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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: iztok.tiselj@ijs.si
April 2006
Basic equations of two-phase flow 1
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
2) Basic equations of two-phase flow
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
Basic equations of two-phase flow 2
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 flow 3
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 flow 4
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 flow 5
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)
Basic equations of two-phase flow 6
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
Basic equations of two-phase flow 7
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
Basic equations of two-phase flow 8
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
Basic equations of two-phase flow 9
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
Basic equations of two-phase flow 10
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)
Basic equations of two-phase flow 11
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
Basic equations of two-phase flow 12
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:
Basic equations of two-phase flow 13
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.
Basic equations of two-phase flow 14
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
Basic equations of two-phase flow 15
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.
Basic equations of two-phase flow 16
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.
Basic equations of two-phase flow 17
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
Basic equations of two-phase flow 18
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
Basic equations of two-phase flow 19
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.
Basic equations of two-phase flow 20
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
Basic equations of two-phase flow 21
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
Basic equations of two-phase flow 22
Typical averaged equations of two-phase flow 6-Equation Two-Fluid Model
• 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
Basic equations of two-phase flow 23
Typical averaged equations of two-phase flow 6-Equation Two-Fluid Model
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
Basic equations of two-phase flow 24
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 25
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
Two-phase flow modelling, seminar at UPC, 2006
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
Two-phase flow modelling, seminar at UPC, 2006
Table of contents, part 2
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
Two-phase flow modelling, seminar at UPC, 2006
Table of contents, part 3
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:
• Momentum balances
gfff A =
x v )-(1 A
+ t
)-(1 AΓ−
∂
∂
∂
∂ ραρα
gggg A =
xv A
+ t
AΓ
∂
∂
∂
∂ ραρα
wallfgravityffigrri
2f
ff
f FFvvvvCCVM xp )-(1
x v
)-(1 21 +
t v )-(1 ,,)(|| ++−Γ−=−
∂∂+
∂
∂
∂∂
αραρα
wallggravityggigrri2g
gg
g FFvvvvCCVM xp
x v
21 +
t v ,,)(|| ++−Γ+−=+
∂∂+
∂∂
∂∂
αραρα
Introduction 13
6-Equation Two-Fluid Model
• Energy balances:
• Two additional equations of state for each phase k are:
• Numerous closure relations...
• Additional models relevant for nuclear thermal-hydraulics (neutronics...)
( )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
. 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
7-Equation Two-Fluid Model
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
3) 1D two-fluid models conservation equations
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
1D 2-fluid models - consrv eqs 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION Lectures 1-2
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
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
1D 2-fluid models - consrv eqs 3
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 2-fluid models - consrv eqs 4
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)
1D 2-fluid models - consrv eqs 5
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.
1D 2-fluid models - consrv eqs 6
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
1D 2-fluid models - consrv eqs 7
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
1D 2-fluid models - consrv eqs 8
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
1D 2-fluid models - consrv eqs 9
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
1D 2-fluid models - consrv eqs 10
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
1D 2-fluid models - consrv eqs 11
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
1D 2-fluid models - consrv eqs 12
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 −=
1D 2-fluid models - consrv eqs 13
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
1D 2-fluid models - consrv eqs 14
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
1D 2-fluid models - consrv eqs 15
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
1D 2-fluid models - consrv eqs 16
6-Equation Two-Fluid Model
• 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)
1D 2-fluid models - consrv eqs 17
6-Equation Two-Fluid Model
• Momentum 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
iC
CVM
iv
Virtual mass term, contains derivatives!
Interface friction coefficient
Interface velocity
1D 2-fluid models - consrv eqs 18
6-Equation Two-Fluid Model
• Energy balances:
( )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
*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
1D 2-fluid models - consrv eqs 19
6-Equation Two-Fluid Model
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
1D 2-fluid models - consrv eqs 20
6-Equation Two-Fluid Model
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 ,
1D 2-fluid models - consrv eqs 21
6-Equation Two-Fluid Model
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−=
1D 2-fluid models - consrv eqs 22
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
1D 2-fluid models - consrv eqs 23
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
1D 2-fluid models - consrv eqs 24
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
1D 2-fluid models - consrv eqs 25
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.)
