase Flows With Droplets and Particles, CRC Press,1997.

- gas-liquid - main topic of the seminar

- immiscible liquid-liquid mixture (not a two-phase flow, but is treated with the same approach as a two-phase mixture.)


Basic equations of two-phase flowsIntroduction, cont.

Two-phase flows according to the structure of the interface:

- Separated flows.Examples: horizontally or vertically stratified flows, jets.Modelling often possible with interface tracking methods.

- Transitional flows.Examples: Slug and annular flows in the pipes.Modelling problematic...

- Dispersed flows.Examples: bubbly, droplet, particle.Modelling with two-fluid models (particles - Lagrangian models)


Navier-Stokes equationsFluid k, that occupies the observed domain, is described with equations:

continuity equation

momentum equation

density

velocity

kρ

kvr( ) 0=⋅∇+ kk

k vt

rρ∂ρ∂

( ) ( )kkkkkkk IpFvv

tv τρρ

∂∂ρ +⋅∇−=⋅∇+

rrrr

kτ viscous stress tensorF

r I unit tensorvolumetric forces

kµ viscositykp pressure


Navier-Stokes equations, cont.

internal energy equation (also found in enthalpy or total energy form)

( ) kkkkkkkkkkk Qvvpqvu

tu +∇+⋅∇−⋅−∇=⋅∇+

rrrr:τρ

∂∂ρ

specific internal energyku kqr heat flux

volumetric source termskQ


Constitutive equations

Equation of state:

Viscous stress tensor for Newtonian fluids:

Heat flux - Fourier's law of heat conduction:

( )kkkk Tpp ,ρ=( )kkkk upp ,ρ= or

( )( ) Ivvv kkkT

kkkkrrr

⋅∇⎟⎠⎞

⎜⎝⎛ −−∇+∇= λµµτ

32

kkk Tkq ⋅∇=r


Boundary conditions at the interface

Local boundary conditions at the interface i. Interface is assumed to be a discontinuity.

Parentheses denote jump in the quantity w on the interface.

Interfacial mass balance:

( )[ ][ ] 0=⋅− nvv ikkrrrρ

ivr

[ ][ ] 21 == −= kk www

nr

interface velocity

unit vector normal to the interface, direction: from fluid 1 to fluid 2


Boundary conditions at the interface, cont.Interfacial momentum balance

Interfacial energy balance (simplified: neglected kinetic energy, neglected work of the surface tension, assumed σ=const, see Ishii, Hibiki for details):

( ) ( )[ ][ ] nnIpnvvu kkikkkrrrrrr σκτρ =⋅−+⋅−

( )[ ][ ] ikikkk qnqnvve =⋅+⋅−rrrrrρ

σ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

21

1121

RRκ

surface tension

local curvature of the interface:

iq

nir

⋅∇−=21κ

surface energy source term, usually zero (nonzero if chemical reaction runs at the interface)


Boundary conditions at the interface, cont.

θ θ

Wetting system 0°<θ<90°

Non-wetting system90°<θ<180°

Wetting angle model near the contact of the interface and solid surface

θθ sincos wallwall tnnrrr

+=

nr

wallnr

walltr


Navier-Stokes equations and interface jump conditions - problems

In theory, interface reconnection may create surface with singularities (non-smooth surface), immediately after the reconnection.

Curvature of the surface is not well-defined in such points.

Before: After:


Navier-Stokes equations and single-to-two phase flow transition

Unlike in the single-phase flow, Navier-Stokes equations (with all the boundary conditions) are not sufficient to describe arbitrary two-phase flows.

Problem that cannot be described with N-S equations is onset of boiling (cavitation) in a single-phase liquid or onset of the condensation in the pure gas phase.

Phase transition may start on the impurities in the bulk of the fluid or at the walls.

Additional information/models are needed (sometimes on molecularscales) to specify the density of the impurities in the liquid or the structure of the wall where the cavitation starts.


Navier-Stokes equations, whole-domain formulation

In some cases, the Navier-Stokes equations can be applied in modelling. Equations are often assumed to be incompressible, heat transfer neglected. The N-S equations and the interface jump conditions can be simplified and extended to the whole computational domain:

Continuity equation for the whole domain

Equation for interface tracking (form continuity eq.)

Momentum equation

Dirac delta function equation of interface

0=⋅∇ vr

( ) ( ) )),(( trfIpFvvtv

srrrr

r

σκδτρρ∂

∂ρ ++⋅∇−=⋅∇+

0=∇+ ρ∂

ρ∂ vt

r

),( trfsrδ


Navier-Stokes equations, applicability of local instant formulation

In general: mathematical and numerical difficulties in modelling of two-phase flows with the local instant formulation are insurmountable in the near future.

- Turbulent fluctuations - even in single-phase flows resolvable only at low Reynolds numbers.

- Existence of the multiple deformable moving interfaces. Motion of the interface is an integral part of the solution (except in particulate flows). Problems with break-up and coallescence of the surfaces.

Characteristic length scales of the interface motion can be much larger than the characteristic scales of turbulent flows, example: turbulent flume.

Characteristic length scales of the interface motion can be much smaller than the characteristic scales of turbulent flows: example turbulent bubbly flows.


Averaging of the Navier-Stokes equations

Why averaging?

Microscopic details of turbulent motions and interfacial geometry are seldom relevant for the engineering problems.

Averaged equations result in mean values of the two-phase flow motion.

Problem: scales eliminated with the averaging influence the meanvalues. That must be taken into account in the closure relations of the averaged equations.


Averaging of the Navier-Stokes equations Most common types of averaging - theory

Eulerian averaging of function :

Temporal (equivalent to Reynolds averaging in turbulent single-phase flow):

Spatial:

Ensemble (statistical):

),( trF r

dttrFt

t∫∆

∆),(1 r

dVtrFV

V∫

∆∆

),(1 r

∑=

N

nn trF

N 1

),(1 r


Averaging of the Navier-Stokes equations Most common types of averaging - theory...

Eulerian averaging of function :

Area (cross-sectional) for 1D two-fluid models:

Other, more "exotic" types of averaging exist (Lagrangian, Boltzmannstatistical averaging). See Ishii, Hibiki for discussion.

"Phenomenological averging" - not averaging at all, averaged equations built on phenomenological approach.

),( trF r

dStrFS

S∫ ),(1 r


Averaging of the Navier-Stokes equations Types of averaging - practical approach

From practical point of view the type of averaging isn't important.

Various types of averaging results in slightly different equations, however, the differences are minor comparing to the typical uncertainty of the closure relations required to close the averaged system of conservation laws.

What is important:- averaging smoothes out the turbulent fluctuations,- "transforms" two phases that alternately occupy the observed point into two continuous fields that exist in that point with a given probability.


Volume fraction, void fraction...

The function is a new fundamental variable produced by the averaging.

is a local time fraction of the phase k after temporal averaging,is a local volume fraction of the phase k after spatial averaging,is a probability for the presence of the phase k after ensemble averaging, etc...

When the averaged equations are solved, detailed definition of is not important anymore. In this seminar is mainly called k-th phase volume fraction.

DETAILS OF THE AVERAGING PROCEDURE SKIPPED (see Ishii, Hibiki for details).

kα

kαkα

kα

kα

kα


Typical averaged equations of two-phase flow 6-Equation Two-Fluid Model

• Represents a basis for the safety analyses of the two-phase flows in water-cooled nuclear reactors. Allows thermal and mechanical non-equilibrium.

• Requires several closure relations that are mainly based on empirical approach.

• Mass balances:

gfff A =

x v )-(1 A

+ t

)-(1 AΓ−

∂

∂

∂

∂ ραρα

gggg A =

x v A

+ t

AΓ

∂

∂

∂

∂ ραρα



• Momentum balances

• Energy balances:

wallfgravityffigrri

2f

ff

f FFvvvvCCVM xp )-(1

x v

)-(1 21 +

tv )-(1 ,,)(|| ++−Γ−=−

∂∂+

∂

∂

∂∂

αραρα

wallggravityggigrri2g

gg

g FFvvvvCCVM xp

x v

21 +

t v ,,)(|| ++−Γ+−=+

∂∂+

∂∂

∂∂

αραρα

( )wallfffgifffffff FvhQA

xv)-A(1

pt

Apx

v u )-A(1+

tu )A(1

,* +Γ−=

∂∂

+∂

∂−∂

∂

∂

−∂ ααραρα

( )wallggggiggggggg FvhQA

xv Ap

tA p

xvu A

+ t

u A ,

* +Γ+=∂

∂+∂

∂+∂

∂

∂

∂ ααραρα



Closure relations:• Two additional equations of state for each phase k are:

• Correlations for inter-phase momentum transfer.

• Correlations for interphase heat and mass transfer.

• Wall friction correlations.

• Correlations for wall-to-fluid heat transfer

• ... others ...

. u d u

+ p d p

= d kk

k

p

k

uk

k

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ ρρρ


Typical averaged equations of two-phase flow CFX-5.6 – homogeneous two-fluid model

Homogeneous two-fluid model in CFX code, contains viscous terms andsurface tension force (if the interface can be found):

• Two continuity eqs.:

• One momentum equation:

• Surface tension:

( ) ( ) 01111 =⋅∇+∂∂ vt

rραρα ( ) ( ) 02222 =⋅∇+∂∂ vt

rραρα

( )( )( ) ( ) pgFvvvvtv

refT ∇−−+=∇+∇−∇+

∂∂ rrrrrrr

ρρµρρ12

)),((12121212 trfnF srrr

δκσ−=


Basic equations of two-phase flow

Two-fluid models of two-phase flow are today's standard for modelling of industrial multiphase flows and will (in my opinion) play an important role in the foreseen future, despite the rapid progress in the field of the more accurate interface tracking methods.

Development and improvement of the empirical closure relations for ensemble, volume, time, or cross-section averaged Navier-Stokes equations of two-fluid models will remain an important research field.

From the stand point of the industrial applications: there are several types of piping flows in nuclear and chemical engineering, oil or water transport, where one-dimensional two-fluid models still present a sufficiently accurate and efficient option.

2D/3D two-fluid models - can be found in CFD codes (CFX, FLUENT) - in development - to be used with caution.

Introduction 2

Catalonia, Slovenia

50 km

Area: 20.000 km2

Population: 2.000.000GDP per capita: 21.000$ (2005)

32.000 km2

7.000.00025.500$ (2004)

Introduction 3

SLOVENIA

Introduction 4

“Jožef Stefan” Institutewww.ijs.si

The Jožef Stefan Institute is named after the distinguished 19th century physicist Jožef Stefan.

JSI is the leading Slovene research organisationresponsible for a broad spectrum of basic and applied research in the fields of natural sciences and technology.

The staff of around 700 specialize in research in physics, chemistry and biochemistry, electronics and information science, nuclear technology, energy utilization andenvironmental science.

Introduction 5

“Jožef Stefan” Institute - Nuclear Research

–Reactor Engineering Division• Thermal-Hydraulics • Structural Mechanics• Reliability, Industrial Hazard and Risk

–Nuclear Physics Division

• Theoretical, experimental and applied reactor physics

–Dept. of Environmental Sciences

• Radiochemistry and Radioecology

–Research Reactor TRIGA Mark-II,

• pool, 250 kW, 1000MW pulse mode

Introduction 6

Reactor Engineering Division of JSIThermal-hydraulics ~12 out of 20 researchers

Introduction 7

Overview of Thermal-hydraulics research at Reactor Engineering Division

– Simulations of transients and accidents in nuclear and experimental installations with computer codes RELAP5, CONTAIN, MELCOR:• 1999-2000 verification of the new full-scope NPP Krško simulator

with RELAP5• Standard experiments PMK, BETHSY (RELAP5), • OECD ISP-44 KAEVER (CONTAIN)

– Modelling of single and two-phase flows (“home-made” codes, CFX, Fluent, NEPTUNE CFD packages):

LES and DNS simulations of single phase turbulent heat transferCharacteristic upwind schemes for fast 1D transients in two-phase flow Numerical schemes for 2D, 3D two-phase flows: two(three)-fluid models and interface tracking models

Introduction 8


Table of contents, part 1

INTRODUCTION1) Introduction2) Basic equations of two-phase flows.

TWO-FLUID MODELS3) 1D two-fluid models - conservation equations4) 1D two-fluid models - flow regime maps and closure equations5) Characteristic upwind schemes for two-fluid models6) Pressure-based solvers for two-fluid models

Introduction 9



INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows - mathematical background8) Interface tracking models9) Coupling of two-fluid models and interface tracking methods10) Simulations of Kelvin-Helmholtz instability

ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS11) WAHA code - mathematical model and numerical scheme12) WAHA code - simulations13) Hands on: simulation of two-phase water hammer transient

and two-phase critical flow.14) Fluid-structure interaction in 1D piping systems

Introduction 10



DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME

(This is not a two-phase flow modelling chapter but...)

15) Mathematical model of DNS16) Pseudo-spectral numerical scheme, general results17) DNS of passive scalar heat transfer at various thermal boundary

conditions, conjugate heat transfer, high Prandtl numbers18) Hands-on. Running of the DNS code.

Introduction 11

Two-Fluid Models1D 6-equation equal pressure two-fluid model for inhomogeneous non-equilibrium two-phase flow – heart of the codes used for simulations in today’s nuclear thermal-hydraulics.

Introduction 12

1D, 6-Equation Two-Fluid Model

• Mass balances:


gfff A =

x v )-(1 A

+ t

)-(1 AΓ−

∂

∂

∂

∂ ραρα

gggg A =

xv A

+ t

AΓ

∂

∂

∂

∂ ραρα

wallfgravityffigrri

2f

ff


x v

)-(1 21 +

t v )-(1 ,,)(|| ++−Γ−=−

∂∂+

∂

∂

∂∂

αραρα


gg

g FFvvvvCCVM xp

x v

21 +

t v ,,)(|| ++−Γ+−=+

∂∂+

∂∂

∂∂

αραρα

Introduction 13

6-Equation Two-Fluid Model


• Two additional equations of state for each phase k are:

• Numerous closure relations...

• Additional models relevant for nuclear thermal-hydraulics (neutronics...)


xv)-A(1

pt

Apx

v u )-A(1+

tu )A(1

,* +Γ−=

∂∂

+∂

∂−∂

∂

∂

−∂ ααραρα


xv Ap

tA p

xvu A

+ t

u A ,

* +Γ+=∂

∂+∂

∂+∂

∂

∂

∂ ααραρα

. u d u

+ p d p

= d kk

k

p

k

uk

k

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ ρρρ

Introduction 14

1D simulations of two-phase flow fast transientsSimulation of water hammer in piping system

Past 4 years: development of computer code for simulations of water hammer transients in 1D piping networks. (WAHALoads project of 5th EU research program.)

Code development performed in cooperation with UCL and CEA.

One of the WAHALoads experiments (UMSICHT, Oberhausen):

Total pipelinelength: 137 m TANK

VALVE

Introduction 15

1D simulations of two-phase flow fast transientsSimulation of water hammer in piping system

Past 4 years: development of computer code for simulations of water hammer transients in 1D piping networks. (WAHALoads project of 5th EU research program.)

Code development performed in cooperation with UCL and CEA.

One of the WAHALoads experiments (UMSICHT, Oberhausen):

VALVE

P09TANKP18P03

GS

P06

P15P04

Introduction 16

Water hammer simulation of UMSICHT experiment

Pressure near the valveP03 - Pressure [MPa]

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 1 2 3 4 5 6 7 8 9 10

Time [sec]

UMSICHTWAHARELAP5

Introduction 17

Water hammer simulation of UMSICHT experimentVapour volume fraction near the valve

GS - Vapor volume fraction

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5 6 7 8 9 10

Time [sec]

FZRWAHARELAP5

Introduction 18

Fluid dynamics at negative pressuresUnder special conditions, i.e., cold and purified liquid, negative pressures could appear briefly near the valve in UMSICHT experiment, due to the delayed cavity growth.

Small negative pressures are actually measured in a few cases, but are within the uncertainty of the measurements.

Negative pressures were measured in water hammer experiment by Bergantand Simpson (1999, Proc. IAHR congress, Graz) and tube-arrest experiment (designed specially for that purpose by Williams & Williams, 2002, J.Phys. D, 35, 2222-2230)

Tube-arrest experiment:

Tube half-filled with purified water is accelerated upward and stopped suddenly. Water hammer-like transient follows.

How to model transients with negative pressures? spring

Introduction 19


Alternative approach to 6-eq. two-fluid model: • 7-equation "two-pressure" two-fluid model (Saurel, Abgrall, 1999). Very

similar equations like 6-eq. model but with two separate phasic pressures. Additional equation for volume fraction completes the system of equations:

7-eq. vs. 6-eq. : several advantages, several drawbacks…

• 7-equation model allows simulations of liquid phase at negative pressure, while the pressure of the vapor phase remains positive.

)( lgm pp = x

v + t

−∂∂

∂∂ µαα

Introduction 20

1D Simulation of tube-arrest experiment

Introduction 21

Two-phase flow modelling: Interface tracking algorithms

• Rising bubble in theviscous fluid flattens thecircular shape andcauses vorticity in andbehind the bubble

• VOF method explicitlytracks the interface between fluids andenables the streamlinelocation

Streamlines around the bubble -experiment (left) simulation (right)

Introduction 22

Introduction - Coupling of interface tracking method (VOF) and two-fluid model

Fluid dispersion and stratification during the Rayleigh-Taylor instability

(Černe, Petelin, Tiselj, 2001, J. Comput.Phys 171, 776)

Introduction 23

Kelvin-Helmholtz instability - Inviscid linear analysis: step velocity and step density profiles assumed

z=H

Results:

Critical relative velocity

Critical wave number

Critical wave length

Immiscible fluids

Velocity and density profiles for linear inviscid analysis

U2

U1

ρ2

ρ1

fluid 2

fluid 1

z=0

ρU

z=-H

σρρρρρ gU ∆+>∆21

212 2

σρ /2* gk ∆=

** /2 kπλ =

21 ρρρ −=∆

Introduction 24

Tilted tube experiment (Thorpe, 1969)

ρ2

ρ1

H

h1

h2

31 kg/m1000=ρ

32 kg/m780=ρ

N/m04,0=σ

2m/s10=g

sPa001,01 ⋅=µ

L=1,83 (0,2) m

H=0,03 m

sPa0015,02 ⋅=µ

Initial conditions

γ

u2

u1

zx

g

z=0Tube tilted for a small angle

S.A. Thorpe, 1969. Experiments on the instability of stratified shear flows: immiscible fluids. Journal of Fluid Mechanics, 39. 25-48

Introduction 25

K-H instability - CFX simulation complete tube length simulated

Temporal development of the interface predicted by CFX. K-H instability in experiment is observed in the middle section of the tube after ~1.8 s.

Viscosity not neglected, surface tension neglected in particular simulation.

Introduction 26

CFX simulation of Kelvin-Helmholtz instability

Volume fraction of lighter fluid.

(computational domain =20cmx3cm, time~2s).

Introduction 27

K-H instability – tough case for CFX code

-Simulation of experiment with K-H instability with two immiscible fluids is very tough task for CFX code.

- “Structured” grid was used and quasi-2D simulations performed. (No reasonable results on unstructured grid)

-Surface tension terms in CFX destabilize the surface contrary to the actual physics of the surface tension force, which plays a stabilizing role in the K-H instability development.

- CFX model without surface tension is more stable than predictedby the linear inviscid analysis and experiment.

- Never trust “beautiful” pictures produced by CFD codes.

Introduction 28

Computational domain and boundary conditions.⇒Boundary conditions:

0)1( =−=+

ydy

dθ

0 , =surfacefreenormalv

Solid – fluid interface

0)1( ==+

ydy

dθand

0)1( ==+ yθ or ( ) 01 ==+ yθ

ISOTHERMAL ISOFLUX

Outer wall boundary is adiabatic.

FLOW

X

Y

-h

h

Z

0

FREE SURFACE

HEATED WALL - CONST. POWER DENSITY

L2=2h L1

L3

Free surface

DNS of turbulent heat transfer with isoflux BC

Introduction 29

DNS of turbulent heat transfer with isoflux BC

Instantaneous dimensionless temperature field on the heated wall with isoflux BC (i.e. – wall of negligible thermal capacity and negligible thickness).

1D 2-fluid models - consrv eqs 1



3) 1D two-fluid models conservation equations

by




Table of contents

INTRODUCTION Lectures 1-2






1D two-fluid models - conservation equations Contents

- Introduction - classification of two-fluid models

- Homogeneous equilibrium model.

- Drift-flux model.

- 6-equation two-fluid models.- Hyperbolicity

- Two-pressure two-fluid models.

- Interfacial area transport equation.


1D two-fluid models - conservation equations Selected references

- M. Ishii, T. Hibiki, Thermo-fluid dynamics of two-phase flows, Springer, 2006.

- G.B. Wallis, One-dimensional two-phase flowm McGraw-Hill, 1969

- RELAP5 computer code manuals: http://www.edasolutions.com/RELAP5/manuals/index.htm

- Materials of the "Short Courses on Multiphase Flow nad Heat Transfer", annual 1-week seminar at ETH Zurich, (Lead lecturers: S. Banerjee, M.L. Corradini, G. Hetsroni, G.F.Hewitt, G. Tryggvason, G. Yadigaroglu, S. Zaleski)


Introduction - Classification of two-fluid models

General form of the two-fluid model equations:

vector of n independent variablesn*n matrix of terms with time derivativesn*n matrix of terms with spatial derivativessource term vector - closure relations without derivativesn*n matrix (preferably with n real eigenvalues and n linearly independent eigenvectors)

1−⋅∂∂

∂∂ A P =

x B +

tA

rrr ψψ

ψr

A

P r

S = x

C + t

rrr

∂∂

∂∂ ψψ

B

C

"Standard" two-fluid models do not contain terms with second order derivatives.


