Two-Phase Flow Modeling

Technical University of Catalonia and Heat and Mass Transfer Technological Center, 2006Seminar on Two-phase flow modelling 1) Introduction by Iztok Tiselj "Jožef Stefan“ Institute, Slovenia Email: [email protected] April 2006 Introduction 1 Technical University of Catalonia and Heat 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 1 Two-phase flow modelling, seminar at UPC, 2006 Table of contents INTRODUCTION 1) Introduction 2) 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 2 Basic equations of two-phase flows Contents - 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 3 Basic equations of two-phase flows Introduction 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 4 Basic equations of two-phase flows Introduction, 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 5 Navier-Stokes equations Fluid k, that occupies the observed domain, is described with equations: continuity equation r ∂ ρk + ∇ ⋅ (ρ k vk ) = 0 ∂t ρk r vk density velocity momentum equation r r r ∂ρ k vk + ∇ ⋅ (ρvk vk ) = ρ k F − ∇ ⋅ ( pk I + τ k ) ∂t r F r volumetric forces pressure I unit tensor τk pk µk viscosity Basic equations of two-phase flow viscous stress tensor 6 cont. internal energy equation (also found in enthalpy or total energy form) r r r r ∂ρ k uk + ∇ ⋅ (ρ k uk vk ) = −∇ ⋅ qk − pk ∇ ⋅ vk + τ k : ∇vk + Qk ∂t uk specific internal energy Qk r qk heat flux volumetric source terms Basic equations of two-phase flow 7 .Navier-Stokes equations. u k ) or p k = p k (ρ k . Tk ) Viscous stress tensor for Newtonian fluids: r r T ⎛2 ⎞ r τ k = µk ∇vk + (∇vk ) − ⎜ µk − λk ⎟∇ ⋅ vk I ⎝3 ⎠ Heat flux .Constitutive equations Equation of state: p k = p k (ρ k .Fourier's law of heat conduction: ( ) r qk = k k ∇ ⋅ Tk Basic equations of two-phase flow 8 . Parentheses interface. [[w]] = wk =1 − wk =2 denote jump in the quantity w on the Interfacial mass balance: r r r [[ρ k (vk − vi )⋅ n ]] = 0 r vi r n interface velocity unit vector normal to the interface. direction: from fluid 1 to fluid 2 Basic equations of two-phase flow 9 . Interface is assumed to be a discontinuity.Boundary conditions at the interface Local boundary conditions at the interface i. Interfacial momentum balance r r r r r r [[ρ k uk (vk − vi )⋅ n + ( pk I − τ k )⋅ n ]] = σκn local curvature of the interface: κ= ⎜ + ⎟ 2 ⎜ R1 R2 ⎟ ⎝ ⎠ 1⎛ 1 1 ⎞ κ = − ∇i ⋅ n 1 2 r σ surface tension Interfacial energy balance (simplified: neglected kinetic energy. assumed σ=const. see Ishii.Boundary conditions at the interface. cont. usually zero (nonzero if chemical reaction runs at the interface) Basic equations of two-phase flow 10 . neglected work of the surface tension. Hibiki for details): r r r r r [[ρ k ek (vk − vi )⋅ n + qk ⋅ n ]] = qi qi surface energy source term. cont. Wetting angle model near the contact of the interface and solid surface r r r n = nwall cosθ + twall sin θ θ Wetting system 0°<θ<90° r nwall r twall θ r n Non-wetting system 90°<θ<180° Basic equations of two-phase flow 11 .Boundary conditions at the interface. 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.problems In theory. Before: After: Basic equations of two-phase flow 12 .Navier-Stokes equations and interface jump conditions . Basic equations of two-phase flow 13 . Phase transition may start on the impurities in the bulk of the fluid or at the walls. Additional information/models are needed (sometimes on molecular scales) to specify the density of the impurities in the liquid or the structure of the wall where the cavitation starts. Navier-Stokes equations (with all the boundary conditions) are not sufficient to describe arbitrary two-phase flows.Navier-Stokes equations and single-to-two phase flow transition Unlike in the single-phase flow. 