1D 2-fluid models - consrv eqs 26
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
4) 1D two-fluid models flow regime maps and closure equations
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
1D 2-fluid models - closures 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION Lectures 1-2
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
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
1D 2-fluid models - closures 3
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 2-fluid models - closures 4
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.
1D 2-fluid models - closures 5
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
1D 2-fluid models - closures 6
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"...
1D 2-fluid models - closures 7
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 =
1D 2-fluid models - closures 8
Horizontal flow regime map in RELAP5 code(drawing from RELAP5/mod3.3 manual)
1D 2-fluid models - closures 9
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 =
1D 2-fluid models - closures 10
Vertical flow regime map in RELAP5 code(drawing from RELAP5/mod3.3 manual)
1D 2-fluid models - closures 11
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? ???
1D 2-fluid models - closures 12
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.
1D 2-fluid models - closures 13
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 −=
1D 2-fluid models - closures 14
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 +=
1D 2-fluid models - closures 15
1D 2-fluid models - closures 16
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
1D 2-fluid models - closures 17
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...
1D 2-fluid models - closures 18
1D 2-fluid models - closures 19
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
1D 2-fluid models - closures 20
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
1D 2-fluid models - closures 21
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.
1D 2-fluid models - closures 22
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
1D 2-fluid models - closures 23
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
1D 2-fluid models - closures 24
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
1D 2-fluid models - closures 25
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
5) Characteristic upwind schemes for two-fluid models
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
characteristic-upwind schemes 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION Lectures 1-2
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
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
characteristic-upwind schemes 3
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 4
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)
characteristic-upwind schemes 5
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)
characteristic-upwind schemes 6
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.
characteristic-upwind schemes 7
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
characteristic-upwind schemes 8
High resolution shock capturing schemes for Euler equations
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 +=
characteristic-upwind schemes 9
High resolution shock capturing schemes for Euler equations
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:
characteristic-upwind schemes 10
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(/ +−∆<∆
characteristic-upwind schemes 11
High resolution shock capturing schemes for Euler equations - discrete form
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
characteristic-upwind schemes 12
High resolution shock capturing schemes for Euler equations - discrete form
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
characteristic-upwind schemes 13
High resolution shock capturing schemes for Euler equations - discrete form
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)
characteristic-upwind schemes 14
1- shock wave, 2- rarefaction wave, 3 - contact discontinuity
p ρ v
length (m)
Sod, JCP 27, 1978
characteristic-upwind schemes 15
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...)
characteristic-upwind schemes 16
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
characteristic-upwind schemes 17
characteristic-upwind schemes 18
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 19
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
characteristic-upwind schemes 20
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
characteristic-upwind schemes 21
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
characteristic-upwind schemes 22
1st substep of operator splitting: basic variables
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
characteristic-upwind schemes 23
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
characteristic-upwind schemes 24
1st substep of operator splitting: basic variables
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
characteristic-upwind schemes 25
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
see Tiselj, Petelin, JCP 136, 1997
characteristic-upwind schemes 26
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
see Tiselj, Petelin, JCP 136, 1997
characteristic-upwind schemes 27
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).
characteristic-upwind schemes 28
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
characteristic-upwind schemes 29
2nd substep of operator splitting: integration of stiff relaxation source terms
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.
characteristic-upwind schemes 30
2nd substep of operator splitting: integration of stiff relaxation source terms
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
characteristic-upwind schemes 31
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.
characteristic-upwind schemes 32
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.