Introduction - Classification of two-fluid models"Standard" two-fluid model equations:

do not contain terms with second order derivatives.

Viscous stresses and heat conduction are described with constitutive equations that do not contain derivatives. Their inclusion would notimprove the accuracy of these models.

(Diffusive terms can be found in two-fluid models of CFD codesTheir accuracy is questionable, but they certainly have a positive influence on the stability of the numerical schemes.)

S = x

C + t

rrr

∂∂

∂∂ ψψ


Introduction - Classification of two-fluid models

Classification according to the number of equations - dimension of the vector :

3-equation two-fluid models (example: HEM model)4-equation two-fluid models (example: drift flux model)5-equation models (example: older version of RELAP5 code)6-equation models (widely used in nuclear thermal-hydraulic codes:

RELAP5, TRAC, CATHARE)7-equation models (two-pressure models, additional equation for

interfacial area concentration)8+ - equation models (multi-field models, example: different types of

bubbles modelled with separate balance equations)

S = x

C + t

rrr

∂∂

∂∂ ψψ

ψr


Homogeneous equilibrium model(3-equation model)

n=3 (HEM model should not be called two-fluid model)

conservative variables orbasic variables (m - mixture)

(Choice of variables is discussed in lessons on numerics)

Homogeneous Equilibrium Model (HEM model) assumes thermal equilibrium (both phases always at saturation conditions) and mechanical equilibrium between both phases .

Important from the theoretical point of view - represents a limit of higher two-fluid models.

),,( mmmmm uv ρρρψ =r

),,( mmm pv ρψ =r

),,( fg vv αψ =rOther possibilities exist for

3-equation two-fluid model: inhomogeneous model without heat transfer


Homogeneous-equilibrium model

The simplest averaged model of two-phase flow (works in 1D, 2D, 3D). Very strong interaction between both phases assures equal phasicvelocities and equal phasic temperatures. Such approximation is seldom acceptable.

•Mass balance for mixture:

•Momentum balance

•Energy balance

0= x v +

tmmm

∂∂

∂∂ ρρ

wallfgravityfm

mmm

m FF xp

x v v +

t v ,, +=

∂∂+

∂∂

∂∂ ρρ

wallwallmmmmmmmm qFv

xvp

xv u +

tu +=

∂∂+

∂∂

∂∂

,ρρ


Homogeneous-equilibrium model

• Equation of state (probably the most complicated part of the HEMmodel).

Sonic velocity exhibits strong discontinuity between the single-phase and two-phase flow.

• Complicated calculation from equations of state:

• Closure relations needed for wall friction and wall heat flux.

• No special model needed for single-to-two-phase flow transition.

( )saturationmsaturationmmm u −−= ,ραα

( )saturationmsaturationmmm upp −−= ,ρ


Drift flux model (4-equation model)

n=4 (drift flux model - again not called two-fluid model)

(m - mixture, g - gas)

Drift flux model or 4-equation two fluid model: one phase in saturation conditions (usually vapor), other phase not necessarily in saturation.

Mixture velocity obtained from the balance equations, relative velocity also available, but not from differential equation but from the empirical correlations.

Very popular model in the early days of nuclear thermal-hydraulics.(see Ishii, Hibiki for details).

(Other types of 4-equation two-fluid models can be constructed. )

),,,( mmmmgm uv ρρρρψ =r


Drift-flux model

Drift flux model takes into account the relative velocity of two phases:

The relative velocity depends on the type of the two-phase flow (flow regime) and must be supplied with appropriate correlations).

40-years old model - still useful in engineering applications (Zuber, Findlay, 1965, J. Heat Transfer 87)

• Mixture mass balance:

• Gas-phase mass balance

⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂−Γ

∂∂

∂∂

rf

gfg

ggg vx

= x

v +

t

ρρρ

ααραρα

)1(

0= x v +

tmmm

∂∂

∂∂ ρρ

fgr vvv −=


Drift-flux model

• Mixture momentum balance

• Mixture energy (phases in thermal equilibrium):

• Closure relations:- correlation for relative velocity - correlation for inter-phase mass transfer- equation of state- wall friction, wall heat flux- conductive heat flux, viscous stress tensor in 2D, 3D versions

(not written in balance equations)

wallggravitygm

rf

gf2mmmm FF

xp v

x

x v

+ t v

,,2)1( +=

∂∂+⎟

⎟⎠

⎞⎜⎜⎝

⎛−

∂∂+

∂∂

∂∂

ρρρ

ααρρ

wallwallmmmmmmmm qFv

xvp

xv u +

tu +=

∂∂+

∂∂

∂∂

,ρρ

rvgΓ

wallgF , wallq


5-equation two-fluid modelsn=5a)

Thermal non-equilibrium between both phases possible, mechanical equilibrium - homogeneous flow (not very realistic and not used in practise)

b)

One phase in saturation conditions, the other one in non-equilibrium, mechanical non-equilibrium possible. This type of two-fluid model was built into the computer code RELAP5/MOD1. Version of the computer code for nuclear thermal-hydraulics analyses from ~1985.

),,,,( ggffmmfg uuv ρρρρρψ =r

),,,,( mmffggfg uvv ρρρρρψ =r


6-equation two-fluid models

n=6

Both phases can exhibit departure from saturation conditions. Mechanical non-equilibrium possible. Both pressures equal. This type of two-fluid model is built into the nuclear thermal-hydraulics computer codes that are still in use today and RELAP5, TRAC, TRACE (RELAP5 and TRAC merged 2-3 years ago) - all codes made in USA, CATHARE code - France.

References:- manuals of the RELAP5 computer code (available online on internet)- D.Bestion, The physical closure laws in the CATHARE code, Nuclear Engineering and Design 124 (3), 1990.

),,,,,( ffggffggfg uuvv ρρρρρρψ =r



• Requires even more closure relations than the drift flux model. Closure relations are mainly based on empirical approach. Thus, more experiments needed.

Mass balances:

gfff A =

x v )-(1 A

+ t

)-(1 AΓ−

∂∂

∂∂ ραρα

gggg A =

x v A

+ t

AΓ

∂

∂

∂

∂ ραρα gΓ vapor mass generation per unit volume

)(xA pipe cross-section (streamwise variations allowed)




wallfgravityffigrri

2f

ff


x v

)-(1 21 +

tv )-(1 ,,)(|| ++−Γ−=−

∂∂+

∂

∂

∂∂

αραρα


gg

g FFvvvvCCVM xp

x v

21 +

t v ,,)(|| ++−Γ+−=+

∂∂+

∂∂

∂∂

αραρα

iC

CVM

iv

Virtual mass term, contains derivatives!

Interface friction coefficient

Interface velocity





xv)-A(1

pt

Apx

v u )-A(1+

tu )A(1

,* +Γ−=

∂∂

+∂

∂−∂

∂

∂

−∂ ααραρα


xv Ap

tA p

xvu A

+ t

u A ,

* +Γ+=∂

∂+∂

∂+∂

∂

∂

∂ ααραρα

*gh

igQ ifQ gas-interface and liquid-interface heat fluxes per unit volume

specific gas and liquid enthalpies at the interface (usually saturation enthalpies)

*fh



Closure relations:• Two additional equations of state for each phase k are:

• Correlations for inter-phase momentum transfer.

• Correlations for inter-phase heat and mass transfer.

• Wall friction correlations.

• Correlations for wall-to-fluid heat transfer

• ... others ...

. u d u

+ p d p

= d kk

k

p

k

uk

k

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ ρρρ

CVM iC iv

igQ ifQ gΓ



Closure relations:• non-diferential closures - no derivatives - contribute to vector .• differential closure equations - contain temporal and/or spatial

derivatives of the variables contribute to matrices, examples:- virtual mass term - in dispersed flows (motion of the bubble/droplet

causes motion of the neighbouring mass of the opposite phase )- interface pressure term - stratified flows in 1D approximation- unsteady wall friction terms (in single-phase 1D flows), ...

• The same physical phenomena can be sometimes described with differential or non-differential model

• closure equations with second-order derivatives - not found in 1D two-fluid models. Insufficient accuracy of the two-fluid model and errors of the numerical schemes (mainly first-order accurate) do not justify inclusion of the closure equations with second-order derivatives.

P = x

B + t

A rrr

∂∂

∂∂ ψψ

P r

BA ,



Closure equations with first order derivatives influence the matrices andand mathematical character of the equations. Standard 6-equation

two-fluid model is non-hyperbolic (ill-posed, i.e. has "slightly" complexeigenvalues of the matrix ).

Differential terms (virtual mass, interface pressure) may be used toimprove hyperbolicity (interface pressure term added into CATHAREcode two-fluid model without physical background, with purpose toremove non-hyperbolicity).

Even a small term with second-order derivatives removes ill-posednessof the two-fluid equations. In practice such diffusion terms are notexplicitly added, but come in the form of the numerical diffusion of thefirst-order accurate schemes.

P = x

B + t

A rrr

∂∂

∂∂ ψψ

BA ,

BAC 1−=


7-equation two-fluid models

n=7

Possibilities for 7th variable: - vapor volume fraction model assumes phasic pressure non-equilibrium (two-pressure two-fluid model).- transport equation for interfacial area concentration - interfacial area concentration is a basis for all the closure laws describing inter-phase heat, mass and momentum transfer (Ishii, Hibiki).- concentration of non-condensable gas (RELAP5)- ....

)variable7,,,,,,( thffggffggfg uuvv ρρρρρρψ =

r

α


7-equation two-fluid modelTwo-pressure two-fluid model

Alternative approach to 6-eq. two-fluid model: • 7-equation "two-pressure" two-fluid model (Saurel, Abgrall, J. Comput.

Physics 150 (2), 1999). Very similar equations like 6-eq. model but with two separate phasic pressures. Additional equation for volume fraction completes the system of equations:

• New terms in total energy equations.

)( lgm pp = x

v + t

−∂∂

∂∂ µαα

( )wallfffgiflgiiifff fff FvhQppp

x)-(1vp

x

p E v)-(1+

tE )(1

,*)(

)(+Γ−+−−=

∂∂+

∂

+∂

∂

−∂µαραρα

( )wallggggiglgiiigggggg FvhQppp

xvp

x

p E gv+

tE

,*)(

)(+Γ++−=

∂∂+

∂

+∂

∂

∂µααρα


7-equation two-fluid modelTwo-pressure two-fluid model

Advantages of the 7-eq. model comparing to standard 6-eq. model:

- No problems with hyperbolicity (no need for virtual mass or empirical interfacial pressure term)

- Much simpler eigenstructure of the equations (simple analytical expressions for eigenvalues and eigenvectors)

- Less problems with numerics (allows calculations of extremely large pressure and volume fraction gradients without oscillations)

- The "two-pressure" model can be used as a single pressure model if instantaneous pressure relaxation is assumed ( ).

Problems:- Unknown relaxation time for the pressure non-equilibrium.- Pressure relaxation term is very stiff (very short relaxation time).

∞=µ


8+ -equations two-fluid models

n=8 and more

- multi-field models (see lecture notes of S. Banerjee at Modelling and Computation of Multiphase Flows, ETH Zurich, annual seminars)

The same phase, for example liquid in annular flow, is modelled with a separate conservation equation for the liquid film at the wall and aseparate equation for the droplets in the vapor code of the flow.

- multi-group models: for bubbly flows: bubble size spectra divided intovarious classes. Each class of bubbles treated with a separate balanceequation (see publications by U. Rohde, Forschungszentrum Rossendorf and CFX5 code manual.)


Interfacial area transport equation

- Interfacial area is the most important parameter that governs the inter-phase heat, mass and momentum exchange in two-phase flows.

- Like all other variables - is flow regime dependent - it can actually serves as a quantity describing the flow regime.

- Advantage of the transport equation for

Advantage of additional equation - more accurate closure relations in transients that change the flow regimes. Advantage of the transport equation over the "standard" (non-differential) closures for is more continuous transition between the correlations of different flow regimes. (reference: Ishii, Hibiki)

SINKSS SOURCE= x a v +

ta ii +

∂∂

∂∂

ia

ia

ia

ia

1D 2-fluid models - closures 1



4) 1D two-fluid models flow regime maps and closure equations

by




Table of contents







1D two-fluid models - flow regime maps and closure eqautions - Contents

- Flow regime maps- vertical flow regimes- horizontal flow regimes- correlations for flow regime transitions

- Non-differential closure equations- inter-phase friction- inter-phase heat and mass transfer- wall friction- wall-fluid heat transfer

- Differential closure equations- virtual mass- interface pressure- unsteady wall friction


1D two-fluid models - flow regime maps and closure equtions - Reference

- RELAP5 manual - a complete set of 1D flow regimes and closure laws applied in one of the leading codes for analyses in nuclear thermal-hydraulics.


Flow regime maps

Closure laws of the 1D two-fluid models depend on the flow regime of the two-phase flow.

Example of flow regimes in vertical upward flow (Photo from Mayinger, Stromung und Warmeubergang in Gas-Flussigkeits-Gemischen, Springer-Verlag, 1982):

Flow regimes fromleft to right:- 2*Bubbly flow- Slug or plug flow- Annular- Annular-whisp


Flow regime maps

Flow regime is an integral "quantity", which is based on geometry of the flow.

Inter-phase heat, mass and momentum transfer and wall-to-fluid transfer strongly depend on the flow regime.

Closure laws are developed separately for each flow regime.

Thus - the first step in development of the closure laws for 1D two-fluid models is to draw an accurate flow regime map, which determines borders between different flow regimes.

Flow regime maps - not directly applicable in 2D, 3D two-phase flow modelling: local closure laws in 2D, 3D cannot base on "integral quantity"...


Example of horizontal flow regime map

Flow regime map for horizontal flow. From Mandhane et al. 1974, Int. J. Multiphase Flow 1.

gg vj α=

ll vj )1( α−=

( )lf jj =


Horizontal flow regime map in RELAP5 code(drawing from RELAP5/mod3.3 manual)


Example of vertical flow regime map

Flow regime map for cocurrent vertical upward flow. From Hewit, Roberts. 1969.

gg vj α=

ll vj )1( α−=

( )lf jj =


Vertical flow regime map in RELAP5 code(drawing from RELAP5/mod3.3 manual)


Correlations for flow regime transitions

Various flow regime maps exist. They are based on a wide range of experiments but are are limited to the measurements and experimental conditions (type of fluid, pressure, temperature, pipe diameter, pipe inclination...).Flow regime maps in the computer codes must operate in much wider range of parameters. Flow regime maps are believed (I. Tiselj) to be the major source of uncertainty in the computer codes based on two-fluid models.

Typical simulation of the transient in the nuclear power plant coolant loop; how much time is code using proper flow regime correlations in each particular volume of the system filled with two-phase flow? ???


Differential and non-differential closure laws

• Non-diferential closures - no derivatives - Derived from steady-state experiments. - Easier to develop from the experimental data.- Validity in transient conditions questionable.

• Differential closure equations - contain temporal and/or spatial derivatives of the variables.

- Can take into account history or spatial distribution of the variables.- Difficult to develop (experiments in transient conditions needed). Can be obtained with theoretical approach.- Influence the mathematical character of the equations and the speed of sound in the two-phase flow.


Inter-phase friction frictionNon-differential closure equations

Physical background - stress terms due to the relative motion of both phases:Liquid and gas phase momentum equations:

Examples of :- Bubbly flow (RELAP5). Assumptions: all bubbles of the same size, bubble diameter=half of the max. bubble stable at thelocal relative velocity .

- Horizontally stratified flow (RELAP5). Assumption: interface isa flat plate. Standard laws for friction near the flat wall are applied.

wallfgravityffigrri FFvvvvC ,,)(||termsalDifferenti ++−Γ−=

wallggravityggigrri FFvvvvC ,,)(||termsalDifferenti ++−Γ+−=

iC

fgr vvv −=


Example - bubbly flow inter-phase friction (RELAP5):Drag coefficient of the bubble:

interfacial area concentration:

Reynolds number in is defined

The product of the critical Weber number and surface tension is:

Modified square of the relative velocity is defined as:

Average bubble diameter is:

⎟⎠⎞

⎜⎝⎛= 1.0,

81max gfDgi aCC ρ

( )5.0,Re/)Re1.01(24min 75.0bubblebubbleDC +=

0/6.3 da bubgf α=

2

)1()(Refgf v

We

µασ −⋅=

)10,5max()( 10−⋅=⋅ σσWe

⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅=)005.0,min(

)(,max 3/122

bubfrfg D

Wevvαρ

σ

20

)(

fgf v

Wed

ρ

σ⋅=

Example - stratified flow friction (WAHA code):Force of f on g = - Force of g on f:

Friction factors near the flat wall

Inter-phase friction coefficient:

Approximate interfacial areaconcentration inthe circular pipe

2rigf vCFF == 22 )(

81)(

81

igggifff vvfvvf −=− ρρ

2)64.1)ln(Re79.0( −−= fff 2)64.1)ln(Re79.0( −−= ggf

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −=

f

ffiff

Avv

µρ

,1000maxRe ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −=

g

ggigg

Avv

µρ

,1000maxRe

))()(

81(

)()(

81

2

2

2

2

gffg

ifffigf

fg

igggi a

vvvv

fCoravvvv

fC−−

=−−

= ρρ

Aagf

))1(,min(2 αα −=

Iterative procedure starts with initial guess

)(5.0 gfi vvv +=



Inter-phase heat and mass transferNon-differential closure equations

Physical background:

gggg A =

xv A

+ t

AΓ

∂

∂

∂

∂ ραρα

( )wallfffgif FvhQA ,*termsalDifferenti +Γ−=

( )wallggggig FvhQA ,*termsalDifferenti +Γ+=

gfff A =

x v )-(1 A

+ t

)-(1 AΓ−

∂

∂

∂

∂ ραρα

gΓ vapor mass generation per unit volume

*gh

igQ ifQ gas-interface and liquid-interface heat fluxes per unit volume

specific gas and liquid enthalpies at the interface (usually saturation enthalpies)

*fh


Inter-phase heat and mass transfer

The vapor generation rate is calculated from known heat fluxes as:

The liquid-to-interface and gas-to-interface volumetric heat fluxes

Interface temperature is assumed to be a saturation temperature at the local pressure. Fluxes , are flow regime dependent (interfacial area dependent). Details - elsewhere (RELAP5).

**fg

igifg hh

QQ−+

−=Γ

ρ/puh +=

0if, ** >Γ== − gsaturationggff hhhh

0if, ** <Γ== − gggsaturationff hhhh

)( fSifif TTHQ −= )( gSigig TTHQ −=

STigQ ifQ

Wall frictionNon-differential closure equations

Simple model - calculate single phase friction for two-phase mixture and split the friction between both phases:

Colebrook, White correlation (for single phase flow):Laminar flow:

Turbulent flow׃

wallfgravityffigrri FFvvvvC ,,)(||termsalDifferenti ++−Γ−=

wallggravityggigrri FFvvvvC ,,)(||termsalDifferenti ++−Γ+−=

m

ffffwfwallf D

vvfF

ρραρ )1(

2,−

=m

ggggwgwallg D

vvfF

ραρρ

2, =

Darcy equations modified for the two-phase flow:

Re64=wf

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−=

Dk

ff ww

27.0Re

51.2log21

Differential correlations to take into account transient effects...



Wall-to-fluid heat transfer

Non-differential closures

Physical background: wall-to-fluid heat transfer -important in the flow around the fuel elements of the nuclear power plant.

( ) wfwallfffgif QFvhQA ++Γ−

=

,*

termsaldifferenti eq.energy -f

( ) wgwallggggig QFvhQA ++Γ+

=

,*

termsaldifferenti eq.energy -g


Virtual mass term (added mass) Differential closure equation

Physical background: in the dispersed flow acceleration of the bubble (droplet) accelerates also the gas (liquid) around the bubble (droplet) -so called added mass effect. This can be taken into account with a new term in momentum equation:

Simplified term for the 1D two-fluid models (one of the possibilities):

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂∂

∂∂

∂∂

∂xv v-

tv -

xv

v+tv C = CVM f

gfg

fg

VM

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>+−

−−+⎟⎠⎞

⎜⎝⎛ −

≤−

+⋅

−=

6.0)/1(

)12)(1(2

23

4.01

2121

)1(

2

2

αραρα

ααα

α

αα

ααρ

fg

mVM

a

C

termsaldifferentinon −=+∂∂+

∂∂

∂∂

CVM xp

x v

21 +

t v

2g

gg

g αραρα


Virtual mass termDifferential closure equation

Problem of the virtual mass term:

- Clearly and accurately defined only for spherical particles. Bubbles/droplets are often non-spherical. Moreover, size of the bubbles is not known...

- Even less than in the bubbly and droplet flow regimes is known about the virtual mass term in other flow regimes.

- Historical reason for inclusion of the VM term: more stable numerics.Virtual mass term can make equations of the 6-equation two-fluid model hyperbolic.


Interface pressure termDifferential closure equation

Physical background:interface pressure term allows simulations of the horizontally stratified flows with 1D two-fluid model - appears in momentum equations:

Interface pressure must be:

to obtain solutions that behave like solutions of the shallow water equation

gDP gfi ))(1( ρραα −−=

termsaldifferentinon −=∂∂+

∂∂+

∂∂

∂∂

x P

xp

x v

21 +

t v i

2g

gg

gααραρα

D pipe diameter

termsaldifferentinon −=∂∂−

∂∂+

∂∂

∂∂

x P

xp )-(1

x v

)-(1 21 +

t v )-(1 i

2f

ff

fααραρα


Interface pressure termDifferential closure equation

Mathematical background:like virtual mass term, interface pressure term can make the two-fluid model hyperbolic. CATHARE code is using interface pressure term in stratified flow:

and in all other flow regimes an expression which makes equations hyperbolic (almost hyperbolic):

gf

rfgi

vP

ρααρρρ

)1(

2

−+=

gDP gfi ))(1( ρραα −−=

hyperbolicity can be lost when relative velocity becomes comparable with the speed of sound in the two-phase mixture

this term is sufficient to make equations hyperbolic in horizontally stratified flows


Unsteady wall frictionDifferential closure equation

Physical background:Standard wall friction correlations are developed from the steady-state measurements. Such correlations are insufficient for some of the fast transients with pressure waves in the piping systems.