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. ) Momentum equation r ∂ρ r + v ∇ρ = 0 ∂t r rr r ∂ρv + ∇ ⋅ (ρv v ) = ρF − ∇ ⋅ ( p I + τ ) + σκδ ( f s ( r . the Navier-Stokes equations can be applied in modelling. t ) δ Basic equations of two-phase flow 14 . t )) Dirac delta function ∂ t equation of interface r f s (r . heat transfer neglected. Equations are often assumed to be incompressible. 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 r ∇⋅v = 0 Equation for interface tracking (form continuity eq.Navier-Stokes equations. whole-domain formulation In some cases. applicability of local instant formulation In general: mathematical and numerical difficulties in modelling of twophase flows with the local instant formulation are insurmountable in the near future.Turbulent fluctuations . . example: turbulent flume.even in single-phase flows resolvable only at low Reynolds numbers. Characteristic length scales of the interface motion can be much larger than the characteristic scales of turbulent flows. .Navier-Stokes equations.Existence of the multiple deformable moving interfaces. Characteristic length scales of the interface motion can be much smaller than the characteristic scales of turbulent flows: example turbulent bubbly flows. Motion of the interface is an integral part of the solution (except in particulate flows). Problems with break-up and coallescence of the surfaces. Basic equations of two-phase flow 15 . Averaged equations result in mean values of the two-phase flow motion.Averaging of the Navier-Stokes equations Why averaging? Microscopic details of turbulent motions and interfacial geometry are seldom relevant for the engineering problems. That must be taken into account in the closure relations of the averaged equations. Problem: scales eliminated with the averaging influence the mean values. Basic equations of two-phase flow 16 . t ) 17 Basic equations of two-phase flow .Averaging of the Navier-Stokes equations Most common types of averaging . t ) dV Ensemble (statistical): 1 N ∑ n =1 N r Fn ( r . t ) dt Spatial: 1 ∆V ∆V ∫ r F ( r . t ) : Temporal (equivalent to Reynolds averaging in turbulent single-phase flow): 1 ∆t ∆t ∫ r F ( r .theory r Eulerian averaging of function F ( r . more "exotic" types of averaging exist (Lagrangian. t ) : Area (cross-sectional) for 1D two-fluid models: r 1 F ( r . Basic equations of two-phase flow 18 . t ) dS S ∫ S Other. See Ishii. "Phenomenological averging" .theory.. Boltzmann statistical averaging).. averaged equations built on phenomenological approach. Hibiki for discussion. r Eulerian averaging of function F ( r .Averaging of the Navier-Stokes equations Most common types of averaging .not averaging at all. "transforms" two phases that alternately occupy the observed point into two continuous fields that exist in that point with a given probability. Various types of averaging results in slightly different equations. . however.practical approach From practical point of view the type of averaging isn't important. Basic equations of two-phase flow 19 . What is important: . the differences are minor comparing to the typical uncertainty of the closure relations required to close the averaged system of conservation laws.averaging smoothes out the turbulent fluctuations.Averaging of the Navier-Stokes equations Types of averaging . detailed definition of is not important anymore. In this seminar is mainly called k-th α k phase volume fraction. When the averaged equations are solved. etc.. The function averaging. Basic equations of two-phase flow 20 . α k is a new fundamental variable produced by the αk αk αk is a local time fraction of the phase k after temporal averaging.. is a probability for the presence of the phase k after ensemble averaging. void fraction. Hibiki for details). is a local volume fraction of the phase k after spatial averaging.Volume fraction... αk DETAILS OF THE AVERAGING PROCEDURE SKIPPED (see Ishii. Mass balances: • ∂A (1 .