characteristic-upwind schemes 33
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)
characteristic-upwind schemes 34
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
characteristic-upwind schemes 35
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
characteristic-upwind schemes 36
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
characteristic-upwind schemes 37
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
characteristic-upwind schemes 38
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 39
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
6) Pressure-based solvers for two-fluid models
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
pressure-based schemes 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION Lectures 1-2
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
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
pressure-based schemes 3
Pressure-based solvers for two-fluid models Contents
- Introduction
- Numerical scheme of RELAP5
- Numerical diffusion, accuracy
pressure-based schemes 4
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
pressure-based schemes 5
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
pressure-based schemes 6
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)
pressure-based schemes 7
Introduction - pressure-based methods
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
pressure-based schemes 8
Introduction - pressure-based methods
Overview of the segregated solver (from Fluent manual):
pressure-based schemes 9
Introduction - pressure-based methods
Overview of the coupled solver (from Fluent manual):
pressure-based schemes 10
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
pressure-based schemes 11
RELAP5 numerical scheme (simplified)
pressure-based schemes 12
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
pressure-based schemes 13
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
pressure-based schemes 14
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
pressure-based schemes 15
"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
pressure-based schemes 16
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 schemes 17
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modeling
7) 3D two-phase flows -mathematical background
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
7-3D-two-phase-flows 2
Two-phase flow modelling, seminar at UPC, 2006
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
ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS Lectures 11-14
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
7-3D-two-phase-flows 3
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.
7-3D-two-phase-flows 4
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
7-3D-two-phase-flows 5
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."
7-3D-two-phase-flows 6
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.
7-3D-two-phase-flows 7
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.
7-3D-two-phase-flows 8
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 ???
7-3D-two-phase-flows 9
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
( ) ( ) )),(( trfIpFvvtv
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)
7-3D-two-phase-flows 11
( ) ( )( ) ( ) ( ) 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
7-3D-two-phase-flows 12
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?)
7-3D-two-phase-flows 13
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....)
7-3D-two-phase-flows 14
3D two-fluid models -energy equations
2 Total Energy equations, 1 Momentum equation
7-3D-two-phase-flows 15
( )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
7-3D-two-phase-flows 16
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)
7-3D-two-phase-flows 17
7-3D-two-phase-flows 18
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)
7-3D-two-phase-flows 19
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
7-3D-two-phase-flows 20
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
7-3D-two-phase-flows 21
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
8) Interface tracking models
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
8-interface-tracking 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION 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
ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS Lectures 11-14
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
8-interface-tracking 3
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
8-interface-tracking 4
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.
8-interface-tracking 5
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)
8-interface-tracking 6
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.
8-interface-tracking 7
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
8-interface-tracking 8
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)
8-interface-tracking 9
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.
8-interface-tracking 10
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...
( ) ( ) )),(( trfIpFvvtv
skkkrrrr
r
σκδτρρ∂
∂ρ ++⋅∇−=⋅∇+
8-interface-tracking 11
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
8-interface-tracking 12
8-interface-tracking 13
Volume of fluid (VOF)Interface reconstruction
Different types of interface reconstruction:- Simple Line Interface Reconstruction with
Calculation (SLIC) step function
First-order reconstructions.
8-interface-tracking 14
Volume of fluid (VOF)Interface reconstruction
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
8-interface-tracking 15
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
8-interface-tracking 16
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
8-interface-tracking 17
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.
8-interface-tracking 18
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
8-interface-tracking 19
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
8-interface-tracking 20
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
8-interface-tracking 21
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
8-interface-tracking 22
Conservative L-S - Dam break
( ) 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
8-interface-tracking 23
Conservative L-S - Dam break
• 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.
8-interface-tracking 24
Conservative Level-Set Conclusion
Very promising method - seems to allow natural transition from whole-field interface tracking mode into the two-fluid model.
8-interface-tracking 25
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
9) Coupling of two-fluid models and VOF method
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
9-VOF+two-fluid 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION 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
ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS Lectures 11-14
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
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
VOF errors IIINumerical dispersion error
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
VOF errors IIINumerical dispersion error
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
VOF errors IIINumerical dispersion error
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 12x60-fcoupled 24x120
fcoupled 24x120-fcoupled 48x240
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
10) Simulations of Kelvin-Helmholtz instability
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
10 - K-H instability 1
Two-phase flow modelling, seminar at UPC, 2006
Table of contents
INTRODUCTION 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
ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS Lectures 11-14
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
10 - K-H instability 2
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
10 - K-H instability 3
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.
10 - K-H instability 4
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 =
10 - K-H instability 5
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.
10 - K-H instability 6
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 +∇∇−−∇=∇+∇−⋅∇+
∂∂
10 - K-H instability 7
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.