Simplified single-phase momentum equation:

Unsteady wall fricton equation:

More details in lectures on 1D simulations of fast transients.

sDD t

τ ττθ−=

r rr

vv

D

xp

x v

21 +

t v

2 rrτρρ 4−=

∂∂+

∂

∂

∂∂

sτr

θsteady state wall friction

relaxation time correlation


Closure equations - conclusions

• Closure equations describing inter-phase heat, mass and momentum transfer and wall-to-fluid transfer depend on the flow regime.

Flow regime is integral "quantity". Application of "integral quantity" on the local scale of partial differential equations is questionable. It "works" in 1D, but,

how to transport the flow regime information to 2D, 3D ?

• Closure relations are the main source of uncertainty in the two-fluid models.

Results are especially questionable in simulations of the transients with flow regime transition.

Applicability of a specific two-fluid model with a given set of closure equations for the particular transient in the nuclear power plant, must be tested with "integral experiments".

characteristic-upwind schemes 1



5) Characteristic upwind schemes for two-fluid models

by




Table of contents







Characteristic-upwind schemes for two-fluid models - Contents

- Pressure-based and characteristic upwind schemes.- Introduction to high resolution shock capturing schemes for

Euler equations of single-phase compressible flows.- Riemann solvers- second-order accurate solutions

- Characteristic-upwind schemes for two-fluid models:- Two-fluid models: conservative or non-conservative form?- Eigenvalues, eigenvectors of the two-fluid model equations.- Integration of the geometric source terms.- Integration of the stiff source terms.

- Characteristic-upwind schemes for two-fluid models, yes or no?


Characteristic upwind schemes for two-fluid models - Selected references

Books:C. Hirsch, Numerical computation of internal and external flow, Vol. 1-2, John Wiley & Sons, (1988).J. D. Anderson, Computational Fluid Dynamics, McGraw-Hill, New York, (1995).R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, ETH, Zurich, (1992).Papers:R. Saurel, R. Abgrall, A Multiphase Godunov method for compressible multifluid and multiphase flows, J. Comp. Physics 150, 425-467, 1999.I. Tiselj, S. Petelin, Modelling of two-phase flow with second-order accurate scheme, J. Comp. Physics 136 (2) 503-521, 1997.R. B. Pember, Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation I. Spurious Solutions", SIAM J. Appl. Math. 53, No. 5, 1293 (1993)


Pressure-based and characteristic upwind schemes

Pressure-based schemes: pressure is a "privileged" variable comparing to density. Suitable for incompressible flows. Characteristic upwind schemes: pressure treated like all other variables (velocity, density, temperature) - suitable for Euler equations of compressible flows.

Is two-phase flow compressible or incompressible?Main criteria for separation of compressible and incompressible flows is fluid velocity, which must be smaller that ~30% of the sound velocity in the fluid.- Effective sound velocities in two-phase flows depends on closure equations and can be as low as 10 to 20 m/s (argument for characteristic upwind schemes)- Pressure based schemes are not limited only to incompressible but can usually handle "slightly" compressible flows... (argument for pressure-based schemes)


Pressure-based and characteristic upwind schemes

Characteristic upwind approach vs. pressure-based methods:- Pressure-based methods - longer history - older versions were first-

order accurate in time and space, robust and efficient. Their weak side is numerical dissipation, which tends to smear discontinuities on coarse grids.

- New pressure-based schemes are improved also for slightly compressible flows, second-order accurate versions available (CFD codes).

- Characteristic upwind scheme can be easily upgraded into second-order accurate scheme, which means reduced numerical diffusion.

- Advantage of characteristic upwind approach: for fast transients with pressure waves. Pressure-based approach might be sufficient for a wide range of transients where the convection terms play a minor role comparing to the source terms.


High resolution shock capturing schemes for Euler equations

Euler equations of single-phase compressible quasi-1D flow of ideal gas:

Conservative form

Non-conservative vectorial form:

Conservative variables are used in vector

Equation of state (ideal gas):

S = x

C + t

rrr

∂∂

∂∂ ψψ

dxdAp

pEAvpvA

vA

AEvA

A

xt⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+++

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

0

0

)()( 2ρ

ρρρ

⎟⎠⎞

⎜⎝⎛ +=== 2

21],[ vu e Ee A v A ,A ρρρρρρψr

v

p

cc

vpE =+−

= γργ

2

21

1



Jacobian matrix:

Diagonalized:

Eigenvalues: Eigenvectors

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−=vvhvhv

vvCγγγ

γγγ23

2

)1(2/)1(1)3(2/)3(

010

LLC 1−⋅Λ⋅=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

+=Λ

vcv

cv

000000

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−+−+=

2/

111

2vcvhcvhvcvcvL

⎟⎟⎠

⎞⎜⎜⎝

⎛=

ργ pc2 ρ/peh +=



Equation:

rewritten:

Modified characteristic variables introduced:

S = x

C + t

rrr

∂∂

∂∂ ψψ

xAR

xLL +

t01 =

∂∂+

∂∂⋅Λ⋅

∂∂ − rrr ψψ

xAR L

xL +

tL 01111 =

∂∂⋅Λ⋅Λ+

∂∂⋅Λ

∂∂ −−−− rrr ψψ

A R L L δψδδξrr 11 Λ −−− ⋅+= 1

0 = x

+ t ∂

∂∂∂ ξξ

rr

Λ

CHARACTERISTIC FORM OF EQUATIONS:


High resolution shock capturing schemes for Euler equations - discrete form

Vectorial equations

are numerically solved with explicit time integration (n - time, j - space):

S = x

C + t

rrr

∂∂

∂∂ ψψ

CFL limit on time step:

( ) ( ) 0 =

xA - A )R( +

xA - A )R(

x -

C + x -

C + t -

j1+jn1/2+j

--1-jjn1/2-j

++

nj

n1+jn

1/2+j

n1-j

njn

1/2-j

nj

1+nj

∆∆+

∆∆∆−−++

rr

rrrrrr ψψψψψψ

( ) ( )( ) ( )n

j

nj

n

j

nj

LLC

LLC

2/11

2/1

2/11

2/1

Λ

Λ

+−−−

+−−

−−++

−++

⋅⋅=

⋅⋅=

nj

nj

nj

nj

--

RLFL R

RLFL R

2/11

2/1

2/11

2/1

)()(

)()(

+−++

+++

+−−−

+

⋅⋅=

⋅⋅=rr

rr

cvcvxt ),max(/ +−∆<∆



Matrices

Flux (slope) limiters:MINMOD

Van Leer

Superbee

⎟⎠⎞

⎜⎝⎛

∆∆

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

∆∆

⎟⎟⎠

⎞⎜⎜⎝

⎛

1min

1max

- xt

2 -

|| ,0 = f

- xt

2 +

|| ,0 = f

kk

k

k--k

kk

k

k++k

λφ

λλ

λφ

λλ

3,1

3,1

=⋅=

=⋅=−−−−

++++

kf

kf

kkk

kkk

λλ

λλ

:,ΛΛ −−++−−++ F ,F ,

)) ,(1 ,(0 = kk θφ minmax

)1/()( ++= kkkk θθθφ

))2,min(),1,2min(,0max( kkk θθφ =

||

=m ,--

= 1/2+jk,

1/2+jk,

1+jk,

m-1+jk,

jk,1+jk,

m-jk,m-1+jk,1/2+jk,

λλ

ξξ

ξξξξ

θ∆

∆=

2/

2/

( ) 2/111

2/1 Λ +−−−

+ ∆⋅+∆=∆ jAj A R L L rrr 1ψξ

SECOND-ORDER CORRECTIONS

0=kφ

1=kφ1st-order upwind

2nd-order Lax-Wendroff



Jacobian matrix averaging (Roe's approximate Riemann solver):

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−−−−−−=+

aveaveaveaveaveave

aveavej

vvhhvvvvC

γγγγγγ

23

22/1

)1(2/)1(1)3(2/)3(

010

11

1112/1)(

++

++++ +

+=

jjjj

jjjjjjjave AA

vAvAv

ρρ

ρρ

11

1112/1)(

++

++++ +

+=

jjjj

jjjjjjjave AA

hAhAh

ρρ

ρρ

12/1)( ++ = jjjave ρρρ 12/1)( ++ = jjjave AAA



Jacobian matrix averaging with Roe's approximate Riemann solver guarantees proper propagation velocities of the discontinuities (shock waves) in the solutions. Rankine-Hugoniot conditions are satisfied at the discontinuities of the numerical solution:

(entropy fix procedure - see LeVeque for details - must be added to remove the discontinuities that violate entropy law - rarefaction shock waves.)

( )( )( )

( )( )

( )( )ωρ

ρ

ωρωρω

dxdAppEAvpvA

vA

AEvA

A/

0

0

)()( 2

===

+∆+∆

∆

+++

∆∆∆

ω propagation velocity of the shock wave

∆ difference between the quantities ahead and behind the shock

( )ωdxdA / cross-section derivative in point of the discontinuity

High resolution shock capturing schemes for Euler equations - solutions (Sod's shock-tube)


1- shock wave, 2- rarefaction wave, 3 - contact discontinuity

p ρ v

length (m)

Sod, JCP 27, 1978


High resolution shock capturing schemes for Euler equations - shock-tube solutions (100 grid points)

length (m)

velo

city

(m/s

)

upwind 1st-order analytical

Lax-Wendroff 2nd-order high resolution 2nd-order

(Not Sod's shock tube - Lax Wendroff fails for Sod's case due to the very large discontinuity...)


High resolution shock capturing schemes for Euler equations - what is applicable for two-fluid models?

Problems of two-fluid models:

- Equations are "Euler-like" but not necessarily hyperbolic.

- Diagonalization of the Jacobian matrix of 6-equation two-fluid model is a difficult task:

- diagonalization can be performed with analytical approximations.

- diagonalization can be performed numerically.

(Details: Tiselj, Petelin, JCP 136, 1997, WAHA code manual, 2004)

High resolution shock capturing schemes for Euler equations - what is applicable for two-fluid models?

Problems of two-fluid models:

- Equations cannot be written in conservative form (although they are derived from conservation equations), i.e., Rankine-Hugoniotconditions are unknown.

Moreover - shocks in two-phase flow are not discontinuities...(See example of shock wave in bubbly mixture, Kameda, Matsumoto, Phys. Fluids 8 (2), 1996)

experiment

time (ms)pr

essu

re (b

ar)

analiticalsolution of hypothetical two-fluid model



High resolution shock capturing schemes for Euler equations - what is applicable for two-fluid models?Problems of two-fluid models:

Regarding the numerical integration source terms can be divided into three groups:

1) Sources due to the variable cross-section - can be treated with

2) Source terms describing interphase mass, momentum, and energy transfer, which tend to establish mechanical and thermal equilibrium – i.e., RELAXATION source terms. These source terms are STIFF (their time scale can be much shorter than the time scale of the sonic waves). SPECIAL TREATMENT REQUIRED.

3) Other source terms, which represent external forces (gravity, wall friction) and wall heat transfer - not stiff (probably).

characteristic upwind in the convection part of equations.


Characteristic-upwind schemes for two-fluid models

Example of numerical scheme for two-fluid model based on characteristic upwind methods and operator splitting with explicit time integration.

Operator splitting:1) Convection and non-relaxation source terms - source terms due to the smooth area change, wall friction and volumetric forces are solved in the first sub step with upwind discretisation:

2) Relaxation (inter-phase exchange) source terms:

,S= x

B + t

A RELAXATIONNON_rrr

∂∂

∂∂ ψψ

S = dtd A RELAXATION

rrψ


1st substep of operator splitting: convection terms with non-relaxation source terms

Equation solved:

Eigenvalues and eigenvectors of Jacobian matrix are found:

Source terms are rewritten:

contains source terms due to the variable pipe cross-section contains wall friction and volumetric forces (no derivatives).

This part of the scheme is the same as for the Euler equations of the single-phase compressible flow.

. SA = x

C + t RN−

− ⋅∂∂

∂∂ rrr

1ψψ

LLC 1Λ −⋅⋅=

. R xAR

xLL +

t FA 0Λ 1 =+∂∂+

∂∂⋅⋅

∂∂ − rrrr ψψ

ARr

FRr


1st substep of operator splitting: basic variables

Basic variables are ~ primitive variables,( replaced with )

The preferred set of variables would be conservative variables:

Conservative equations + and -:

1)+ Numerical conservation of mass and energy can be assured with conservative variables. No conservation of momentum: equations of two-fluid model cannot be written in conservative form, due to the pressure gradient terms, virtual mass terms, interfacial pressure terms, and possibly other correlations that contain derivatives... (Conservation of momentum is less important than conservation of mass/energy.)

],[ e ,e )-(1 v ,v )-(1 , ,)-(1 ggffggffgf ραραραραραραϕ =r

)u ,u v,v , p, ( = gfgf ,αψr , gf ρρ u ,u gf



Conservative variables + and -:

2)- "Non-standard" water property subroutines are required that calculate two-phase properties ( ) from the conservative variables ( ).

3)- Primitive variables are very convenient for evaluation of eigenvaluesand eigenvectors.

4)+/-The conservative quantities as components of vector,

are more sensitive to the numerical oscillations than the primitive variables:

)e ,e ,v ,v , , ( = f fggffg gfg ρααρρααρρααρψ )1()1()1( −−−r

)u ,u ,v,v ,p, ( = gfgfαψr

ρρα gf ,, ,pu ,u )-(1 , ,)-(1 ggffgf ραραραρα


1st substep of operator splitting: basic variablesConservative variables + and -:4) CONTINUED- Specific numerical oscillations are induced near the property discontinuities (Karni, 1994, Abgrall, 1996) when conservative variables are used.

+ Non-conservation of mass and energy can also cause numerical oscillations near the strong pressure and volume fraction discontinuities.

+/- The optimal set of variables might be a mixture of conservative and nonconservative variables:

)e ,e ,v , , ( = f fggffg ρααρρααρψ )1()1( −−r



Influence of the basic variables on the solution of the Toumi'sshock tube problem for the 6-equation two-fluid model.

see Tiselj, Petelin, JCP 136, 1997


1st substep of operator splitting: basic variables - examples

Influence of the basic variables on the solution of the Toumi'sshock tube problem for the 6-equation two-fluid model.

Initial vapor volume fraction discontinuity: αLEFT=0.25, αRIGHT=0.1



1st substep of operator splitting: basic variables - examples

Influence of the basic variables on the solution of the Tiselj'sshock tube problem for the 6-equation two-fluid model.

Initial vapor volume fraction discontinuity: αLEFT=0.9, αRIGHT=0.1



1st substep of operator splitting: basic variables - conclusions

Optimal scheme for the convective part of equations remains to be found...

Implicit time schemes might be preferred.

Problem: transition from single-phase to two-phase flow. 3 equations in single-phase volume, 6 (5,7) equations in two-phase volume.

Degeneration of eigenvectors for zero relative velocity in two-fluid models with two velocity fields (a small artificial relative velocity maintained everywhere solves the problem).


2nd substep of operator splitting: integration of stiff relaxation source terms

Relaxation source terms: inter-phase heat, mass and momentum exchange terms are stiff, i.e., their characteristic time scales can be much shorter that the time scales of the hyperbolic part of the equations. Integration of the relaxation sources within the operator-splitting scheme is performed with variable time steps, which depend on the stiffness of the source terms.

Upwinding is not used (difficult to use) for calculation of the relaxation source terms.

S = dtd RELAXATION

rrψ A



Smmmm tS ∆+= −+ )()(11 ψψψψ rrrrr

A

Second equation of the operator splitting scheme

is integrated over a single time step with variable time steps that depend on the stiffness of the relaxations and can be much shorter that the convective time step .

The time step for the integration of the source terms is controlled by the relative change of the basic variables. The maximal relative change of the basic variables in one step of the integration is limited to 0.01 to obtain results that are "numerics" independent. Time step is further reduced when it is necessary to prevent the change of relative velocity direction, or to prevent the change of sign of phasic temperature differences.Probably the best solution: implicit integration of relaxation sources.



Relaxation source terms of the WAHA two-fluid model do not affect the properties of the mixture in a given point: mixture density, mixture momentum, and mixture total energy should remain unchanged afterthe integration of the relaxation source terms. It is in principle possible to choose a set of basic variables:

that enables simplified integration of the relaxation source terms. Only a system of three differential equations is solved instead of the system of six.

It is difficult to calculate the state of the fluid from the variables that are result of such relaxation.

)T ,T vv ,e ,v, ( = gffgmmmmmM ,−ρρρψr


Numerical schemes for hyperbolic equation with stiff source terms

- LeVeque and Yee (1990) tested a simple convection equation with a stiff source term and showed that a general stiff source term affects the propagation velocity of the discontinuous solutions and can cause non-physical numerical oscillations. - Pember's conjecture from (1993): stiff relaxation source terms do not produce spurious solutions, when the solutions of the original hyperbolic model tend to the solution of the equilibrium equations as the stiffness of the relaxation source terms is increased.- Numerical tests with the 6-eq. two-fluid model confirmed the results of Pember: the stiff sources describing inter-phase mass, energy and momentum exchange in two-fluid models do not produce spurious solutions and do not modify the propagation velocity of the discontinuities.- Stiff source terms are integrated with variable time step depending on the stiffness.


Numerical scheme for the convection equation Integration of the source terms

Current test cases for numerics and physics:• 1) Shock tube with large pressure and void fraction jumps (test of

numerics).• 2) Simple water hammer experiments (Simpson, 1989).• 3) Two-phase flow in the nozzle. Especially important as a test of

closure laws (physics): very accurate steady-state solutions can be easily calculated from steady-state ordinary differential equations for subcritical flows (experiment Abuaf et. al. 1981, Brookhaven Nat. Lab.). Also very though test for numerics.


05

1015202530354045

0 20 40 60 80 100

Vhem Vf Vg

Ci=10Hif=Hig=10^3

610

615

620

625

630

635

640

0 20 40 60 80 100

Them Tf Tg

Ci=10Hif=Hig=10^3

Propagation velocities of shock and rarefaction waves in two-fluid models

Shock waves of two fluid model with various inter-phase momentum (Ci), heat and mass transfer (Hif,Hig)


05

1015202530354045

0 20 40 60 80 100

Ci=10^3Hif=Hig=10^6

610

615

620

625

630

635

640

0 20 40 60 80 100

Ci=10^3Hif=Hig=10^6


05

1015202530354045

0 20 40 60 80 100

Ci=10^4Hif=Hig=10^7

610

615

620

625

630

635

640

0 20 40 60 80 100

Ci=10^4Hif=Hig=10^7


05

1015202530354045

0 20 40 60 80 100

Ci=10^5Hif=Hig=10^9

610615

620625630

635640

0 20 40 60 80 100

Ci=10 5̂Hif=Hig=10 9̂


05

1015202530354045

0 20 40 60 80 100

Ci=10^6Hif=Hig=10^11

610

615

620

625

630

635

640

0 20 40 60 80 100

Ci=10 6̂Hif =Hig=10 1̂1


Integration of the stiff relaxation source terms

• The arbitrary stiff source terms can affect the propagation velocity of the discontinuous solutions and can produce spurious numerical solutions.

• Results with the two-fluid model confirm the Pember's conjecture from (1993), which states that the stiff relaxation source terms do not produce spurious solutions, when the solutions of the original hyperbolic model (6-equation two-fluid model) tend to the solution of the equilibrium equations (Homogeneous-Equilibrium model) as the stiffness of the relaxation source terms is increased.

• Stiffness of the neglected wall-to-fluid heat transfer sources cannot be excluded in advance in some extreme conditions in nuclear thermal-hydraulics – that would cause a new problem for numerics.


Characteristic upwind schemes for two-fluid models - conclusions

Is it reasonable to develop new codes based on characteristic upwind schemes?

New code for simulation of water hammer transients - WAHA - has been developed using characteristic upwind scheme within the WAHALoads project financed by EU's 5th research program.

Authors: Jozef Stefan Institute, Slovenia, Universite Catholique de Louvain, Belgium, Comissariat a l'Energie Atomique, Grenoble, France.

pressure-based schemes 1



6) Pressure-based solvers for two-fluid models

by




Table of contents


TWO-FLUID MODELS 3) 1D two-fluid models - conservation equations4) 1D two-fluid models - flow regime maps and closure equations5) Characteristic upwind schemes for two-fluid models6) Pressure-based solvers for two-fluid models





Pressure-based solvers for two-fluid models Contents

- Introduction

- Numerical scheme of RELAP5

- Numerical diffusion, accuracy


Pressure-based solvers for two-fluid models -Selected references

Book:Ferziger, Peric, Computational methods for fluid dynamics,

Springer, 1997.

Internet:http://www.cfd-online.com/Wiki/Numerical_methods

RELAP5, CFX, Fluent, NEPTUNE manuals


Introduction - pressure-based methodsPressure equation arises from the requirement that the solution of the momentum equation also satisfies continuity.

"Standard" two-fluid model equations:

Equations are discretised "directly". Often in the conservative form.

Such discretisation is often unstable - especially if diffusive terms (second-order derivatives) are absent. (CFX is known to have problems with inviscid flows)

S = x

C + t

rrr

∂∂

∂∂ ψψ


Introduction - pressure-based methods

RELAP5 - 30 years old numerical scheme - no second-order terms in RELAP5 two-fluid model. Stability comes from the numerical diffusion of first-order accurate discretisation and artificial viscosity term.