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 watercooled nuclear reactors. Requires several closure relations that are mainly based on empirical approach. Allows thermal and mechanical non-equilibrium.α ) ρ f ∂t ∂A α ρ g ∂t + + ∂A (1 .α ) ρ f v f ∂x ∂x = A Γg = − A Γg ∂A α ρ g v g Basic equations of two-phase flow 21 . gravity + Fg . gravity + F f .wall ∂t ∂x ( ) ∂Aα ρ g u g v g ∂ Aα ∂A α v g * +p = A Qig + Γg hg + v g Fg .wall 2 ∂t ∂x ∂x (1 .α ) ρ f + (1 .α ) ρ f u f v f ∂x ∂x +p −p ∂A(1 .α ) − CVM = Ci | vr | vr − Γg (vi − v f ) + F f .α ) ρ f α ρg • ∂ vg ∂ v2 ∂p 1 g +α + CVM = −Ci | vr | vr + Γg (vi − v g ) + Fg .Typical averaged equations of two-phase flow 6-Equation Two-Fluid Model • Momentum balances ∂ vf ∂ v2 1 ∂p f + (1 .α ) v f ∂Aα * +p = A Qif − Γg h f + v f F f . wall + α ρg ∂t ∂x ∂x 2 Energy balances: ∂A(1 − α ) ρ f u f ∂t ∂ Aα ρ g u g ∂t + + ∂A(1 .wall ∂t ∂x ( ) 22 Basic equations of two-phase flow . Typical averaged equations of two-phase flow 6-Equation Two-Fluid Model Closure relations: • Two additional equations of state for each phase k are: ⎛∂ρ ⎞ ⎛∂ρ ⎞ d ρ k = ⎜ k ⎟ d p + ⎜ k ⎟ d uk . ⎜∂ ⎟ ⎜ ∂p ⎟ ⎝ uk ⎠ p ⎝ ⎠u k • • • • • Correlations for inter-phase momentum transfer. Correlations for interphase heat and mass transfer.. others .. Wall friction correlations.. Correlations for wall-to-fluid heat transfer .. Basic equations of two-phase flow 23 . Typical averaged equations of two-phase flow CFX-5. t )) Basic equations of two-phase flow 24 .: r ∂ (α1ρ1) +∇⋅(α1ρ1v) = 0 ∂t • r ∂ (α2ρ2 ) + ∇⋅ (α2ρ2v ) = 0 ∂t One momentum equation: r r rr r rT r ∂ρv + ∇ ρvv − µ ∇v + (∇v ) = F12 + ρ − ρref g − ∇p ∂t ( ( )) ( ) • Surface tension: r r r F12 = −σ12κ12n12δ ( f s (r . contains viscous terms and surface tension force (if the interface can be found): • Two continuity eqs.6 – homogeneous two-fluid model Homogeneous two-fluid model in CFX code. or cross-section averaged Navier-Stokes equations of two-fluid models will remain an important research field.can be found in CFD codes (CFX.to be used with caution. oil or water transport. 2D/3D two-fluid models .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. Basic equations of two-phase flow 25 . where one-dimensional two-fluid models still present a sufficiently accurate and efficient option. volume. FLUENT) . despite the rapid progress in the field of the more accurate interface tracking methods.in development . From the stand point of the industrial applications: there are several types of piping flows in nuclear and chemical engineering. time. Development and improvement of the empirical closure relations for ensemble. 000$ (2005) 50 km Introduction 2 .500$ (2004) Slovenia Area: 20.Catalonia.000 km2 Population: 2. 32.000.000 km2 7.000 25.000.000 GDP per capita: 21. SLOVENIA Introduction 3 . si The Jožef Stefan Institute is named after the distinguished 19th century physicist Jožef Stefan. nuclear technology.“Jožef Stefan” Institute www. chemistry and biochemistry. electronics and information science.ijs. Introduction 4 . energy utilization and environmental science. JSI is the leading Slovene research organisation responsible 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. Industrial Hazard and Risk –Nuclear Physics Division • Theoretical.Nuclear Research –Reactor Engineering Division • Thermal-Hydraulics • Structural Mechanics • Reliability. • pool.