10 - K-H instability 9
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
10 - K-H instability 10
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.
10 - K-H instability 11
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.
10 - K-H instability 12
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
10 - K-H instability 13
CFX - Kelvin-Helmholtz instability
10 - K-H instability 14
• 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
10 - K-H instability 15
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
10 - K-H instability 16
10 - K-H instability 17
Kelvin-Helmholtz instability with surface tension
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
Kelvin-Helmholtz instability with surface tension
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.
10 - K-H instability 18
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).
10 - K-H instability 19
Kelvin-Helmholtz instability without surface tension
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]
10 - K-H instability 20
Kelvin-Helmholtz instability without surface tension
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.
10 - K-H instability 21
Kelvin-Helmholtz instability without surface tension
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.
10 - K-H instability 22
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
10 - K-H instability 23
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
10 - K-H instability 24
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]
10 - K-H instability 25
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
10 - K-H instability 26
Direct Contact Condensation
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
10 - K-H instability 27
Direct Contact Condensation
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−
−=,
&
10 - K-H instability 28
Direct Contact Condensation
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
10 - K-H instability 29
Direct Contact Condensation
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
10 - K-H instability 30
Direct Contact Condensation
3D simulationVoid fraction
Grid:4000 volumes, time step=0.03 s, CPU time=7 h @3.0 GHz Pentium
10 - K-H instability 31
Direct Contact Condensation
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
10 - K-H instability 32
Direct Contact Condensation
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
?
10 - K-H instability 33
Direct Contact Condensation
Small increase of water temperature -> Small condensation rate
Heat transfer coefficient was increased by factor 20 -> better agreement with experiment
10 - K-H instability 34
Direct Contact Condensation
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
10 - K-H instability 35
Direct Contact Condensation
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
10 - K-H instability 36
Direct Contact Condensation
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
10 - K-H instability 37
Direct Contact Condensation
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
10 - K-H instability 38
Direct Contact Condensation
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
10 - K-H instability 39
Direct Contact Condensation
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
10 - K-H instability 40
WAHA-maths-numerics 1
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
11) WAHA code - mathematical model and numerical scheme
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
WAHA-maths-numerics 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contentsINTRODUCTION Lectures 1-2
TWO-FLUID MODELS Lecture 3-6INTERFACE TRACKING IN 3D TWO-PHASE FLOWS
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 flow.14) Fluid-structure interaction in 1D piping systems
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
WAHA-maths-numerics 3
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-maths-numerics 4
WAHA code - mathematical model and numerical scheme - reference
- WAHA code manual, available on internet
www2.ijs.si/~r4www/waha3_manual.pdf
WAHA-maths-numerics 5
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:
WAHA-maths-numerics 6
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:
LHS: cdifferential terms
RHS: sources
WAHA-maths-numerics 7
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:
Pipe elasticity (Wylie, Streeter):)),(()(),( txpAxAtxA e+= dpK
Edp
dD
xAdAe ==
)(
WAHA-maths-numerics 8
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:
Additional closure relations:1) Equations of state (more later):
. du u
+ p d p
= d kk
k
p
k
uk
k
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂ ρρρ
WAHA-maths-numerics 9
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:
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(
WAHA-maths-numerics 10
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:
Additional closure relations:3) Interfacial pressure term exists only in stratified flow.
gDSp gfi ))(1( ρραα −−=
WAHA-maths-numerics 11
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:
Additional closure relations:4) Source terms are flow regime dependent. Source terms are:4.1) Terms with Ci - inter-phase drag
WAHA-maths-numerics 12
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:
Additional closure relations:4.2a) Terms with inter-phase exchange of mass and energy with:
Γg=-(Qif+Qig)/(hg-hf) - vapor generation term
WAHA-maths-numerics 13
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
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:
Additional closure relations:4.2b) Terms with inter-phase exchange of mass and energy with:
Qik=Hik (Ts-Tk) - interface heat transfer terms
WAHA-maths-numerics 14
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
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:
Additional closure relations:4.3) Terms due to the variable pipe cross-section.
WAHA-maths-numerics 15
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
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:
Additional closure relations:4.4) Ff,wall , Fg,wall - wall friction (Dynamical wall friction model available too).