Schemes developed for conservation laws in single-phase flow are usually applied also for two-phase flows - especially in 2D, 3D CFD codes. Number of conservation laws not important...

Pressure-velocity coupling:- avoid checker-board of pressure-velocity field:

- use staggered grid, - Rhie-Chow type of velocity interpolation on coincident grids (used in general-purpouse CFD codes)



Segregated Solver (RELAP5, NEPTUNE, CFX, Fluent - for two-phase flows)

1) Solve Momentum equations (u,v,w) 2) Solve pressure correction equation (SIMPLE...)

– Correct fluxes and velocities 3) Solve transport equations for other scalars

Coupled Solver (CFX, Fluent - for single-phase flows)1) Solve the Momentum equations- Pressure equation system in one

go (u,v,w,p) 2) Solve transport equations for other scalars



Overview of the segregated solver (from Fluent manual):



Overview of the coupled solver (from Fluent manual):


RELAP5 numerical scheme (simplified)RELAP5 continuity and momentum equation for single-phase flow:

RELAP5 code discretisation properties: - Staggered grid - velocities calculated at the boundaries of the control volumes.

- Implicit for the acoustic terms, explicit for non-acoustic terms (semi-implicit scheme)

- Acoustic terms:

- Artificial viscosity term added for stability in the momentum equation.

02

02

=∂∂+

∂∂+

∂∂=

∂∂+

∂∂

xp

xv

tv

xv

tρρρρ

00 =∂∂+

∂∂=

∂∂+

∂∂

xp

tv

xv

tρρρ

RELAP5 numerical scheme (simplified)Staggered grid in RELAP5:

scalar node p,α,ρf,g,uf,g

mass, energy control volume

momentum control volume

velocity node vf,vg

j j+1

j+1/2

vf

vg


RELAP5 numerical scheme (simplified)


Donor-cell discretisation of the convective terms, density for example:

the same for velocity

Difference equations obtained for the positive velocities in the grid points i and i+1/2:

01

2/111

2/11

=∆−

+∆− +

−−+

++

xvv

t

ni

ni

ni

ni

ni

ni ρρρρ

artificial viscosity term

00

2/1

2/112/1 >

<⎪⎩

⎪⎨⎧

=+

+++

i

i

i

ii v

vρρρ

1st-order accurate difference

2nd-order accurate difference

0

2)()(2)()()(

211

1

22/1

22/1

22/3

221

22/1

12/1

2/1

=∆−

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ∆⎟⎟⎠

⎞⎜⎜⎝

⎛

∆+−

−∆

−+

∆−

+++

−+++++

++

xpp

xx

vvvx

vvtvv

ni

ni

ni

ni

ni

ni

ni

ni

ni

nin

iρρ


RELAP5 numerical scheme (simplified)

Two-equations written in each point. Velocity is eliminated and a linear system of N-equations is solved with unknown pressure pn+1. (N number of volumes)

After calculation of the pressure field, the velocity field is updated.

Other variables - calculated in two steps - mainly due to the stiff inter-phase exchange source terms. Inter-phase exchange terms are also calculated implicitly, other sources - with explicit integration.

01

2/111

2/11

=∆−

+∆− +

−−+

++

xvv

t

ni

ni

ni

ni

ni

ni ρρρρ

0

2)()(2)()()(

211

1

22/1

22/1

22/3

221

22/1

12/1

2/1

=∆−

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ∆⎟⎟⎠

⎞⎜⎜⎝

⎛

∆+−

−∆

−+

∆−

+++

−+++++

++

xpp

xx

vvvx

vvtvv

ni

ni

ni

ni

ni

ni

ni

ni

ni

nin

iρρ


RELAP5 and other codes in nuclear thermal-hydraulics

TRAC, CATHARE - even more implicit treatment of equations.

CATHARE - fully implicit:

Multi-dimensional codes (NEPTUNE, CFX) - fully implicit...

More implicit approach means more stability, but not more accuracy (stability is a result of numerical diffusion of the implicit schemes).

More implicit approach allows use of longer time steps - however, time step longer than the characteristic time of the physical phenomena means non-accurate simulation of the phenomena.

S = x

f + t

nnnn

)()( 111

+++

∆∆

∆− ψψψψ rrrrr


"Water hammer due to the valve closure" simulation

Stiff source term -integration problematic also in RELAP5 (implicit time integration of source terms)

Calculated vapor volume fraction near the valve: RELAP5 1 ∆t=∆x/cRELAP5 2 ∆t=0.01∆x/c2F - WAHA ∆t=∆x/cadaptive time step for relaxation source terms.0

0.001

0 .002

0 0 .05 0 .1 0 .15 0 .2 0 .25

tim e (s )

Vap

or v

ol. f

ract

ion

R E L A P 5 1 R E L A P 5 22 F 2 nd -o rd e r


RELAP5 at very small time steps

Quasi second-order pressure waves are predicted by the RELAP5 when a very small time step is used. The resolution of the steep gradients is improved; however, numerical oscillations appear near the shock wave. 8

10

12

14

16

18

0 2 4 6 8 10Length (m)

Pres

sure

(MPa

)

RELAP5 1st RELAP5 2nd 2nd-order scheme

This is a consequence of the 2nd-order central differencing of the pressure gradient:

0...2/1 =∆

++∆+ x

CONVECTIONt

iρ11

12/11

2/1 −− ++++

++ ppvv n

ini

ni

nin


Pressure-based methods - conclusions

Work fairly well, although the various numerical artifacts are less controlled than in the characteristic upwind schemes.

Advantages of the characteristic based schemes seem to be insufficient to justify development of the codes based on the characteristic upwind schemes.

Characteristic upwind or pressure-based schemes - it is not very important - the main problem of the two-phase flows is not numericsand numerical errors but physics and physical models.

7-3D-two-phase-flows 1


Seminar on Two-phase flow modeling

7) 3D two-phase flows -mathematical background

by




Table of contentsINTRODUCTION Lecture 1-2TWO-FLUID MODELS Lectures 3-6

INTERFACE TRACKING IN 3D TWO-PHASE FLOWS7) 3D two-phase flows - mathematical background8) Interface tracking models9) Coupling of two-fluid models and VOF method10) Simulations of Kelvin-Helmholtz instability




3D two-phase flows - mathematical background - Contents

- Introduction, computer codes CFX, Fluent, NEPTUNE (new 3D code for nuclear thermal hydraulics).

- 3D two-fluid models in CFX, Fluent, NEPTUNE

- 3D closure laws in CFX, Fluent, NEPTUNE

- Turbulence in two-fluid model codes.


3D two-phase flows - mathematical background - References

- Ishii, Hibiki (book, 2006)

- NEPTUNE, CFX, Fluent manuals

Additional:- Tsai & Yue Annu. Rev. Fluid. Mech. 1996.28:249-78 - about free-surface flows in oceanography

- Detailed surface modelling (non-zero thickness of the interface...): Anderson, McFadden, Wheeler, Annu. Rev. Fluid Mech. 1998. 30:139–65


ECORA - project of 5th research program of EU

ECORA document: Recommendation on use of CFD codes for nuclear reactor safety analyses - Conclusions:

"Two-phase CFD is much less mature than single phase CFD. The flows are much more complex and myriads of basic phenomena may take place at various scales. Thus it is clear that the physical modellingwill have to be improved over a long time period. Fundamental questions related to the averaging or filtering of equations are not yet as clearly formalised as they are for RANS or LES methods in single phase. This makes that the separation between physics and numericsis not always well defined...... ECORA strongly recommends further investigations on thistopic."


NURESIM - project of 6th research program of EU

The European Platform for NUclear REactor SIMulations, NURESIM is planned to become common European standard software platform formodeling, recording, and recovering computer data for nuclear reactors simulations. Key objectives of NURESIM:

(i) integration of advanced physical models in a shared and opensoftware platform; (ii) promoting and incorporating the latest advances in reactor and core physics, thermal-hydraulics, and coupled (multi-) physics modeling; (iii) progress assessment by using deterministic and statisticalsensitivity and uncertainty analyses, verification and benchmarking;(iv) training, dissemination, best practice and quality assurance.


NURESIM - project of 6th research program of EU

The specific objectives of NURESIM are to initiate the development of the next-generation of experimentally validated, “best-estimate” tools for modeling (thermal-hydraulics, core physics, and multi-physics) of the present and future reactors.

The improved prediction capabilities, standardization and robustness of the envisaged NURESIM European Platform would address current and future needs of industry, reactor safety organizations, academic, government, and private institutions.

Thermal hydraulics - NEPTUNE code.


Computer codes

CFX, Fluent - commercial CFD codes - academic licenses ~1000 EU per CPU - major players on the market of CFD codes.

CFX and Fluent used to be competitors, but have recently got the same owner (ANSYS). Future ???

Both codes have a strong two-phase flow modules. Especially useful for particles (bubbly flows).

Neptune - nuclear thermal-hydraulics oriented code - in development.Future dissemination ???


Navier-Stokes equations, whole-domain formulation

Approach available in Fluent with VOF technique - computed surface will always remain sharp (even when it has nothing to do with the actual shape of the surface)

Continuity equation for the whole domain

Equation for interface tracking (form continuity eq.)

Momentum equation

Dirac delta function equation of interface

0=⋅∇ vr


srrrr

r

σκδτρρ∂

∂ρ ++⋅∇−=⋅∇+

0=∇+ ρ∂

ρ∂ vt

r

),( trfsrδ

3D two-fluid models - homogeneous (equal velocity) model

2 Continuity equations

( ) ( ) GLGGGG Ut

Γ=⋅∇+∂∂ r

ραρα ( ) ( ) LGMLLLLL SUt

Γ+=⋅∇+∂∂ r

ραρα

Density

( ) ( )( )( ) MT SgpUUUUU

t

rrrrr++−∇=∇+∇−⋅∇+

∂∂ ρµρρ

( ) GLLL µαµαµ −+= 1( ) GLLL ραραρ −+= 1

1 momentum equation1=+ GL αα

User specified mass source

Interphase mass transfer

User specified momentum sourceViscosity

G - Gas L - Liquid

Volume conservation

Model available in CFX5 and Fluent CFD codes (3,4, or 5 eqs. two-fluid model)7-3D-two-phase-flows 10

3D two-fluid models - inhomogeneous model (different velocities)


( ) ( )( ) ( ) ( ) LMLLGLGLGT

LLLLLLLLLLLL MSUUUUpUUUt

rrrrrrrrr++Γ−Γ+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞⎜

⎝⎛ ∇+∇⋅∇+∇−=⊗⋅∇+

∂∂ µααραρα

( ) ( ) LGMLLLLL SUt

Γ+=⋅∇+∂∂ r

ρρα ( ) ( ) GLMGGGGG SUt

Γ+=⋅∇+∂∂ r

ρρα

2 Countinity equations

2 momentum equations

User specified mass source

Inter-phase mass transfer

( ) ( )( ) ( ) ( ) GMGGLGLGLT

GGGGGGGGGGGG MSUUUUpUUUt

rrrrrrrrr++Γ−Γ+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞⎜

⎝⎛ ∇+∇⋅∇+∇−=⊗⋅∇+

∂∂ µααραρα

User specified momentum source

Interfacial forces acting on phase L due to

presence of other phaseModel available in CFX5, Fluentand Neptune CFD codes (4,5,6eqs. two-fluid model)

( )LGLGL UUCMrrr

−=Drag Force

Mixture model (for droplets)

Mixing length scale - user specified - interfacial area is supposed to be a part of solution and

not a user defined parameter...

( )4/;

21

22

DAAUU

DC droplet

GLL

D πρ

=−

= rr

Dimensionless drag force coefficient

GGLGiD

LG UUaCCrr

−= ρ8

LG

GLi d

a αα=

Interfacial area per unit volumeGGLLLG ραραρ +=

Similar model found in CFX, Fluent, Neptune

3D two-fluid models - inter-phase momentum transfer in dispersed flows

ia


Dimensionless drag force coefficient for spherical particles (bubbles, droplets)

High Re: Schiller-Naumann drag modelRe24=DCLow Reynolds Re<<1

( )687.0Re15.01Re24 +=DC

Transitional area at medium Reynolds numbers.

3D two-fluid models - inter-phase momentum transfer in dispersed flows

CFX and Fluent offer drag forces for non-spherical bubbles, but should be switched on by user... (How do one knows that bubbles changed their shape?)


CFX, Fluent and Neptune can take into account also the followinginter-phase momentum transfer in dispersed flows: lift, virtual mass, turbulent dispersion force. Approach probably useful for particle flows (and allows numerous user defined parameters to fit the experiments...)

Neptune - separate correlations

CFX, Fluent - no correlations

3D two-fluid models - inter-phase momentum transfer in stratified flows

3D two-phase flows- inter-phase momentum transfer in dispersed-to-stratified flows

??? (user defined....)


3D two-fluid models -energy equations

2 Total Energy equations, 1 Momentum equation


( )UUhh stattot

rr⋅+=

21

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ ∇−∇+∇⋅∇−∇−⋅∇+∂∂−

∂∂ UUUUThU

tph

tT

LLLLLtotLLLLtotLLL

rrrrrδµαλαρααρα

32

,,

LLtotLGLtotGLG SQhh ++Γ−Γ ,,

Total enthalpy

External heat source

Interphase heat transfer

Heat transfer induced by

interphase mass transfer

( ) ( ) ( ) =⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛ ∇−∇+∇⋅∇−∇−⋅∇+∂∂−

∂∂ UUUUThU

tph

tT

GGGGGtotGGGGtotGGG

rrrrrδµαλαρααρα

32

,,

GGtotGLGtotLGL SQhh ++Γ−Γ ,,

Static enthalpy

3D two-fluid models -energy equations

2 Thermal Energy equations, 2 Momentum equations

External heat sourceInterphase heat

transfer

Heat transfer induced by

interphase mass transfer

( ) ( ) GGGLGLGLGGGGGGGGGG SQhhThUht

++Γ−Γ=∇−⋅∇+∂∂ λαραρα

r

( ) ( ) LLtotLGLtotGLGLLLLLLLLLL SQhhThUht

++Γ−Γ=∇−⋅∇+∂∂

,,λαραραr


3D two-fluid models -inter-phase heat & mass transfer

Interfacial heat transfer – Thermal phase change model

GLGLGL Am&=Γ

LSGS

GLLGGL HH

qqm−+=&

Interfacial area density

Interfacial mass flux

Heat flux from phase L to G

( )LSATLGL TThq −=Heat flux from phase G to L

SATLLSGGSGL HHHHm ,,0 ==→>&

LLSSATGGSGL HHHHm ==→< ,0 ,&

( )GSATGLG TThq −=

Heat transfer coefficientsLG hh ,

BASIC MODEL THE SAME AS IN 1D TWO-FLUID MODELS

Problem: unknown interfacial area and heat transfer coefficients (flow regime dependent)



3D two-fluid modelswall-to-fluid heat transfer

Single phase type of heat transfer assumed in CFX and Fluent. Acceptable if the wall-fluid area known for each phase ... again -part of the solution is expected as a user defined parameter...

Neptune:

- nucleate boiling correlations (important for nuclear simulations)

- flashing flow model (flashing delay possible in Neptune)


3D two-phase flows - turbulence

Characteristic length scales of the interface motion can be much larger than the characteristic scales of turbulent flows, example: turbulent flume.

Characteristic length scales of the interface motion can be much smaller than the characteristic scales of turbulent flows: example turbulent flow of very small bubbles.

3D two-fluid models - turbulence


Turbulence k-ε, for one phase or both phases

Turbulence production

Effective Viscosity

Modified pressure

( ) ( ) ρεσµµρρ −+⎟

⎟⎠

⎞⎜⎜⎝

⎛∇⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅∇=⋅∇+

∂∂

kk

t PkkUkt

r

( ) ( ) ( )ρεεεσµµερρε εε

ε21 CPC

kU

t kt −+⎟

⎟⎠

⎞⎜⎜⎝

⎛∇⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅∇=⋅∇+

∂∂ r

44.11 =εC92.12 =εC3.1=εσ0.1=kσ

( ) ( )kUUUUUP tT

tk ρµµ +⋅∇⋅∇−∇+∇⋅∇=rrrrr

332

teff µµµ +=ε

ρµ µ

2kCt =

kpp ρ32+=′

09.0=µC

Turbulent eddy dissipation

Turbulent kinetic energy

ε

k

Turbulent viscosity


3D two-fluid models - turbulence

Turbulence (NEPTUNE)

Model of dispersed phase kinetic energy transport and fluid/particle fluctuating movement covarianceModel of dispersed phase kinetic stress and fluctuating movementcovariance

Fluent, CFX: user can apply various turbulence models in every phase that he/she wants...

8-interface-tracking 1



8) Interface tracking models

by




Table of contents

INTRODUCTION Lecture 1-2TWO-FLUID MODELS Lectures 3-6





Multi-dimensional two-fluid models -Contents

- Review of the interface tracking methods- Lagrangian (moving-grid) methods- Eulerian (fixed-grid) methods (Marker-And-Cell, Embedded interface methods, VOF, Level set)

- Volume-of-Fluid method

- Level set method- Simulation of the K-H instability with "conservative level set"

method- Dam-break simulation.

- Interface sharpening in two-fluid models


Interface tracking methods- References 1Lagrangian methods:- Hyman 1984, Physica D 12:396-407- Hirt, Amsden, Cook, 1974, J. Comput. Phys. Vol. 14, 227-253.Eulerian:- MAC:

Harlow, Welch, 1965, Phys. Fluids 8: 2182-89,- Embedded interface methods:

Unverdi, Tryggvason, J. Comput. Phys. 100 (1) 1992)Tryggvason et al., J. Comput. Phys. 169 (2) 2001

- VOF: Hirt and Nichols 1981, J. Comput. Phys. 39:20 1-25,Scardovelli & Zaleski, DNS of free-surface and interfacial flow, Annu. Rev. Fluid Mech. 1999. 31:567–603.

- Level set: Sethian & Smereka, Annu. Rev. Fluid Mech. 2003. 35:341–72.


Interface tracking methods- References 2

Interface sharpening:???

Other interesting papers:- Recent review of the methods for free-surface flows: Caboussat, Arch. Comput. Meth. Eng. 12 (2), 2005.

Book: Validation of Advanced Computational Methods for Multiphase FlowLemonnier , Jamet, Lebaigue, Begell House, 2005. (test cases for interface tracking methods)


Lagrangian interface tracking methods

– The grid moves with fluid.– Suitable for small displacements of the surface. The grid

automatically follows free surface. Suitable for Fluid-structure interaction.

– Remeshing required for large surface distortions.– Severe limitation: cannot track surfaces that break apart or

intersect.


Eulerian interface tracking methods

Marker methods:- Marker-And-Cell (MAC)- Embedded interface methodsAll use surface markers, allow very accurate representation of the surface (accurate surface tension calculations).

Volume-Of-Fluid (VOF), Level-set:Each fluid is treated with function tracing the amount of each phase in the given point. Similar to the volume fraction of a given phase in two-fluid model

All methods need a basic solver for Navier-Stokes equations


Eulerian interface tracking methods -Solution of N-S equations

Algorithms for interface reconstruction are built into the basic numerical scheme for solution of Navier-Stokes equations:

- Choice of the basic numerical scheme must take into account large gradients in the material properties at the interface.

- The most efficient single-phase schemes are not necessarily successful in two-phase flow...

Useful schemes:- segregated solvers (Fluent, NEPTUNE, CFX4), pressure correction

schemes- coupled solvers - available for two-phase flow in CFX5 (not in CFX4)


Marker and cell (MAC)

– One of the first methods for time dependent flow

– Based on fixed Eulerian grid of control volumes

– The location of free surface is determined by a set of zero-mass and zero-volume marker particles that move with the fluid and are traced with Lagrangian approach.


Embedded interface methods (Tryggvason)

–Fixed Eulerian grid–Whole-domain formulation

–Interface is being tracked with the surface markers connected into the surface.

Front-tracking methods - not further discussed in this seminar. Volume-tracking preferred - closer relation with two-fluid models...


skkkrrrr

r

σκδτρρ∂

∂ρ ++⋅∇−=⋅∇+


Volume of fluid (VOF)

– To compute time evolution of free surface continuity equation for void fraction is solved

– Due to the step function nature of void fraction this equation must be solved in a way that retains the step function nature.

– With ordinary first or second order accurate discretization scheme step function gets smeared due to numerical diffusion

– A special procedure must be used to assure sharp free surface.

( ) ( ) 0=∇+∂∂ Ut

rαα

Volume of fluid (VOF)Interface reconstruction

– Reconstructs surface from volume fraction with geometrical elements.

0.4

1.0

0.07

0.95

0.0

0.2

1.0 1.0 0.7

Position of the interface in the Eulerian grid and void fractions.

Many different reconstructionschemes...

all based on geometry




Different types of interface reconstruction:- Simple Line Interface Reconstruction with

Calculation (SLIC) step function

First-order reconstructions.



Different types of interface reconstruction:

- Flux Line-Segment for Advection and Interface Reconstruction (FLAIR)

j,y(i,j)(i-1,j)

(i,j-1)

- Least-squares Volume-of-Fluid Interface Reconstruction Algorithm (LVIRA)

i,x ∆x

∆y

nr

Second-order approaches but very complicated in 3D


Level-Set

Use of a continuous level-set function φ, which is positive in the space occupied by the first fluid, negative in the space occupied by the second fluid.Value of φ in a point is distance from point to the surface

Free surface position is defined with the zero value of level set function φ (distance function)

( ) Interfacexxx II ∈−=Φrrr

;min

1=φ 0=φ

2=φ

1−=φ2−=φ

xr

xr


Level-SetTemporal development equation

–Heavy side function is used to represent density and viscosity over interface

–To achieve numerical robustness a smeared out Heavy side function is often used

–Where ε corresponds to the half of the interface thickness.

( )⎩⎨⎧

>Φ<Φ

=Φ0,10,0

H

( ) ( )⎪⎪⎩

⎪⎪⎨

⎧

>Φ

≤Φ≤−+Φ+

−<Φ

=Φ Φ

ε

εεπε

ε

επ

,1

,sin21

221,0

H

0=∇⋅+∂∂ φφ v

tr


Level-Set vs. VOF

Mass conserved in VOF but not in Level-Set (special additional algorithms needed).

Level-set - problems with φ near the steep gradients (bigger than in VOF).

3D - easier implementation of level-set, VOF more problematic.


Conservative Level-SetOlsson & Kreiss, J. Comput. Phys. 210, 2005

• After advective step - a different level-set function is defined:

α=0.5 on the surface, α does not measure distance from the surface but volume fraction.

• Equation which acts as artificial compression is solved until steady state is reached

• is normal at the interface and is calculated only once at the beginning of the second step. We denote time variable by τ to stress that this is an artificial time, not equivalent to an actual time t. Artificial compression flux α(1-α) acts in the regions where 0<α<1. Small amount of “viscosity” ε∆α is added to smear discontinues.

0=∇⋅+∂∂ αα u

tr

( )( ) αεαατα ∆=−⋅∇+

∂∂ nv1 α

α∇∇=nv

nv


Conservative Level-Set - Our implementation

System of Navier-Stokes eq.

SIMPLE pressure correction procedure to get divergence free velocity field

• Solving momentum equation to obtain intermediate velocity • Solving pressure correction equation • Solving momentum equation only with the contribution of pressure

part to get • Solving continuity equation for volume fraction to obtain

0=∇⋅+∂∂ αα u

tr

( ) ( )( )( ) guupuutu T rrrrrr

+∇+∇⋅∇+∇−=∇⋅+∂∂ µ

ρρ1 ( ) 21 1 ρααρρ −+= ( ) 21 1 µααµµ −+=

*ur

*1 ut

p r⋅∇⎟

⎠⎞

⎜⎝⎛

∆−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ′∇∇ρ

1+nur

1+nα


Conservative Level-SetOur implementation

Staggered grid to avoid checkerboard distribution of the variables All equations discretized with fluxes to ensure conservation

upwind and Lax Wendroff scheme) -> decreased numerical diffusion and dispersion, second order accurate in space and time

CGSTAB algorithm to solve pressure correction eq. (5-diagonal matrix in 2D)

Second order discretization with Van Leer limiter (combination of

yGG

xFF

tn

jin

jin

jin

jinji

nji ∆

−∆−

∆−= −++++ 2/1,2/1,,2/1,2/1,

1, ααp

uv

i,j i+1/2,j

i,j+1/2

i-1/2,j

i,j-1/2

Gi,j+1/2

Gi,j-1/2

Fi-1/2,j Fi+1/2,j


Conservative L-S - Dam break

• see: Validation of Advanced Computational Methods for MultiphaseFlow for details of the benchmark

• Surface tension was neglected due to the scale of the problem

• Two problems, dam break on dry and wet surface

• Water-air system

g=9.81 m/s2

L=1.2 m

H=0.14 mhl=0.1 m

hr=0.01 m



( ) or mmmm /0 −=∆• Mass conservation

1,00E-08

1,00E-07

1,00E-06

1,00E-05

1,00E-04

1,00E-030 0,1 0,2 0,3 0,4 0,5

t

mas

s

1,0E-081,0E-071,0E-06

ur⋅∇=residualmax



• Dry ground

• Grid:512x64, time step=1e-2 s, CPU time=1.5 h @3.0 GHz Pentium 4

• Wet ground – jet is formed

• Grid:1024x128, time step=1e-3 s, CPU time=15 h @3.0 GHz Pentium 4• Most of the CPU time for pressure correction eq.


Conservative Level-Set Conclusion

Very promising method - seems to allow natural transition from whole-field interface tracking mode into the two-fluid model.


Interface sharpening in two-fluid models

CFX

The implementation of free surface flow involves some special discretisation options to keep the interface sharp. These include:

• A compressive differencing scheme for the advection of volumefractions in the volume fraction equations. • A compressive transient scheme for the volume fraction equations (if the problem is transient). • Special treatment of the pressure gradient and gravity terms to ensure that the flow remain well behaved at the interface.

Neptune - interface sharpening supposed to exist - not documented yet (similar mechanism as in CFX).

9-VOF+two-fluid 1



9) Coupling of two-fluid models and VOF method

by


9-VOF+two-fluid 2


Table of contents





9-VOF+two-fluid 3

9) Coupling of two-fluid models and VOF method - Contents

-VOF method

-“Two-fluid” model

-Model Coupling

-Simulation of the Rayleigh-Taylor instability

References: Cerne, Petelin, Tiselj

J. Comp. Phys. 171, 776–804 (2001), Coupling of the interface Tracking and the Two-Fluid Models .....

Int. J. Numer. Meth. Fluids 2002; 38:329–350 Numerical errors of the VOF...

9-VOF+two-fluid 4

Description of the problemVarious two-phase flow regimes

0=⋅∇ ur

( )

( ),Dpg

uutu

µρ

ρ∂∂ρ

⋅∇+∇−

=∇⋅+

r

rrr

( ) 0ut kk

k =⋅∇+∂

∂ rαα

( )

rrhkkkk

kkkkk

kk

uuCpg

uut

u

rrr

rrr

−∇−

=∇+∂

∂

αρα

ραρα

1=∑k

kα

interfacial drag:

Chk =Ch1 =−Ch2

separated flow - model VOF

dispersed flow -two-fluid model

( ) 0=⋅∇+∂

∂u

trα

α

9-VOF+two-fluid 5

Description of the problemFlow regime change

v v v

9-VOF+two-fluid 6

VOF method - I

0=⋅∇ ur

( ) ( ),Dpguutu µρρ

∂∂ρ ⋅∇+∇−=∇⋅+ rrrr

⎪⎩

⎪⎨

⎧

<<=

,1,,

001

,

,

ji

ji

αα ( ) .0=⋅∇+ α

∂α∂ ut

rfluid 1

fluid 2

both fluids

Whole domain formulation of basic equations (no surface tension term):

9-VOF+two-fluid 7

VOF method II

0.4

1.0

0.07

0.95

0.0

0.2

1.0 1.0 0.7

j,y

→n

(i,j)(i-1,j)

(i,j-1)

i,x ∆x

∆y

simulated structures are larger than the grid distance

9-VOF+two-fluid 8

VOF errors - I

reconstruction error

9-VOF+two-fluid 9

VOF errors - I

S

dh

reconstruction error - bubble on a coarse grid

9-VOF+two-fluid 10

VOF errors - I

0

0,2

0,4

0,6

0 2 4 6 8 10d/h

reconstruction error

( ) ( )( ) ( )tttN Vji

jTEDiRECONSTRUCjACTUALi∑∈

−=),(

2,,

1 ααδ

d bubble diameterh distance between the grid points

yx ∆=∆

9-VOF+two-fluid 11

VOF errors - II Advection error

Initial state:- different bubbles flows together with the surrounding liquid in a constant velocity field

Final state:- bubbles with d<2.5h move faster- shapes of the bubbles are changed

9-VOF+two-fluid 12

VOF errors IIINumerical dispersion error

u

d

Shear flow test-the horizontal velocity changes linearly in vertical direction-a vertical strip of fluid perpendicular to the velocity is stretched to the infinity -(periodic boundary conditions)

9-VOF+two-fluid 13


Numerical DispersionWhen the strip width is close to the grid size, the tension of the reconstruction algorithm to keep the fluid chunk as compact as possible results in dispersion.

Several fluid chunks with the characteristic size h<d<3h are provided,the fluid chunks are stable despite the shear velocity field.

9-VOF+two-fluid 14


black coloured spot in the prescribed prescribed velocity field - vortex shear flow with the zero velocity in the origin and boundaries and maximum velocity in the middlecircle bubble is put on the position of the maximum velocity gradient (point(0.5,0.85))

bubble is deformed into the spiral whirling to infinity

9-VOF+two-fluid 15


Numerical dispersion: left - solution on finer grid, right -numerical solution on coarse grid

9-VOF+two-fluid 16

“Two-fluid” model - I

fluid 1

fluid 2

fluid 2

fluid 1

0.90.9 0.8

0.2 0.3 0.3

0.7 0.6 0.7

simulated structures are smaller than the grid distance

9-VOF+two-fluid 17

“Two-fluid” model - II

( ) 0=⋅∇+∂

∂kk

k ut

rαα 1=∑k

kα

interfacial drag

2121 ααρdcCC =−=

( ) ( ) ( )kkkkkkkkkkkk

kk DuuCpguut

uf µααραραρ ⋅∇+−+∇−=∇+∂

∂21rrrrr

r

2121 81 vvacCC icd

rr −=−= ρ

9-VOF+two-fluid 18

Model coupling - I

VOF

two-fluid

VOF

9-VOF+two-fluid 19

Model Coupling - II

∑∈++

++++ −=jiVljki

ljkijiljkiji

ji fHV

,1),(,,,

,, )(1 ξγ

definition of the "dispersion"

fluidsstratifiedji 0, =γ fluidsmixedji 0, >γ

(i,j)(i+1,j)

)ofondistributi(local, αγ funcji =

practical implementation:

measured on 3x3 number of cells

9-VOF+two-fluid 20

Model Coupling III

00 =γ

• Tests on simple two-fluid states

Switch criteria between models

• the interface in the cell (i,j) is reconstructed

• , the fluids in the cell (i,j) are calculated with the "two-fluid" model

two-fluid model

γγ max0 =VOF model

8.03.00 −=γ

0, γγ <ji

0, γγ >ji

9-VOF+two-fluid 21

Transition between VOF and two-fluid model

Wrong reconstruction:

8.0, =jiγ

9-VOF+two-fluid 22

Advantage of the coupled model

0

0,1

0,2

0,3

0,4

0,5

0,6

0 5 10 15 20

VOF 28x28

coupled 28x28

VOF swithed to 56x56

0.5 2.01.51.0 t

-the distributions of the volume fraction are compared to the exact solution- in the moment of numerical dispersion the VOF model significantly increases the error - the switch to denser nodalization model may delay the error increase- at switch to two-fluid model the error is increased due to the numerical diffusion, but long time its prediction of volume fraction distribution is better than at VOF model

δ

9-VOF+two-fluid 23

t=7t=4.8t=3.6t=2.6t=1.6t=0.8t=0.4t=0

Result - VOF simulationRayleigh-Taylor instability

9-VOF+two-fluid 24

Comparison of VOF results for different grid densities

( ) ( )( ) ( )tttN Vji

jLijMinod ∑∈

−=),(

2,,

1 ααδ

0

0,2

0,4

0,6

0,8

1

1,2

1 10 100

nod

t

f 6x30 -f 12x60

f 12X60 -f 24X120

f 24X120 -f 48X240

f 48X240 -f 96x480α=f

9-VOF+two-fluid 25

t=0 t=0.4 t=0.8 t=1.6 t=2.6 t=3.6 t=4.8 t=7

Results - coupling of VOF and two-fluid models - Rayleigh-Taylor instability

9-VOF+two-fluid 26

Comparison of coupled VOF+two-fluid model -results for different grid densities

0

0,1

0,2

0,3

0,4

0,5

0,6

1 10 100

nod

t

fcoupled 6x30-fcoupled 12x60



fcoupled 48x240-fVOF 48x240

α=f

( ) ( )( ) ( )tttN Vji

jLijMinod ∑∈

−=),(

2,,

1 ααδ

9-VOF+two-fluid 27

Conclusions VOF-two-fluid couplingCONCLUSIONS•The grid cell limitation causes some errors in the VOF model, like reconstruction error, advection error and numerical dispersion. Such errors cannot be reduced by applying better and more accurate interface tracking algorithm.•The numerical dispersion can be avoided either by grid refinement of the mesh or switching to the two-fluid model during the simulation. The first solution is effective, when the characteristic size of the chunks does not change much during the transient. On the other hand, when the physical dispersion of the fluids is very fine, the second solution is better.•The study in this paper was performed with the VOF method and the LVIRA piecewise linear reconstruction algorithm, however the results can be applied also for the other VOF reconstruction algorithms.

9-VOF+two-fluid 28

Accuracy of the interface reconstruction - I

Estimate for the accuracy of the interface reconstruction:

nr

jiji ,, αγ ∇=

fluidsseparated0, ≈jiγ

9-VOF+two-fluid 29

fluids mixed1, >jiγ

Accuracy of the interface reconstruction - II

nr

nr



10) Simulations of Kelvin-Helmholtz instability

by


10 - K-H instability 1


Table of contents






Simulations of Kelvin-Helmholtz instability -Contents

Same phenomena simulated with:

CFX - two-fluid model with and without interface sharpening

Fluent - VOF simulation and two-fluid model simulation

Conservative level-set (home-made code)

Additional simulation:Condensation induced water hammer in horizontal pipe


Kelvin-Helmholtz instability VOF model in Fluent

• Kelvin-Helmholtz (K-H) instability is one of the basic instabilities of the two-fluid flows and affects the interface.

• Small density difference and negligible influence of the viscosity allow accurate inviscid linear analysis of the phenomena.

• K-H instability is one of the test cases for the interface tracking methods in: Validation of Advanced Computational Methods for Multiphase Flow Lemonnier , Jamet, Lebaigue, Begell House, 2005.


Thorpe’s experiment(Thorpe, J. Fluid Mech. 39, 1969)

γ=4.13 °

U2

U1

z x

g

z=0

ρ2=780 kg/m3

µ2=0.0015 Pa·s σ=0.04 N/mρ1=1000 kg/m3

µ1=0.001 Pa·s

H=30 mm

h1=15 mm

h2=15 mm

L=1830 (200) mm

Wall, u=v=0

Wall, u=v=0

σρρρρρ gUcr ∆+≥∆21

212 2

21 ρρρ −=∆

σρ /2 gkcr ∆=

crcr k/2πλ =


Thorpe’s experiment vs. analytical solutions

( ) thh

ghU1221

2211

sinρρ

γρρ+

−=

• Undisturbed velocity field (far from closed ends, neglected viscosity):

( )t

hhgh

U1221

1212

sinρρ

γρρ+

−−=

• Experimental onset of instability is 1.88 s (analytical 1.5 s).• Experimental critical wavelength is 25-45 mm (analytical 27 mm).• Thorpe’s experiment is in agreement with results of the inviscid

linear analysis.• Linear analysis is appropriate due to the small density ratio, linear

inviscid theory is insufficient at higher density ratios.• Linear analysis is valid until amplitude is small.


Fluent simulation of K-H instability

• Continuity equation:

• Momentum equation with volumetric surface force:

• Implicit (first order accurate) time scheme was used to calculate velocity field and SIMPLE pressure correction.

• Two simulations were done:– Simulation with explicit time scheme for volume fraction with

geometric VOF surface reconstruction.– Simulation with implicit time scheme for volume fraction without

surface reconstruction.

( ) ( ) 011 =∇+∂∂ Ut

rαραρ

( ) ( )( )( ) ( ) gnnpUUUUUt

T rrrrrrρασµρρ +∇∇−−∇=∇+∇−⋅∇+

∂∂


Fluent simulation - VOF

• Volume fraction field from 0.0 s to 3.55 s.

• Explicit time scheme, with geometric surface reconstruction used.

• Surface is always sharp.

• Grid:29x196, time step=1e-4 s, CPU time=39 h @ 2.4GHz Opteron10 - K-H instability 8

Fluent simulation - VOF

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01x10-9

1x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

dt=1e-3 dt=5e-3 dt=1e-4

measured and analytical time for onset of instability

Growth of instability on mesh with 29x196 volumes and double precision, explicit time scheme for volume fraction, geometric surface reconstruction.


Fluent simulation - no surface reconstruction(4-equation two-fluid model)

• Volume fraction field from 2.0 s to 3.0 s.• Implicit time scheme, without surface reconstruction.• Numerical diffusion of surface can be seen.• Grid:29x196, time step=1e-4 s, CPU time=46 h @2.4 GHz Opteron


Fluent simulation - no surface reconstruction

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51x10-9

1x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

Fluent, dt=1e-3 Fluent, dt=1e-4 CFX, dt=1e-4

Growth of instability on mesh with 29x196 volumes and double precision, implicit time scheme for volume fraction, without surface reconstruction.


Fluent - VOF - conclusions

• With linearised Navier-Stokes equations one can analytically predictonset of K-H instability and the critical wavelength.

• Problem was simulated with Fluent CFD program, solving non-linear Navier-Stokes equations.

• VOF surface tracking in Fluent code was tested

• Fluent simulations:– Onset of instability can be predicted without surface

reconstruction, but there is a significant diffusion of the surface. – With surface reconstruction, surface is always sharp, but onset

of instability cannot be predicted.


CFX - Kelvin-Helmholtz instability

• Homogeneous two-fluid model, with surface sharpening• Viscosity not neglected, 2D• Continuity equation:

• Momentum equation:

• Volumetric surface tension force:

• Additional force as generator of the flow:

( ) ( )( )( ) ADSTFT FgFpUUUUU

t

rrrrrr++−−∇=∇+∇−⋅∇+

∂∂ ρµρρ

( ) ασ ∇∇= nnFSTFrrr

( ) 21 1 ρααρρ −+=

( ) 21 1 µααµµ −+=

( ) ( ) 0=∇+∂∂ Ut

rρρ

( ) γρρρρρρ sin

2/2 21

21 gFAD +⎟⎠⎞

⎜⎝⎛ +−=

r


CFX - Kelvin-Helmholtz instability


• Equations are solved with implicit second order accurate time scheme.

• Space derivates are discretized with high resolution scheme (combination of first and second order accuracy), which reduces numerical diffusion and dispersion.

• CFX uses some special discretization options to keep interface sharp:– A compressive differencing scheme for volume fraction– Special treatment of the pressure gradient and gravity terms to

ensure that flow remain well behaved at the interface• Equations are solved iteratively until prescribed residual is achieved

in each timestep.• Structured grid was used.• Only a section of the tube was simulated with periodical boundary

conditions.

K-H instability - CFX simulation complete tube length simulated

Temporal development of the interface predicted by CFX. K-H instability in experiment is observed in the middle section of the tube after ~1.8 s. Viscosity not neglected, surface tension neglected in particular simulation. Grid:29x1790, time step=1e-4 s, CPU time=20 h @2.4 GHz Opteron


Kelvin-Helmholtz instability with surface tension

•Volume fraction field from 2.0 s to 3.25 s.

•Most unstable wavelength in simulation is 40 mm.

•In experiment λcr is 25-45 mm.

•Analytically predicted λcr is 27 mm.•Grid: 29x196, time step=1e-4 s, CPU time=50 h @3.0 GHz Pentium




Growth of instability on mesh with 29x196 volumes and max residual = 1e-5, double precision, different dt [s].

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

analitical prediction dt=1e-4 dt=1e-3

Visible from volume fraction field

Tough case for CFX-5.7, very small timestep must be used


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01x10-9

1x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

dt=1e-3 dt=5e-3

Growth of instability on mesh with 29x196 volumes and max residual = 1e-5, double precision. There is no need for small timestep in CFX-10.


Kelvin-Helmholtz instability without surface tension

•Volume fraction field from 2.0 s to 3.35 s.

•Most unstable wavelength (λcr) in simulation is 30 mm.

•Analytically predicted λcr is infinitely small (in simulation λcr = 2∆x=2 mm).



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

analitical prediction dt=1e-1 dt=1e-2 dt=1e-3 dt=1e-4

Growth of instability on mesh with 29x196 volumes and max residual = 1e-5, double precision, different dt [s]



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.01x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

double precision single precision

Growth of instability on mesh with 29x196 volumes, dt = 0.01 s and max residual = 1e-4.



0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.51x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

1x10-2

1x10-1

Am

plitu

de [m

]

Time [s]

without surface tension with surface tension

Growth of instability on mesh with 29x196 volumes dt = 1e-4 and max residual = 1e-5, double precision.


Kelvin-Helmholtz instability with CFX

• With linearised Navier-Stokes equations we can analytically predictonset of K-H instability and critical wavelength.

• Problem was simulated with CFD programs, solving non-linear Navier-Stokes equations

• CFX simulations:– Onset of instability can be predicted with CFX-5.7 but extremely

(inconveniently) small time step must be used. Numerical diffusion of surface is relatively small.