WAHA-maths-numerics 16
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:
Additional closure relations:4.5) Term with g cosθ - volumetric forces.
WAHA-maths-numerics 17
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:
Additional closure relations:4.6) Terms for wall heat transfer are neglected in WAHA code.
WAHA-maths-numerics 18
Two-fluid model of 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
WAHA-maths-numerics 19
WAHA-maths-numerics 20
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
WAHA-maths-numerics 21
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 =−= )(
WAHA-maths-numerics 22
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-maths-numerics 23
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
WAHA-maths-numerics 24
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).
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
WAHA-maths-numerics 25
1st substep of operator splitting: convection terms with non-relaxation source terms
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:
WAHA-maths-numerics 26
1st substep of operator splitting: convection terms with non-relaxation source terms
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:
WAHA-maths-numerics 27
1st substep of operator splitting: basic variables
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
WAHA-maths-numerics 28
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. 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.
WAHA-maths-numerics 29
2nd substep of operator splitting: integration of stiff relaxation source terms
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-maths-numerics 30
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
WAHA-maths-numerics 31
WAHA special models:
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-maths-numerics 32
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
WAHA-maths-numerics 33
WAHA special models:
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
WAHA-maths-numerics 34
WAHA special models:
• 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
WAHA-maths-numerics 35
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:
WAHA-maths-numerics 36
Water properties of the WAHA code:
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.
WAHA-maths-numerics 37
Water properties of the WAHA code:
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
12) WAHA code - simulations
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
WAHA-simulations 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contentsINTRODUCTION Lectures 1-2
TWO-FLUID MODELS Lecture 3-6INTERFACE TRACKING IN 3D TWO-PHASE FLOWS
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 flow.14) Fluid-structure interaction in 1D piping systems
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
Initial flow direction
Measuring point
VALVE
WAHA-simulations 18
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
WAHA code - simulations
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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
13) Hands on: simulation of two-phase water hammer transient and two-phase critical flow
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
WAHA-hands-on 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contentsINTRODUCTION Lectures 1-2
TWO-FLUID MODELS Lecture 3-6INTERFACE TRACKING IN 3D TWO-PHASE FLOWS
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
DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18
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
Initial flow direction
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
- input file for Moby-Dick two-phase critical flow transient 2/3
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
- input file for Moby-Dick two-phase critical flow transient 3/3
*-------------------------------------------------------
* 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
* type press alpha_g velf velg tf tgwch_nods
*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
* type press alpha_g velf velg tf tgwch_nods
*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
Technical University of Catalonia andHeat and Mass Transfer Technological Center, 2006
Seminar on Two-phase flow modelling
14) Fluid-structure interaction in 1D piping systems
by
Iztok Tiselj"Jožef Stefan“ Institute, Ljubljana, Slovenia
1D-piping-FSI 2
Two-phase flow modelling, seminar at UPC, 2006
Table of contentsINTRODUCTION Lectures 1-2
TWO-FLUID MODELS Lecture 3-6INTERFACE TRACKING IN 3D TWO-PHASE FLOWS
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 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 in 1D piping systems - Introduction
• 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
Fluid-structure interaction in 1D piping systems - Introduction
• 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
Fluid-structure interaction in 1D piping systems - Introduction
- 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
Fluid-structure interaction in 1D piping systems - Introduction
• 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
Fluid-structure interaction in 1D piping systems - Introduction
• 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
Fluid-structure interaction in 1D piping systems - Types of FSI
- 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
Fluid-structure interaction in 1D piping systems - Types of FSI
- 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
Fluid-structure interaction in 1D piping systems - Types of FSI
-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
Fluid-structure interaction in 1D piping systems – 1D models
- 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 ]
Case 4, with FSI (soft pipe)Case 5, no FSI (s tiff 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 1, with FSI (soft pipe)Case 5, no FSI (s tiff pipe)
Pressure near the valve:
Left: valve fixed
Right: valve free
1D-piping-FSI 17
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
- 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)
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
- 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,
Initial flow direction
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
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
- 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
Fluid-structure interaction in 1D piping systems – 1D models
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