– There is no need for such small timestep in CFX-10.0


Conservative Level-Set - Thorpe's K-H instability

•Implemented wetting angle to assure proper behavior of free surface in contact with wall

•Still some problem in contact with wall•Onset of instability: 2.35 s (exp.:1.9 s, anal.:1.5 s)•Critical wavelength: 33 mm (exp.:25-45 mm, anal.:27 mm)

Grid:2440x40, time step=1e-3 s, CPU time=2 h @3.0 GHz Pentium 4Not real aspect ratio


Conservative Level-Set - Thorpe's K-H instability

•Real aspect ratio, only the 65 cm in the middle of the channel is shown

2.88

2.98

3.08

3.18

3.28

3.38

3.43

Time [s]


Direct Contact Condensation

KFKI experiment done at PMK-2 facility in Hungary

9

4

2

101010

8,T48,T38,T28,T1

1

6

5

3

1309 1150

578574593142258

2870

water vapour7

11



Simulation of the pipe in CFX

Cold water injection p=14.5 bar

v=0.242 m/sTL=295 K

Steam tank p=14.5 barTV=470 K

Steam, TV=470 K

Pipe length L=2.87 mPipe diameter d=73 mm

d d



2 continuity equations, 1 Momentum eq., 2 Energy eqs.k-ε turbulence model Thermal phase change model for interfacial heat transfer

Both phases modeled as compressible (density and temperature are pressure dependent)

Steam tables with wider range of pressures and temperatures and more interpolation points was used

Main unknown -> liquid-to-interface heat transfer coefficient

iGLGL am&=Γ α∇=ia( )

LsatV

LsatLGL hh

TTHTCm−

−=,

&



Heat transfer coefficient is calculated using surface renewal theory introduced by Hughes and Duffey 1991

Thermal diffusivity

4/12/1

, /2 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛=LL

LLpLL

acHTC

ρµε

πρ

LpL

LL c

a,ρ

λ=

Turbulence eddy dissipation from k-εturbulence model



2D simulation

Void fraction of water

Temperature of water

Mass transfer rate

Heat transfer coefficient

Grid:10x400, time step=0.03 s, CPU time=9 h @3.0 GHz PentiumNot real aspect ratio



3D simulationVoid fraction

Grid:4000 volumes, time step=0.03 s, CPU time=7 h @3.0 GHz Pentium



Interfacial mass transfer rate vs. time

0

0.005

0.01

0.015

0.02

0.025

0.03

0 2 4 6 8 10time [s]

mc

[kg/

s]

ny=10ny=20ny=40

0

0.005

0.01

0.015

0.02

0.025

0.03

0 2 4 6 8 10time [s]

mc

[kg/

s]

CFL=1CFL=0.67CFL=0.33

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 2 4 6 8 10time [s]

mc

[kg/

s]

dx/dy=1dx/dy=2dx/dy=4dx/dy=8

0

0.005

0.01

0.015

0.02

0.025

0.03

0 2 4 6 8 10time [s]

mc

[kg/

s]

3D2D



Temperature at the top of the pipe

Void fraction of steam

00.10.20.30.40.50.60.70.80.9

1

0 5 10 15 20time [s]

volu

me

fract

ion

T1

T2

T3

T4

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

T1

T2

T3

T4

CFX

Exp.

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

T1

T2

T3

T4

00.10.20.30.40.50.60.70.80.9

1

0 5 10 15 20time [s]

volu

me

fract

ion

T1

T2

T3

T4

?



Small increase of water temperature -> Small condensation rate

Heat transfer coefficient was increased by factor 20 -> better agreement with experiment



T1 measuring point

T1

00,10,20,30,40,50,60,70,80,9

1

0 5 10 15 20time [s]

volu

me

fract

ion

cfx

exp

T1

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

cfx

exp



T2 measuring point

T2

00,10,20,30,40,50,60,70,80,9

1

0 5 10 15 20time [s]

volu

me

fract

ion

cfx

exp

T2

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

cfx

exp



T3 measuring point

T3

00,10,20,30,40,50,60,70,80,9

1

0 5 10 15 20time [s]

volu

me

fract

ion

cfx

exp

T3

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

cfx

exp



T4 measuring point

T4

00,10,20,30,40,50,60,70,80,9

1

0 5 10 15 20time [s]

volu

me

fract

ion

cfx

exp

T4

020406080

100120140160180200

0 5 10 15 20time [s]

tem

pera

ture

[°C

]

cfx

exp



Increased heat transfer coefficient by factor 20

Void fraction of water

Temperature of water

Mass transfer rate

Heat transfer coefficient

Grid:10x400, time step=0.03 s, CPU time=9 h @3.0 GHz PentiumNot real aspect ratio



With increased heat transfer coefficient by factor 20 comparison with experiment is much better.

New correlation for heat & mass transfer in stratified flow is being developed within NURESIM project

Different phenomena occurs– Small condensation rate -> reflection of the wave and

bubble entrapping– Large condensation rate -> bubble entrapping due to

instability


WAHA-maths-numerics 1



11) WAHA code - mathematical model and numerical scheme

by




Table of contentsINTRODUCTION Lectures 1-2

TWO-FLUID MODELS Lecture 3-6INTERFACE TRACKING IN 3D TWO-PHASE FLOWS

Lectures 7-10





WAHA code - mathematical model and numerical scheme - Contents

- WAHA code - introduction

- Two-fluid model of WAHA code- "non-standard" terms in WAHA two-fluid model- Closure equations of WAHA code

- WAHA code numerical scheme- operator splitting- convective terms – 1st step- source terms – 2nd step

- WAHA special models: pipe expansion, contraction (abrupt area change), branch, forces

- Water properties of the WAHA code


WAHA code - mathematical model and numerical scheme - reference

- WAHA code manual, available on internet

www2.ijs.si/~r4www/waha3_manual.pdf


Two-fluid model of WAHA codeSix-equation, two-fluid model, similar to codes like RELAP5, TRAC,

CATHARE, TRACE, etc.

dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

Continuityequations:

wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂

∂θραααραρα

wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂

∂θαρααραρα

Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

LHS: cdifferential terms

RHS: sources




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Pipe elasticity (Wylie, Streeter):)),(()(),( txpAxAtxA e+= dpK

Edp

dD

xAdAe ==

)(




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:1) Equations of state (more later):

. du u

+ p d p

= d kk

k

p

k

uk

k

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ ρρρ




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:2) Virtual mass term is used to obtain hyperbolicity of equations

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂∂

∂∂

∂∂

∂−

xv v-

tv -

xv v+

tv )-(1 CS= CVM f

gfg

fg

mvm ραα)1(




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:3) Interfacial pressure term exists only in stratified flow.

gDSp gfi ))(1( ρραα −−=




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4) Source terms are flow regime dependent. Source terms are:4.1) Terms with Ci - inter-phase drag




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.2a) Terms with inter-phase exchange of mass and energy with:

Γg=-(Qif+Qig)/(hg-hf) - vapor generation term


Two-fluid model of WAHA code

Six-equation, two-fluid model, similar to codes like RELAP5, TRAC, CATHARE, TRACE, etc.

dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.2b) Terms with inter-phase exchange of mass and energy with:

Qik=Hik (Ts-Tk) - interface heat transfer terms




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.3) Terms due to the variable pipe cross-section.




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.4) Ff,wall , Fg,wall - wall friction (Dynamical wall friction model available too).




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.5) Term with g cosθ - volumetric forces.




dxxdA

xAwv )-(1 =

xp Kwv )-(1

x wv )-(1

tp K )-(1 +

t)-(1

ffgffff

ff )(

)(1)()(

)(−−Γ−

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα

dxxdA

xAwv =

xp Kwv

x wv

tp K +

t

ggggggg

gg )(

)(1)()(

)(−−Γ

∂∂−+

∂−∂

+∂∂

∂∂

ραραρα

ραρα


wallfffigrriif

fff

f FgvvvvCx

pCVM xp )-(1

x v wv )-(1 +

t v )-(1 ,cos)1()(||)( −−+−Γ−=

∂∂−−

∂∂+

∂∂

−∂


wallgggigrriig

ggg

g FgvvvvCx

pCVM xp

x v wv +

t v ,cos)(||)( −+−Γ+−=

∂∂++

∂∂+

∂∂

−∂


Momentumequations:

dxxdA

xApwv)(1FvuuQ

xp)w-(1

xpKwv)-p(1

xwv)-(1

ptpKp

tp

x u wv)-(1+

tu )(1

fwallffffgif

fff

fff

f

)()(

1)()(

)()(

)1()(

,* −−−+−Γ−

=∂∂−

∂∂−+

∂−∂

+∂∂−+

∂∂−

∂∂

−∂

∂−

α

ααα

ααραρα

dxxdA

xApwvFvuuQ

xpw

xpKwvp

xwvp

tpKp

tp

x u wvt

u gwallgggggigg

gggg

gg

)()(

1)()()()(

)( ,* −−+−Γ+=

∂∂−

∂∂−+

∂−∂

+∂∂+

∂∂+

∂∂

−+∂

∂ααα

αααραρα

Internalenergy

equations:

Additional closure relations:4.6) Terms for wall heat transfer are neglected in WAHA code.



Closure relations are flow regime dependent:

WAHA flow regime map:

vr

Dispersed flowS = 0

α > 0.95 Droplet flow0.95 > α > 0.5 Transitional flow

α < 0.5 Bubbly flow

Horizontally stratified flowS = 1

Transitional area1 > S > 0

0.5 vcrit vcrit

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=fg

gfcrit gDvρ

αραρρ )1()(

Critical velocity (Kelvin-Helmholtz instability)

ααρ −−= 1XXXXXSS vvninclinatioHKStratification factor S:

( ) ( ) smkg

smkgv

vsmkg

smkgv

vX

mm

mm

mm

mmv2

2

2

2

/

/30000

30000/2500

/2500

0

25003000030000

1

≥

≤≤

<

⎪⎪⎩

⎪⎪⎨

⎧

−−=

ρ

ρ

ρ

ρρ

( )criticalr

criticalrcritical

criticalr

critical

rHK

vLv

vLvvL

vLv

LLv

vLS

1

12

2

211

0

1

≥

≤≤

<

⎪⎪⎩

⎪⎪⎨

⎧

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

( ) ( )0

00

0

000

60

6030

30

0

306060

1

≥

≤≤

<

⎪⎪⎩

⎪⎪⎨

⎧

−−=

θ

θ

θ

θninclinatioX

( ) ( ) smsmv

vsmsmv

vX

m

m

m

mv //100

100/25/25

025100100

1

≥≤≤

<

⎪⎩

⎪⎨

⎧−−=

( ) ( )3

36

6

636

1010105

105

010510105

1

−

−−

−

−−−

≥≤≤⋅

⋅<

⎪⎩

⎪⎨

⎧⋅−⋅−=

αα

αααX

( )( ) ( )( )

( )( ) 3

36

6

6361

101101105

1051

0105101051

1

−

−−

−

−−−

≥−≤−≤⋅

⋅<−

⎪⎩

⎪⎨

⎧⋅−⋅−−=

αα

αααX

Dispersed flow:

Bubbly-to-droplet transition:

Horizontally stratified flow:

Dispersed-to-horizontaly stratified:

k = g, fgffg

ikkki a

vvvvfC 2

2

)()(

81

−−= ρ

( ) ( ) )1( qdropleti

qbubblyii CCC −

−− ⋅=

Inter-phase momentum transfer

3.05.095.0

95.0 =⎟⎠⎞

⎜⎝⎛

−−= rq

rα

( ) ( )dispersedistratifiedii CSCSC −− −+= )1(

LEGEND:- abub/ adrp is modified vapor/liquid volume fraction- d0 is average slug diameter- Re is Reynolds number

Vapor volume fraction:

interfacial frictioncoefficient:

drag coefficient:

interfacial areaconcentration:

5.0<α

gfDfi aCC ρ81=

Re/)Re1.01(24 75.0+=DC

0/6.3 dagf α=

Bubbly flow 95.0>α

⎟⎠⎞

⎜⎝⎛= 1.0,

81max gfDgi aCC ρ

⎟⎟⎠

⎞⎜⎜⎝

⎛ += 5.0,Re

)Re1.01(24min75.0

DC

0/)1(6.3 dagf α−=

Droplet flow



Inter-phase heat&mass trans.

Vapor generation rate Γg is calculated as:

The volumetric heat fluxes are calculated as:

Horizontally stratified flow: Dittus-Boelter type of correlation:

Dispersed-to-horizontaly stratified: interpolation

**fg

igifg hh

QQ−+

−=Γ

)( fSikik TTHQ −=

hk* - specific enthalpies,

Qik - liquid-to-interface and gas-to-interface heat fluxes

,

2k kf k

ik

Nu a kH

α=

0.67

1 (Re 1000) Pr8max 4,

1 12.7 (Pr 1)8

k k k

kk

k

fNu

f

⎛ ⎞⎜ ⎟−⎜ ⎟=⎜ ⎟

+ −⎜ ⎟⎝ ⎠

( ) ( )dispersedifstratifiedifif HSHSH −− −+= )1(

k = g, f


Inter-phase heat&mass trans.Dispersed flow (Downar-Zapolski HRM model):

– Homogeneous Relaxation Model (HRM) – vapor generation Γg:

– vapor heat transfer coefficient Hig:

– fluid heat transfer coefficient Hif:

– vapor or fluid volumetric heat flux Qik:

Legend:

ρm – mixture density

X - quality

θ - relaxation time

η - temperature relation

TS – Saturation temp.

Tk – phase temp.

hk – phase enthalpy

θρ Saturation

mg

XX −−=Γ

))25100(1()10,max()10,max(10 9

56 ηη

αα ⋅+⋅+⋅= −

−

igH

( ) ( ))(

**

fS

fgggSigif

TT

hhTTHH

−

−Γ−−−=

forgkTTHQ fSikik =−= )(


Some other capabilities of WAHA code:

Wall friction (steady):

Minor loses at elbows:

Unsteady wall friction:

Instantaneous relaxation available for inter-phase heat, mass, and momentumtransfer - such results are similar to results of HEM model.

Boundary conditions: closed end, constant pressure (tank), and constant mass flow rate (pump). Tank allows modelling of critical flow at the boundary.

,(1 )

2k k k k

k wall wkm

v vF f

Dρ α ρ

ρ−

=

xDfml ∆

=πβ2

)()()( ttt uns τττ +=

( ) ( )t

un un Tt t t e k c vθτ τ ρ−∆

= − ∆ + ∆transient friction coefficient kTrelaxation time Θ


WAHA numerical schemeNumerical scheme is based on characteristic upwind methods and operator splitting.

Operator splitting:1) Convection and non-relaxation source terms - source terms due to the smooth area change, wall friction and volumetric forces are solved in the first sub step with upwind discretisation:

2) Relaxation (inter-phase exchange) source terms:

Relaxation source terms: inter-phase heat, mass and momentum exchange terms are stiff, i.e., their characteristic time scales can be much shorter that the time scales of the hyperbolic part of the equations. Integration of the relaxation sources within the operator-splitting scheme is performed with variable time steps, which depend on the stiffness of the source terms. Upwinding is not used for calculation of the relaxation source terms.

,S= x

+ t

RELAXATIONNON_

rrr

∂∂

∂∂ ψψ BA

S = dtd RELAXATION

rrψ A



Equation solved:

Eigenvalues and eigenvectors of Jacobian matrix are found:

Source terms are rewritten:

contains source terms due to the variable pipe cross-section contains wall friction and volumetric forces (no derivatives).

Equation rewritten:

. S = x

+ t RN −

− ⋅∂∂

∂∂ rrr

1AC ψψ

1−⋅⋅= LΛLC

. R xAR

x +

t FA 01 =+∂∂+

∂∂

⋅⋅∂∂ −

rrrr ψψLΛL

ARr

FRr

. xxR

xAR

x +

t FA 0111111 =∂∂⋅⋅+

∂∂⋅⋅+

∂∂⋅

∂∂ −−−−−−

rrrr

LΛΛLΛΛLΛL ψψ



Equation rewritten:

. xxR

xAR

x +

t FA 0111111 =∂∂⋅⋅+

∂∂⋅⋅+

∂∂⋅

∂∂ −−−−−−

rrrr

LΛΛLΛΛLΛL ψψ

Modified characteristic variables are introduced as :

.111 xR A R FA δδψδδξrrr −−−−− ⋅+⋅+= LΛLΛL 11

characteristic-like form of Eqs:(allows 2nd order accurate discretisationwith application of slope limiters)

. 0 = x

+ t ∂

∂∂∂ ξξ

rr

Λ

xRAR FA δδψδδξδζrrr 111 −−− ++⋅=⋅= LLLΛΛ

Slopes are not measured by “Modified characteristic variables” but rather with variables:



The combination of the first- and the second-order accurate discretisation is(Godunov’s method):

0 = x

- )( +

x

- )( +

t

- nj

n1j+n

1/2j+--

n-1j

njn

-1/2j++

nj

1+nj

∆∆∆

ξξξξξξrrrrrr

ΛΛ

where elements of diagonal matrices are calculated as:−−++ ΛΛ ,

6,1

6,1

=⋅=

=⋅=−−−−

++++

kf

kf

kkk

kkk

λλ

λλ

61min

61max

1,=k , - xt

2 -

|| ,0 = f

1,=k , - xt

2 +

|| ,0 = f

kk

k

k--k

kk

k

k++k

⎟⎠⎞

⎜⎝⎛

∆∆

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛

∆∆

⎟⎟⎠

⎞⎜⎜⎝

⎛

λφ

λλ

λφ

λλ

and

Different slope limiters – second order correction:MINMOD Van Leer Superbee

max min pp = (0 , (1 , )) φ θ ( ) /( 1)p p p pφ θ θ θ= + + max(0,min(2 ,1),min( ,2))p p pφ θ θ=

( ) ( )

2/12/1

11112/111

2/1112/112/111

+−

−−−−−−+++−−−−

−−−+++−−−−−−++−++−

++∆

∆+

+∆

∆+

∆∆

+∆

∆=

∆∆

jj FFj

A

jA

jj

RRx

AR

xA

R xxt

rrr

rrrr

LΛΛLΛΛLΛΛ

LΛΛLΛLΛLψψψ

Difference scheme (basic variables) used in the WAHA code for convective part is:



Basic variables are ~ primitive variables,

(phasic internal energies uf , ug replaced with the phasic densities, due to the applied water property subroutines)

The preferred set of variables would be conservative variables:

Conservative variables were not used due to:1) Equations of two-fluid model cannot be written in conservative form, due to the pressure

gradient terms, virtual mass terms, interfacial pressure terms, and possibly other correlations that contain derivatives...

2) Oscillations appear in the vicinity of particular discontinuities, if complex systems of equations are solved with conservative variables.

3) "Non-standard" water property subroutines are required that calculate two-phase properties ( ) from the conservative variables ( ).

],[ e ,e )-(1 v ,v )-(1 , ,)-(1 ggffggffgf ραραραραραραϕ =r

ρρα gf ,, ,p u ,u )-(1 , ,)-(1 ggffgf ραραραρα

)u ,u v,v , p, ( = gfgf ,αψr



Relaxation source terms: inter-phase heat, mass and momentum exchange terms are stiff, i.e., their characteristic time scales can be much shorter that the time scales of the hyperbolic part of the equations. Second equation of the operator splitting scheme

is integrated over a single time step with variable time steps that depend on the stiffness of the relaxations and can be much shorter that the convective time step .

Smmmm tS ∆+= −+ )()(11 ψψψψ rrrrr

A

The time step for the integration of the source terms is not constant and is controlled by the relative change of the basic variables. Currently, the maximal relative change of the basic variables in one step of the integration is limited to 0.01 to obtain results that are "numerics" independent. Time step is further reduced when it is necessary to prevent the change of relative velocity direction, or to prevent the change of sign of phasic temperature differences.



Relaxation source terms of the WAHA two-fluid model do not affect the properties of the mixture in a given point: mixture density , mixture momentum , and mixture total energy should remain unchanged after the integration of the relaxation source terms. It is in principle possible to choose a set of basic variables:

that enables simplified integration of the relaxation source terms. Only a system of three differential equations is solved instead of the system of six.

This reduction of the system is only partially taken into account in WAHA numerical scheme: only one relaxation equation for inter-phase friction is solved for the relative velocity. Similar reduction of the thermal relaxation source terms is not used, because it is difficult to calculate the state of the fluid from the variables that are result of such relaxation.

)T ,T vv ,e ,v, ( = gffgmmmmmM ,−ρρρψr


WAHA special models:

i2 i2+1

i1 i1-1i1-2A1 A2

i2+2

k n

• Abrupt area change:

• The abrupt area change model is needed, when flow passes through a sudden expansion or contraction area in a channel

• The implemented abrupt area change models are built on 3 basic assumptions:– steady-state balance conditions for conservative variables across the area change– no generation (or loss) of mass, momentum and energy– preservation of characteristics ξ in each pipe

[ ] 0=∂ Avkkkx ρα

( )[ ] [ ] ApApv xkkkkx ∂=+∂ αρα 2

( )[ ] 0=+∂ Apwv kkkkx ρα



9.0

10.0

11.0

12.0

13.0

14.0

15.0

16.0

0.0 1.0 2.0 3.0 4.0 5.0[m]

[MPa] WAHA, simplified WAHA, conservative

WAHA, cons-char Relap5 mod3.2.2g

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0 5.0[m]

[α] WAHA, simplif ied WAHA, conservative

WAHA, cons-char Relap5 mod3.2.2g

Abrupt area change two-phase test case

Expansion at l = 3 m

pressure VVF

Current abrupt area change models do not contain the generation or loss of momentum and energy, where flow passes the abrupt area change. These models -especially momentum losses - must be included in abrupt area change model to obtain more realistic behaviour of flow on the abrupt area change.Important:

– Abrupt area change model was verified for the single-phase flow only.– Reduced CFL number is recommended with values ~0.5– Minor losses are not included in the abrupt area change model.


WAHA special models:• Branch model

– A branch model is applied to connect three pipes in a single point– Model of branch is based on the abrupt area change model.– Branch model in WAHA3 tested in single phase flow only.

INITIAL CONDITIONS (P1/P6/P2):Temperature T = 293/293/293 K

Vapor velocity v = 1/0.769/0.769 m/sPresure p = 80/80/80 bar

Vapor volume fraction - pure liquid

GEOMETRY (P1/P6/P2):Length l = 10/5/3 m

Diameter d = 7.9/7.9/0.7 mm

Pipe 1...1 2 3 4 100999897...Pipe 6...1 2 3 4 50494847...

Pipe 2..1 2 3029..

const.

Closed end

p=const.

7.8

8.0

8.2

8.4

8.6

8.8

9.0

9.2

0.000 0.002 0.004 0.006 0.008 0.010[s]

p1 [MPa] WAHA

RELAP

7.8

8.0

8.2

8.4

8.6

8.8

9.0

9.2

0.000 0.002 0.004 0.006 0.008 0.010[s]

p2 [MPa] WAHA

RELAP

7.8

8.0

8.2

8.4

8.6

8.8

9.0

9.2

9.4

0.000 0.002 0.004 0.006 0.008 0.010[s]

p3 [MPa] WAHA

RELAP



Pipe 1 . ...1Ψv

2Ψv

1−nΨv

nΨv

1Fv

nFv

2Fv

2−nFv

0Fv

1−nFv

• Forces- from American National Standard, ANSI/ANS-58.2-1988, Revision of ANSI/ANS-58.2-1980, “Design basis for protection of light water nuclear power plant against the effects of postulated pipe rupture”,, Appendix A: Derivation of fluid force equations.- WAHA code can calculate forces on the 3D piping system. - Forces are calculated on the edges of the volumes.

is dynamic fluid thrust force vector on pipeFr

( ) ( ). . . . .in out pipe

in out ambient pipec v c.s A A A c v

d v dVF v v dA pdA pdA p dA gdV

dtρ

ρ ρ⎡ ⎤⎢ ⎥= − + ⋅ + + + −⎢ ⎥⎣ ⎦∫ ∫ ∫ ∫ ∫ ∫

rr r r rr r r r

∆ x i

irv

Ai

pi vi

∆ xi+1

1+irv

Ai+1

pi+1 vi+1

iFv



• Forces force on the Edwards's pipe

-40000-35000

-30000-25000-20000

-15000-10000

-50000

0 0.2 0.4 0.6 0.8

time (s)fo

rce

(N)

total forceA*p

Edward'spipe - force on the pipe

-40000

-35000

-30000

-25000

-20000

-15000

-10000

-5000

00 0.005 0.01 0.015

time (s)

forc

e (N

)

Total forcepressure*crossection

GS7

PIPE MEMBRANE

L = 4.097 m, A = 4.2 10-2 m2

CLOSED END

GS5 GS1


Water properties of the WAHA code:

250 300 350 400 450 500 550 600 650-100

-50

0

50

100

150

200

250

Negative pressure area in the Waha

Pre

ssur

e [b

ar]

Temperature [K]

Satura tionVapor spinodal (extended)Liquid spinodal (extended)Negative pressureVapor spinodalLiquid spinodal

kk

k

p

k

u

k du u

+ p d p

= dk

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂ ρρρ

• Thermodynamic propertiesof liquid and steam are based on NBS/NRC-84formulation

• Pre-tabulated and stored in ASCII file:

– 400 pressures (-95 – 1000 bar)– 500 temperatures (273 – 1638 K)

• Extended into negative pressure (up to –95 bar)

• Extended liquid and vapor spinodal lines

EoS:



900000

1100000

1300000

1500000

1700000

1900000

2100000

0 0.5 1 1.5 2length (m)

p(Pa

)

RELAP5 steam tables t=0.60514ms

WAHA steam tables t=0.60008ms

900000

1100000

1300000

1500000

1700000

1900000

2100000

0 0.5 1 1.5 2length (m)

p(Pa

)RELAP5 steam tables t=1.0008ms

WAHA steam tables t=1.0065ms

Single-phase vapor wave - pressure.Single-phase liquid wave - pressure.

- Comparison: WAHA code with the WAHA steam tables and WAHA code with the steam tables of RELAP5/MOD3.2.2 Gamma (internal version of WAHA code)- Propagation of pressure waves in single-phase liquid and in single-phase vapor shock tube- Differences are more due to the slightly different time steps than due to the different water properties.



Edwards pipe problem - vapor volume fraction.Edwards pipe problem - pressure.

- Comparison: WAHA code with the WAHA steam tables and WAHA code with the steam tables of RELAP5/MOD3.2.2 Gamma (internal version of WAHA code)- Edwards pipe problem - rapid depressurization of the hot liquid in a horizontal pipe - Calculations were performed with instantaneous relaxation of inter-phase heat, mass and momentum transfer.

0.E+00

1.E+06

2.E+06

3.E+06

4.E+06

5.E+06

6.E+06

7.E+06

0 0.1 0.2 0.3 0.4 0.5 0.6

time (s)

p (P

a)

RELAP5 steam tables

WAHA steam tables

-2.E-01

0.E+00

2.E-01

4.E-01

6.E-01

8.E-01

1.E+00

1.E+00

0 0.1 0.2 0.3 0.4 0.5 0.6

time (s)

vapo

r vo

lum

e fr

actio

n

RELAP5 steam tables

WAHA steam tables

WAHA-simulations 1



12) WAHA code - simulations

by


WAHA-simulations 2




Lectures 7-10




WAHA-simulations 3

WAHA code - simulations

GS7

PIPE MEMBRANE

L = 4.097 m, A = 4.2 10-2 m2

CLOSED END

GS5 GS1

Transient:- rapid depressurization of the hot liquid from the horizontal pipe

Aim: - test case for codes used to simulate LOCA accidents in NPPs- to verify several WAHA code physical models like:

- propagation of the rapid depressurization wave- the pressure undershoot model- the flashing model- propagation of the void fraction wave- the two-phase critical flow- transition into the horizontally stratified flow

A. R. Edwards, T. P. O'Brien, Studies of phenomena connected with the depressurization of water reactors, Journal of the British Nuclear Energy Society, 9, 125-135, 1970.

Edwards pipe:

WAHA-simulations 4


Edwards pipe:GS7

PIPE MEMBRANE

L = 4.097 m, A = 4.2 10-2 m2

CLOSED END

GS5 GS1

Boundary conditions: - left: closed end- right: constant pressure pT = 1 bar- cross-section of the break is 12.5% smaller than cross-section of the pipe.

Initial conditions:-velocity: stagnant liquid-pressure p = 70 bar-temperature T = 515 K

Conclusion:- The accuracy of the WAHA code predictions is comparable to the accuracy of

the RELAP5 predictions despite a much simpler flow regime map and absence of a special critical flow model.

WAHA-simulations 5


0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

7

Pre

ssur

e in

GS

1 [M

Pa]

Time [s ]

ExperimentWAHA3

Edwards pipe:

Pressure in GS1 [MPa]

WAHA-simulations 6


0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

7

Pre

ssur

e in

GS

7 [M

Pa]

Time [s ]

ExperimentWAHA3

Edwards pipe:Pressure in GS7 [MPa]

WAHA-simulations 7


0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Vap

or v

olum

e fra

ctio

n in

GS

5 []

Time [s ]

ExperimentWAHA3

Edwards pipe:

Vapor volume fraction in GS5

WAHA-simulations 8


0 0.1 0.2 0.3 0.4 0.5 0.6250

300

350

400

450

500

550

Tem

pera

ture

in G

S5

[K]

Time [s ]

ExperimentWAHA3 - liquidWAHA3 - vapor

Edwards pipe:Temperature in GS5 [K]

WAHA-simulations 9


Super Moby Dick exp.

“Transient”:- high pressure “Super Moby Dick”

experiment performed at CEA in Grenoble in 80’s - steady state two-phase critical flashing flow in the convergent-divergent nozzle (Faucher).

Aim: - to verify the Homogeneous-Relaxation

Model (Lemonnier) used in the WAHA code to model inter-phase heat and mass transfer in dispersed flow

- to verify conservation properties of the WAHA code in the variable cross-section geometry.E. Faucher, Simulation numerique des ecoulements unidimensionnels instationnaires avec autovaporisation, Doctorat de l’universe Paris Val de Marne, (2002).

inlet outlet

WAHA-simulations 10


Super Moby Dick exp.

Boundary conditions: -inlet: constant pressure pR = 80 bar, temperature T = 549.6 K (20/465.7)-outlet: constant pressure pL = 47 bar, temperature T = 465.5 K

Conclusion:- advantage of the WAHA code: critical flow is simulated with standard

discretisation and boundary conditions - non-conservative numerical scheme:

- overall loss of mass flow along the nozzle is less than 0.7 %- maximum non-conservation is less than ~1.5% (strong phase changes).

inlet outlet

WAHA-simulations 11


Super Moby Dick exp. Why critical flow?

1

10

100

1000

10000

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[m/s] cVliqVvap

WAHA-simulations 12


Why flashing flow?

(VVF=α)Super Moby Dick exp.

455

460

465

470

475

480

485

490

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[K]

0.0

0.1

0.2

0.3

0.4

0.5[VVF]T sat T liq VVF

410

420

430

440

450

460

470

480

490

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[K]

0.0

0.2

0.4

0.6

0.8

1.0[VVF]T sat T liq VVF

pinlet = 20 bar poutlet = 4 bar < pSAT

pinlet = 20 bar poutlet = 16 bar > pSAT = cca 13 bar

WAHA-simulations 13


Pressure [bar]Super Moby Dick exp.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[MPa] WAHA 1 WAHA 2 WAHA 3

EXP 1 EXP 2 EXP 3

WAHA-simulations 14


Vapor volume fraction αSuper Moby Dick exp.

0.00.10.20.30.40.50.60.70.80.91.0

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[α] WAHA 1 WAHA 2 WAHA 3

EXP 1 EXP 2 EXP 3

WAHA-simulations 15


Mass flow rate [kg/s]Super Moby Dick exp.

10.0

12.0

14.0

16.0

18.0

20.0

0.00 0.15 0.30 0.45 0.60 0.75 0.90[m]

[kg/s] WAHA 1 WAHA 2 WAHA 3

EXP 1 EXP 3 EXP 2

WAHA-simulations 16


TANK PIPE

L = 36 m, A = 2.85 10-4 m2, e = 1.6 mm,E = 120 GPa

Initial flow direction

Measuring point

VALVE

Transient:- column separation water hammer induced due by rapid valve closure.

Aim: - fundamental benchmark for two-phase computer codes because of the simple:

- geometry, - initial conditions and - water hammer initiating mechanism.

A. R. Simpson, 1986, Large water hammer pressures due to column separation in sloping pipes, Ph.D thesis, The University of Michigan, Department of Civil Engineering.

Simpson’s pipe:

WAHA-simulations 17


Simpson’s pipe:

Boundary conditions: -right: closed end (valve)-left: constant pressure pT = 3.419 bar

Initial conditions:-velocity v = 0.4 m/s-pressure p = 3.419 bar-temperature T = 296.3 K

Effect of the elasticity taken into account.

Conclusion: At low temperatures flashing and condensation of the steam are not governed by the heat and mass transfer between both phases, but by the dynamics of the liquid column (energy equations are not needed).

TANK PIPE

L = 36 m, A = 2.85 10-4 m2, e = 1.6 mm,E = 120 GPa


Measuring point

VALVE

WAHA-simulations 18


0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

Pre

ssur

e hi

stor

y ne

ar th

e va

lve

[MP

a]

Time [s ]

ExperimentWAHA3 - e las tic pipeWAHA3 - s tiff pipeWAHA3 - uns teady friction

Simpson’s pipe:

Pressure near the valve [MPa]

WAHA-simulations 19


0 0.05 0.1 0.15 0.2 0.25 0.30

0.002

0.004

0.006

0.008

0.01

0.012

VV

F hi

stor

y ne

ar th

e va

lve

[]

Time [s ]

WAHA3 - e las tic pipeWAHA3 - s tiff pipeWAHA3 - uns teady friction

Simpson’s pipe:Vapor volume fraction α near the valve

WAHA-simulations 20


PPP pipeline (A. Dudlik, FraunhoferInstitut Umwelt-, Siecherheits-, Energietechnik UMSICHT, Oberhausen).

- database with over 400 experimentsperformed at UMSICHT’s test loop (totallength ~ 200 m)

- advanced measuring equipment (wiremesh sensor – void distribution)

WAHA-simulations 21


PPP pipeline:

bridgeclosure valve

turningpoint

FP 1

FP 2

FP 3

B 2

P23 P01 P02 P03 P06

P09 P12

P15

P18

0 m

0.2 m 34.5 m

44.4 m 50.9 m

67.0 m

67.9 m

81.6 m

75.5 m90.7 m

84.6 m

137.0 m

139.4 m

145.5 m

WM 60.8 m

77.5 m88.7 m

142.9 m

146.8 m

149.4 m-18.2 m

-14.5 m

-0.2 m

-8.7 m

VALVE

TANK

PIPELINE

N = 1

WAHA-simulations 22


PPP pipeline:

Modelled section:- L = 149.5 m (valve – tank)

Transient:- column separation water hammer induced due by rapid valve closure.

Boundary conditions: -left: closed end (valve)-right: constant pressure

Initial conditions:-case 135: p = 1.13 bar, v = 3.975 m/s, T = 293.7 K-case 307: p = 9.92 bar, v = 4.009 m/s, T = 392.1 K-case 329: p = 10.18 bar, v = 3.975 m/s, T = 419.6 K

34.50m 6.50m 6.50m

1.00

m

46.50 m3.00m 7.50m

10.0

0m

4.00 m

2.00

m

TANK

VALVE

PIPELINE

2.50

3.50

WAHA-simulations 23


0 5 10 15 201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

135:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

WAHA3 - s teady s ta tePPP pipeline:

Case 135 – Steady state – pressure in P03

WAHA-simulations 24


0 1 2 3 4 5 6 70

10

20

30

40

50

60

135:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

ExperimentWAHA3

PPP pipeline:Case 135: Pressure near the valve [MPa]

WAHA-simulations 25


0 1 2 3 4 5 6 70

10

20

30

40

50

60

307:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

ExperimentWAHA3

PPP pipeline:Case 307: Pressure near the valve [MPa]

WAHA-simulations 26


0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

50

55

329:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

ExperimentWAHA3

PPP pipeline:

Case 329: Pressure near the valve [MPa]

WAHA-simulations 27


0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

135:

Vap

or v

olum

e fra

ctio

n in

P03

[bar

]

Time [s ]

ExperimentWAHA3

PPP pipeline:Case 135: α near the valve [MPa]

WAHA-simulations 28


0 1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

307:

Vap

or v

olum

e fra

ctio

n in

P03

[bar

]

Time [s ]

ExperimentWAHA3PPP pipeline:

Case 307: α near the valve [MPa]

WAHA-simulations 29


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

329:

Vap

or v

olum

e fra

ctio

n in

P03

Time [s ]

ExperimentWAHA3

PPP pipeline:

Case 329: α near the valve [MPa]

WAHA-simulations 30


0 1 2 3 4 50

10

20

30

40

50

60

329:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

ExperimentWaha3Waha HEMWaha neg. pressure

PPP pipeline:

Case 329: pressure near the valve [MPa]

Influence of different

relaxationmodels

WAHA-simulations 31


0 2 4 6 8 100

5

10

15

20

25

30

35

40

45

50

55

329:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

ExperimentWAHA3RELAP5

PPP pipeline:Case 329: pressure near the valve [MPa]

Comparisonto RELAP5

code

WAHA-simulations 32


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

329:

Vap

or v

olum

e fra

ctio

n in

P03

Time [s ]

ExperimentWAHA3RELAP5

PPP pipeline:Case 329: α near the valve [MPa]

Comparisonto RELAP5

code

WAHA-simulations 33


0 1 2 3 4 50

5

10

15

20

25

30

35

40

45

50

329:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

Steady s ta te , T = 419.6 KSteady s ta te , T = 424.0 KSteady s ta te , T = 415.0 K

PPP pipeline:

Case 329: pressure near the valve [MPa]

Influence of different

initialtemperature

WAHA-simulations 34


0 1 2 3 4 50

5

10

15

20

25

30

35

40

45

50

329:

Pre

ssur

e hi

stor

y in

P03

[bar

]

Time [s ]

dx = 0.5 m (N = 299)dx = 0.25 m (N = 598)dx = 1.0 m (N = 150)

PPP pipeline:Case 329: pressure near the valve [MPa]

Gridrefinement

WAHA-simulations 35


CWHTF:

Preferences:- database with 20 experiments

performed at FZR’s cold waterhammer test facility (CWHTF)

- two discontinuities initiallypresent in the pipe that propagate with different velocity.

Transient:- overpressure accelerates a column

of liquid water into vacuum at the closed vertical end of the pipe

TANK

VALVE

CLOSEDEND

p2evacuationpressure

p1

Vapor vol.fract.

α = 1.0

Water onlyα = 0.0

LE

LV

E. Altstadt, H. Carl, R. Weiss, CWHTF - |Cold Water-Hammer Test Facility, Forschungszentrum Rossendorf.

WAHA-simulations 36


CWHTF:TANK

VALVE

CLOSEDEND

p2evacuationpressure

p1

Vapor vol.fract.

α = 1.0

Water onlyα = 0.0

LE

LV

Experiment labeled “150601”:

Boundary conditions: - right: closed end- left: constant pressure (tank)

constant pressure (precise geometry - pipe)

Initial conditions:- p1 = 1 bar- p2 = 29 mbar- v = 0 m/s- T ~ 295 K

Warnings:- absence of non-condensable gas model in the WAHA- no FSI effects considered

WAHA-simulations 37


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

30

35

40

45

50

Pre

ssur

e hi

stor

y ne

ar th

e cl

osed

end

[bar

]

Time [s ]

ExperimentTankPipe

Pressure near the closed end [bar]

CWHTF:

WAHA-simulations 38


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

VV

F hi

stor

y ne

ar th

e cl

osed

end

[bar

]

Time [s ]

TankPipe

α near the closed end [bar]

CWHTF:

WAHA-simulations 39


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

5

10

15

20

25

30

35

40

45

50

Pre

ssur

e hi

stor

y ne

ar th

e cl

osed

end

[ba

r]

Time [s ]

ExperimentTank (ideal gas)Pipe (idea l gas )

Pressure near the closed end [bar]

CWHTF:

Liquid – ideal gas mixture

WAHA-simulations 40


KFKI exp.:Preferences:

- condensation induced waterhammer was observed in the steam-line of the integral experimental device PMK-2 that is located at the Hungarian Atomic Energy Research Institute

Cold waterinjection

Steam tank

Steam

1

...2 3 4 5

59

58575655...

INITIAL CONDITIONS:steam temperature Ts = 470 Kliquid temperature Tl = 295 Kliquid velocity vl = 0.242 m/s

pressure p = 14.5 bar

GEOMETRY:pipe length l = 2.95 m

pipe diameter d = 73 mmnumber of volumes N = 59

Experiment labeled “E22”:

Boundary conditions: - right: steam tank- left: cold water intake (constant velocity)

...very complicated thermally controled transientH.M. Prasser, G. Ezsol, G. Baranyai, PMK-2 water hammer tests, condensation caused by cold water injection into main steam-line of VVER.440-type PWR, WAHALoads project deliverable D48, 2004.

WAHA-simulations 41


KFKI exp.:Transient:

- liquid flows into the pipe (steam)- condensation rate increases and

consequently increases relative vapor velocity over the liquid head

- liquid-vapor surface becomes wavy- amplitude of the waves increase until the

liquid slug is formed that captures the vapor bubble

- condensation of the entrapped vapor bubble accelerates columns of liquid on both sides of the bubble

- strong water hammer appears when the bubble is condensed and two liquid columns collide.

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Profile at t = 3.75 s

E05

: Liq

uid

volu

me

fract

ion

[ ]

Length [m]

Liquid

Vapor

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Profile at t = 5.08 s

E05

: Liq

uid

volu

me

fract

ion

[ ]

Length [m]

Vapor bubble Liquid slug Vapor

WAHA-simulations 42


4.75 4.8 4.85 4.9 4.950

20

40

60

80

100

120

140

160

180

200

Pre

ssur

e [b

ar]

Time [s ]

ExperimentWAHA3

Pressure near the water intake [bar]

KFKI exp.:

detail

dispersed

WAHA-hands-on 1



13) Hands on: simulation of two-phase water hammer transient and two-phase critical flow

by


WAHA-hands-on 2




Lectures 7-10

ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS11) WAHA code - mathematical model and numerical scheme12) WAHA code - simulations13) Hands on: simulation of two-phase water hammer

transient and two-phase critical flow14) Fluid-structure interaction in 1D piping systems


WAHA-hands-on 3

WAHA code - mathematical model and numerical scheme - reference

- WAHA code manual, available on internet

WAHA-hands-on 4

title simpson test - elastic pipe*--------time constants----------------------------------* beg end maj_out min_out diff restarttime00 0 2.0e-1 4.e-3 2.e-3 0.80 0.01time01 0.2 3.0e-1 2.e-3 1.e-3 0.80 0.01* fluid order abr_model eig_val_out extend_out maj_resultsswitch 1 2 3 1 1 1* ambient_press force_outforce 1.e+5 0.001*-------------minor output --------------------------------* pipe volume variableprint00 1 100 1print02 1 100 2print03 1 100 3print04 1 100 5print05 1 100 7print06 1 100 9print07 1 20 1print08 1 20 2print09 1 20 3*-------------pipes --------------------------------* type namecomp001ty pipe cev_01* length elast thick rough w.fr.f p.fr.f h.m.tr. nodscomp001g0 36.0-0 1.2e11 1.6e-3 0.0e0 0 0 0 100* area incl azim f_coeff whichcomp001g1 2.85e-4 0. 0.0 1.0 30+ 2.85e-4 0. 0.0 1.0 100* type press alpha_g velf velg uf ug wch_nodscomp001s0 agpvu 3.419e5 0.0 0.4 0.0 97.67e3 0.0 100* from tocomp001c0 002-99 000-00*-------------------------------------------------------* type namecomp002ty tank tank_01* length elast thick rough w.fr.f p.fr.f h.m.tr. nodscomp002g0 0.0-0 0.0 0.0 0.0e0 0 0 0 0* area incl azim f_coeff whichcomp002g1 2.85e-4 0. 0.0 1.0 0* type press alpha_g velf velg uf ug wch_nodscomp002s0 agpvu 3.419e5 0.0 0.0 0.0 97.67e3 0.0 0* from tocomp002c0 000-00 001-01***************************************************************

end

- input file for Simpson's water hammer transient

TANK PIPE

L = 36 m, A = 2.85 10-4 m2, e = 1.6 mm,E = 120 GPa


Measuring point

VALVE

WAHA-hands-on 5

- input file for Moby-Dick two-phase critical flow transient 1/3

title - two-phase critical flashing flow in the super moby dick nozzle

* case 20B192C: pin=20 bar, Tin=192.3 C

* case 80B276C: pin=80.0bar Tin=275.5 C

* case 120B305C: pin=120.0bar Tin=305.7 C

*--------time constants----------------------------------

* beg end maj_out min_out diff restart

time00 0 0.2e+0 5.0e-3 8.0e-4 0.80 1.0

* fluid order abr_model eig_val_out extend_out maj_results

switch 1 2 3 0 1 1

*-------------pipes --------------------------------

* type name

comp001ty pipe

* length elast thick rough w.fr.f p.fr.f h.m.tr.nods

comp001g0 0.9 0.0 1.588e-3 0.0 0 0 0 90

* crossct inclin azim which_nodes

comp001g1 0.003494 0. 0.0 1.0 1

+ 0.0032 0. 0.0 1.0 2

+ 0.0029 0. 0.0 1.0 3

... slide 2/3

+ 4.126256E-3 0. 0.0 1.0 89

+ 4.266039E-3 0. 0.0 1.0 90

* type press alpha_g velf velg tf tgwch_nods

*comp001s0 agpvt 20.08e5 0.00 0.1e0 0.0 465.5 0.0 90

comp001s0 agpvt 80.00e5 0.00 0.1e0 0.0 549.6 0.0 90

*comp001s0 agpvt 120.06e5 0.00 0.1e0 0.0 578.7 0.0 90

* from

comp001c0 002-99 003-01

... slide 3/3

{

WAHA-hands-on 6


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40

45

Noz

zle

cros

s-se

ctio

n [c

m2 ]

Length [m]

comp001g1 0.003494 0. 0.0 1.0 1

+ 0.0032 0. 0.0 1.0 2

+ 0.0029 0. 0.0 1.0 3

+ 0.0024 0. 0.0 1.0 4

+ 0.0020 0. 0.0 1.0 5

+ 0.0017 0. 0.0 1.0 6

+ 0.0013 0. 0.0 1.0 7

+ 0.0008 0. 0.0 1.0 8

+ 0.0005 0. 0.0 1.0 9

+ 0.000350 0. 0.0 1.0 10

+ 0.0003183 0. 0.0 1.0 46

+ 3.5791826E-4 0. 0.0 1.0 47

+ 3.9990842E-4 0. 0.0 1.0 48

+ 4.4422692E-4 0. 0.0 1.0 49

+ 4.908738E-4 0. 0.0 1.0 50

+ 5.398491E-4 0. 0.0 1.0 51

+ 5.911529E-4 0. 0.0 1.0 52

+ 6.4478494E-4 0. 0.0 1.0 53

+ 7.007455E-4 0. 0.0 1.0 54

+ 7.590344E-4 0. 0.0 1.0 55

+ 8.196517E-4 0. 0.0 1.0 56

+ 8.825974E-4 0. 0.0 1.0 57

+ 9.4787165E-4 0. 0.0 1.0 58

+ 1.0154742E-3 0. 0.0 1.0 59

+ 1.085405E-3 0. 0.0 1.0 60

+ 1.1576643E-3 0. 0.0 1.0 61

+ 1.232252E-3 0. 0.0 1.0 62

+ 1.3091682E-3 0. 0.0 1.0 63

+ 1.3884127E-3 0. 0.0 1.0 64

+ 1.4699855E-3 0. 0.0 1.0 65

+ 1.5538868E-3 0. 0.0 1.0 66

+ 1.6401168E-3 0. 0.0 1.0 67

+ 1.7286747E-3 0. 0.0 1.0 68

+ 1.8195612E-3 0. 0.0 1.0 69

+ 1.9127764E-3 0. 0.0 1.0 70

+ 2.0083198E-3 0. 0.0 1.0 71

+ 2.1061913E-3 0. 0.0 1.0 72

+ 2.2063916E-3 0. 0.0 1.0 73

+ 2.30892E-3 0. 0.0 1.0 74

+ 2.413777E-3 0. 0.0 1.0 75

+ 2.5209623E-3 0. 0.0 1.0 76

+ 2.630476E-3 0. 0.0 1.0 77

+ 2.742318E-3 0. 0.0 1.0 78

+ 2.8564883E-3 0. 0.0 1.0 79

+ 2.9729874E-3 0. 0.0 1.0 80

+ 3.0918144E-3 0. 0.0 1.0 81

+ 3.2129711E-3 0. 0.0 1.0 82

+ 3.336455E-3 0. 0.0 1.0 83

+ 3.4622678E-3 0. 0.0 1.0 84

+ 3.5904084E-3 0. 0.0 1.0 85

+ 3.7208776E-3 0. 0.0 1.0 86

+ 3.8536756E-3 0. 0.0 1.0 87

+ 3.9888015E-3 0. 0.0 1.0 88

+ 4.126256E-3 0. 0.0 1.0 89

+ 4.266039E-3 0. 0.0 1.0 90

inlet outlet

WAHA-hands-on 7


*-------------------------------------------------------

* type name

comp002ty tank tank_01

* length elast thick rough w.fr.f p.fr.f h.m.tr. nods

comp002g0 0.0-0 0.0 0.0 0.0 9 8 8 0

* crossct inclin azim f_coeff which_nodes

comp002g1 0.003494 0.0 0.0 1.0 0


*comp002s0 agpvt 20.08e5 0.0 0.1 0.0 465.5 0.0 0

comp002s0 agpvt 80.00e5 0.0 0.1 0.0 549.6 0.0 0

*comp002s0 agpvt 120.06e5 0.0 0.1 0.0 578.7 0.0 0

* from to

comp002c0 000-00 001-01

*-------------------------------------------------------

* type name

comp003ty tank tank_03

* length elast thick rough w.fr.f p.fr.f h.m.tr. nods

comp003g0 0.0-0 0.0 0.0 0.0 9 8 8 0

* crossct inclin azim f_coeff which_nodes

comp003g1 4.266039E-3 0.0 0.0 1.0 0


*comp003s0 agpvt 7.000e5 0.0 0.1 0.0 465.5 465.5 0

comp003s0 agpvt 47.000e5 0.0 0.1 0.0 465.5 465.5 0

*comp003s0 agpvt 77.000e5 0.0 0.1 0.0 465.5 465.5 0

* from to

comp003c0 001-99 000-00

***************************************************************

*

end

1D-piping-FSI 1



14) Fluid-structure interaction in 1D piping systems

by


1D-piping-FSI 2




Lectures 7-10


and two-phase critical flow.14) Fluid-structure interaction in 1D piping systems DNS OF THE PASSIVE SCALAR TRANSFER

IN THE CHANNEL AND FLUME Lectures 15-18

1D-piping-FSI 3

Fluid-structure interaction in 1D piping systems - Contents

- References- Introduction - types of fluid-structure interactions- Typical mathematical models for 1D FSI in piping systems- Examples- Numerical methods- Two phase FSI

1D-piping-FSI 4

Fluid-structure interaction in 1D piping systems - References

• A. S. Tijsseling, Fluid-structure interaction in liquid-filled pipe systems a review, Journal of Fluids and Structures, 10 109-146, 1996.

• D. C. Wiggert, A. S. Tijsseling, Fluid transients and fluid-structure interaction in flexible liquid-filled piping, ASME Applied Mechanical Review, 54 5 455-481, 2001.

• D. J. Leslie, A. E. Vardy, Practical guidelines for fluid-structure interaction in pipelines a review, Proc. of the 10th international meeting of the work group on the behaviourof hydraulic machinery under steady oscillatory conditions, 2001.

• D. C. Wiggert, Coupled transient flow and structural motion in liquid-filled pipingsystems a survey, Proc. of the ASME Pressure Vessels in Piping Conference, Paper 86-PVP-4, 1986.

• R. A. Valentin, J. W. Phillips, J. S. Walker, Reflection and transmission of fluid transients at an elbow, Transactions of SMiRT5, Paper B 2-6, 1979.

• R. Skalak, An extension of the theory of waterhammer, Transactions of the ASME, 78 105-116, 1956.

• A. Bergant, A. R. Simpson, A. S. Tijsseling, Water hammer with column separation A historical review, Journal of Fluids and Structures, 22 2 135-171, 2006

1D-piping-FSI 5

Fluid-structure interaction in 1D piping systems - Introduction

1D-piping-FSI 6


• Fluid-Structure Interaction = FSI

Moving fluid Deformed structure

Redistribution of the pressure load

Pressure load

• Consequences: Noise, vibration, displacements and stresses (pipe) and extreme pressures (fluid).

• Statistical data USA (1986-2000) "Failed Pipe (Internal Force)“:5979 accidents, 357 deaths, 3494 injuries, costing over $1 billion.

• With appropriate FSI analysis: reduction of the extreme pressures in the fluid and maximum stresses in thestructure, frequency change, energy transfer control and prevention of the failures.

1D-piping-FSI 7


• FSI during fast transients: accidental condition

• Wylie about FSI:– 98% pipelines not subjected – no simple FSI inspection criterion– FSI analysis necessary for all

pipelines!

• FSI analyses have been performed onlyfor the most important pipelines

• Conventional simulation of the fast transient: NO FSI (stiff and supported pipe)

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

2

Pre

ssur

e [M

Pa]

Time [s ]

Case 4, with FSI (soft pipe)Case 5, no FSI (s tiff pipe)

} 50% higher maximal pressure!

Pressure near the valve (rapid valve closuretransient, full axial coupling, free valve)

1D-piping-FSI 8


- Common Sources of FSI 1. Long lengths of unsupported or poorly supported pipework2. Unsupported/unrestrained elbows3. Unsupported/unrestrained valves4. T-junctions5. Transient in the fluid (liquid density)

- Combinations of the above - features numbered 1-5 are independently important while combinations are very important.

- Vibrating machinery can induce vibrations in the pipeline. The intensity will depend on the proximity of the frequency of vibration to a natural frequency of the pipeline.

Practice:vulnerable are

parts submerged to cavitation or

oxidation

1D-piping-FSI 9


• Another example (with and without FSI) – Tank-straight pipe-valve system:

Pressure near the rapidly closed valve (rapid valve closure transient, valve is fixed no jucntion coupling effect, only Poisson coupling – pipe

breathing

PIPE VALVE Pipe properties: L = 20 m, R = 398.5 mm,e = 8 mm, E = 210 GPa, ν = 0.3, ρ s = 7900 kg/m 3

Inital flow direction

Measuring point

TANK

Initial conditions: v = 1 m/s, p = 0 Pa,ρf = 1000 kg/m 3

1D-piping-FSI 10


• Another example (with and without FSI) – Tank-straight pipe-valve system:

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

2Pres s ure his tory near the va lve - de ta il

Pre

ssur

e [M

Pa]

Time [s ]

no FSIwith FSI

0 0.5 1 1.5 2 2.5 3 3.5 4-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3Pres s ure his tory near the va lve

Pre

ssur

e [M

Pa]

Time [s ]

no FSIwith FSI

1D-piping-FSI 11

Fluid-structure interaction in 1D piping systems - Types of FSI

-There are several types of waves that characterize FSI: - axial,- flexural,- rotational,- radial and- torsional stress waves in the pipeline- pressure waves in the fluid.

- According to the interaction between these waves one can differentiate the following types of the coupling:- Poisson coupling: pressure waves in the fluid are coupled with axial waves in the structure and changes of the pipe cross-section.Figurativelly known as pipe breathing- Junction coupling: different waves are appropriately coupled together at geometric changes (elbows, area changes, valves, junctions, etc.).- Friction coupling: axial waves in the structure are initiated due to the difference between fluid and structure velocity – less important.

1D-piping-FSI 12


- Poisson coupling leads to precursorwaves - these are stress wave induceddisturbances in the liquid, which travelfaster than and hence ahead of, theclassical waterhammer waves.

- The interaction is always caused bydynamic forces which act simultaneouslyon fluid and pipe. It is convenient to classify the dynamic forces into twogroups:- distributed forces (Poisson and frictioncoupling)- local forces (junction coupling)

1D-piping-FSI 13


- Classification according to the fact whether the fluid knows for pipe deformations or not:

- One-way coupling or uncoupled calculation (fluid transient is evaluated in undeformed structure). Most of the FSI analyses in the past in fluid-filled systems comprised two separate analyses undertaken sequentially (uncoupled calculation). Fluid-transient code is used to determine pressure and velocity histories in rigid and anchored structure, which are used as input to a structural dynamics code. It is also possible to couple codes in each calculation time step (one-way coupling). The results are identical in both cases.

- Two-way coupling - most recent FSI methods, where FSI is defined with mathematical model or where two computer codesare coupled successively in such way, that the fluid code takes into account also deformations of the structure (Abaqus-Fluent, Ansys-CFX, etc).

1D-piping-FSI 14


-Time-domain and frequency-domainanalyses- Typical outcome of a time-domainanalysis is a series of graphs showing how parameters vary in time.

- Typical outcome of a frequency-domain analysis is a series of graphs highlighting the dominant frequencies in the response of various parameters.

- Mathematically, time-domain and frequency-domain analyses contain the same information. It is possible, for example, to obtain frequency-domain results from a Fourier analysis of the output from a time-domain analysis. Inversely not always true.

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

2Pres s ure his tory near the va lve - deta il

Pre

ssur

e [M

Pa]

Time [s ]

no FSIwith FSI

1D-piping-FSI 15

• Set of four linear first-order PDEs:

fluid:

pipe:

( )21 2 21 - + = 0x

t

NR p ν v+ νK Ed t EA t s

∂∂ ∂⎛ ⎞−⎜ ⎟ ∂ ∂ ∂⎝ ⎠

= yx xt t

p

Qu Nρ A

t s R∂ ∂

−∂ ∂& 1 - - = 0x x

t

N uνR pEA t Ed t s

∂ ∂∂∂ ∂ ∂

&

Fluid-structure interaction in 1D piping systems – 1D models

- Skalak’s basic 4 equation model – axial movement:

- no two-phase flow- no damping- no friction- no convective term- no ...

Poisson coupling

v =uz , A f p = At z , Q y =0 , M x =0

p =const. , uz = 0

Junction coupling relations – pipe end

v =uz , A f p Y rod u z v0, rod = At z

Constant pressure (tank), rigidly anchored structure:

Closed pipe, free structure:

Closed pipe, free structure, rod impact:

+ = 0fv pρt s

∂ ∂∂ ∂

1D-piping-FSI 16


- Valve closure transient

PIPE VALVE

Pipe properties: L = 20 m, R = 398.5 mm ,e = 8 mm, E = 210 GPa, ν = 0.3, ρ s = 7900 kg/m3

Inital flow direction

Measuring point

TANK

Initial conditions:v = 1 m/s, p = 0 Pa, ρf = 1000 kg/m3

0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

2

Pre

ssur

e [M

Pa]

Time [s ]


0 0.05 0.1 0.15 0.2-1.5

-1

-0.5

0

0.5

1

1.5

2

Pre

ssur

e [M

Pa]

Time [s ]


Pressure near the valve:

Left: valve fixed

Right: valve free

1D-piping-FSI 17


- Rod impact experiment

0 0.005 0.01 0.015 0.02 0.025-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Pre

ssur

e rig

ht [M

Pa]

Time [s ]

ExperimentCalcula tion GaleCalcula tion Tijsse ling

0 0.005 0.01 0.015 0.02 0.025-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Pre

ssur

e le

ft [M

Pa]

Time [s ]

ExperimentCalcula tion GaleCalcula tion Tijsse ling

- The experiment performed at University of Dundee-The experimental apparatus is relatively simple:

- no initial flow influence - no initial deformation- no influence of supports- no valve closing timeeffect.

1D-piping-FSI 18


- Valentin’s 8 equation model – axial, rotational and flexuralmovement – for plane pipelines with elbows

- Skalak’s model + Timoshenko’s beam equations (from beam eq.)

21 - = - y y

zt

Q ut sκ GA

ϕ∂ ∂∂ ∂

&&( ) - = 0y y

t t f f

u Qρ A + ρ A

t s∂ ∂∂ ∂

&

- z zt t y

Mρ I = Qt s

ϕ∂ ∂∂ ∂& 1 - = 0z z

t

MEI t s

ϕ∂ ∂∂ ∂

&

A f ,1 v1 uz , 1 = A f , 2 v2 uz,2

p1 = p2 , uz, 1 = u y , 2

A f , 1 p1 At , 1 z , 1= Q y ,2A f ,2 p2 At ,2 z , 2 = Q y , 1

uy , 1 = uz , 2 , x ,1 = x , 2 , M x , 1 = M x ,2

Junction coupling relations – elbow

Singular coupling!(straight sections)


- Rod impact experiment- The experiment performed at

University of Dundee-The experimental apparatus is relatively simple :

- no initial flow influence - no initial deformation- no influence of supports- no valve closing timeeffect.

-2e+006

1.5e+006

-1e+006

-500000

0

500000

1e+006

1.5e+006

2e+006

0 0.005 0.01 0.015 0.02

CalculationExperiment

-2e+006

-1.5e+006

-1e+006

-500000

0

500000

1e+006

1.5e+006

2e+006

2.5e+006

0 0.005 0.01 0.015 0.02

CalculationExperiment

1D-piping-FSI 19

1D-piping-FSI 20


- Valentin’s 8 equation model – smoth model

( ) - - = y y f x

t t f fp

u Q A p Nρ A + ρ A

t s R∂ ∂∂ ∂

&21 - = - - y y x

zpt

Q u ut s Rκ GA

ϕ∂ ∂∂ ∂

& &&

- z zt t y

Mρ I = Qt s

ϕ∂ ∂∂ ∂& 1 - = 0z z

t

MEI t s

ϕ∂ ∂∂ ∂

&

+ = 0fv pρt s

∂ ∂∂ ∂

( ) ( )2 22

2 2

1 21 2 21 - + 2 1 1 = - p yx

t pp

R ν uNR p ν R v+ νK Ed t EA t s RR R

⎛ ⎞ −∂∂ ∂⎛ ⎞ ⎜ ⎟− − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠

&

= yx xt t

p

Qu Nρ A

t s R∂ ∂

−∂ ∂& 1 - - = yx x

t p

uN uνR pEA t Ed t s R

∂ ∂∂∂ ∂ ∂

&&

Junction coupling relations – elbow

Where Rp is curvatureradius of the pipe

1D-piping-FSI 21


- Rod impact experiment

Comparison between singularand smoth coupling - pressure

0 0.005 0.01 0.015 0.02-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500Pres s ure at impact end

Pre

ssur

e [k

Pa]

Time [s ]0 0.005 0.01 0.015 0.02

-2000

-1500

-1000

-500

0

500

1000

1500

2000Pres s ure at remote end

Pre

ssur

e [k

Pa]

Time [s ]

1D-piping-FSI 22


- Valve closure – single elbow pipeValve is free

(multiplication fact. fordeformations is 50)

TANK PIPE 1

L1 = 5 m, R = 0.3985 m, e = 8 mm, E = 210 GPaν = 0.3, ρt = 7900 kg/m3,


P1

VALVE

PIPE 2

ELBOW

L 2, R

= 0

.398

5 m

, e =

8 m

m, E

= 2

10 G

Pa

ν =

0.3,

ρ t = 7

900

kg/m

3

Valve is fixed(multiplication fact. fordeformations is 200)

Valve closure, initialpressure in the pipe is zero,fluid velocity v = 1 m/s

1D-piping-FSI 23


- Valve closure – Tank-pipe-valve system, pipe is arbitrary

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Deformations at t =0.0000 s

Wid

th [m

]

Length [m]

Initia lDeformed

Valve closure, initialpressure in the pipe is zero,fluid velocity v = 1 m/s

Valve is free(multiplication fact. for

deformations is 50)

1D-piping-FSI 24


- Wiggert’s 14 equationsmodel – 1D pipe in 3D space – full coupling:- additional equations for

torsional motion- additional eqs. for x-z

plane- radial deformations still

not included (negligible)

1D-piping-FSI 25

Fluid-structure interaction in 1D piping systems – numerical methods

Vectorial form of the equations – valid for any system:

0 + = t zψ ψ∂ ∂∂ ∂

A Br r

0 + = t zψ ψ∂ ∂∂ ∂

Cr r

⋅-1C = A BCharacteristic lines –

processor demanding withincreasing timeThe Jacobian matrix C has some very important properties:

- it is analytically diagonalizable - the eigensystem is constantduring the simulation due to the assumption of the single-phase flow and constant fluid density. These assumptions are generally not accurate!

Consequence:-The model is suitable for numerical solutions with Method ofCharacteristics (MOC)

MOC is most common method, other methods are mixed MOC-FEM procedure, component synthesis method, and Godunov’s method (WAHA).

1D-piping-FSI 26


Two phase flow modelling (void generally reduces FSI effect): - MOC: column separation concentrated cavity model (Bergant)

- predicts most of the cavitation situations (in cold water – inertially controlled cavitation)

- simple model to implement- Cavitation starts when pressure falls below sat. pressure – the cavity volume Vc

is evaluated using:

- pressure in the cavity fixed at saturation - Condensation - when overpressure wave transverses a cavity, first it has to

cause the cavity to collapse. The delay action associated with this behavior emulates the reduction of fluid wave speed and its dependency on the void fraction

- Godunov method: near future, coupling of Valentine's 8 equation model with WAHA code – the result will be two-phase flow FSI coupling

- Coupling of two codes – using best “market” codes + coupling at fluid-structure interface (Newton’s law)

Vc,old = Vc,new + Af (uf,right – uf,left) ∆t

Date post:	10-Oct-2014
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Two-Phase Flow Modeling

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