“Jožef Stefan” Institute . of Environmental Sciences • Radiochemistry and Radioecology –Research Reactor TRIGA Mark-II. experimental and applied reactor physics –Dept. 250 kW. 1000MW pulse mode Introduction 5 . Reactor Engineering Division of JSI Thermal-hydraulics ~12 out of 20 researchers Introduction 6 . Overview of Thermal-hydraulics research at Reactor Engineering Division – Simulations of transients and accidents in nuclear and experimental installations with computer codes RELAP5. Fluent. • OECD ISP-44 KAEVER (CONTAIN) – Modelling of single and two-phase flows (“home-made” codes. 3D two-phase flows: two(three)-fluid models and interface tracking models Introduction 7 . NEPTUNE CFD packages): LES and DNS simulations of single phase turbulent heat transfer Characteristic upwind schemes for fast 1D transients in two-phase flow Numerical schemes for 2D. CONTAIN. BETHSY (RELAP5). CFX. MELCOR: • 1999-2000 verification of the new full-scope NPP Krško simulator with RELAP5 • Standard experiments PMK. TWO-FLUID MODELS 3) 1D two-fluid models .conservation equations 4) 1D two-fluid models .flow regime maps and closure equations 5) Characteristic upwind schemes for two-fluid models 6) Pressure-based solvers for two-fluid models Introduction 8 .Two-phase flow modelling. 2006 Table of contents. seminar at UPC. part 1 INTRODUCTION 1) Introduction 2) Basic equations of two-phase flows. simulations 13) Hands on: simulation of two-phase water hammer transient and two-phase critical flow. part 2 INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows . 2006 Table of contents. 14) Fluid-structure interaction in 1D piping systems Introduction 9 .Two-phase flow modelling.mathematical model and numerical scheme 12) WAHA code . seminar at UPC.mathematical background 8) Interface tracking models 9) Coupling of two-fluid models and interface tracking methods 10) Simulations of Kelvin-Helmholtz instability ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS 11) WAHA code . part 3 DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME (This is not a two-phase flow modelling chapter but. general results 17) DNS of passive scalar heat transfer at various thermal boundary conditions. seminar at UPC.) 15) Mathematical model of DNS 16) Pseudo-spectral numerical scheme. high Prandtl numbers 18) Hands-on. Introduction 10 .Two-phase flow modelling... conjugate heat transfer. 2006 Table of contents. Running of the DNS code. Introduction 11 .Two-Fluid Models 1D 6-equation equal pressure two-fluid model for inhomogeneous nonequilibrium two-phase flow – heart of the codes used for simulations in today’s nuclear thermal-hydraulics. 6-Equation Two-Fluid Model • Mass balances: ∂A (1 .α ) − CVM = Ci | vr | vr − Γg (vi − v f ) + F f .α ) ρ f + (1 .α ) ρ f v f ∂x ∂x = A Γg = − A Γg + ∂A α ρ g v g Momentum balances ∂ vf ∂ v2 1 ∂p f + (1 . gravity + F f .1D.α ) ρ f + ∂A (1 .α ) ρ f ∂t ∂A α ρ g ∂t • (1 . gravity + Fg . wall + α ρg ∂t ∂x ∂x 2 Introduction 12 .wall ∂t 2 ∂x ∂x α ρg ∂ vg ∂ v2 ∂p 1 g +α + CVM = −Ci | vr | vr + Γg (vi − v g ) + Fg . ..α ) ρ f u f v f ∂x ∂x +p ∂A(1 . ⎜∂ ⎟ ⎜ ∂p ⎟ ⎝ uk ⎠ p ⎝ ⎠u k • • Numerous closure relations.) Introduction 13 ..wall ∂t ∂x ( ) + ∂Aα ρ g u g v g ∂ Aα ∂A α v g * +p = A Qig + Γg hg + v g Fg .α ) v f ∂Aα * −p +p = A Qif − Γg h f + v f F f .wall ∂t ∂x ( ) Two additional equations of state for each phase k are: ⎛∂ρ ⎞ ⎛∂ρ ⎞ d ρ k = ⎜ k ⎟ d p + ⎜ k ⎟ d uk . Additional models relevant for nuclear thermal-hydraulics (neutronics..6-Equation Two-Fluid Model • Energy balances: ∂A(1 − α ) ρ f u f ∂t ∂ Aα ρ g u g ∂t • + ∂A(1 . Oberhausen): Total pipeline length: 137 m TANK VALVE Introduction 14 .) Code development performed in cooperation with UCL and CEA. (WAHALoads project of 5th EU research program. One of the WAHALoads experiments (UMSICHT.1D simulations of two-phase flow fast transients Simulation of water hammer in piping system Past 4 years: development of computer code for simulations of water hammer transients in 1D piping networks. Oberhausen): P09 P03 GS VALVE P06 P18 TANK P04 P15 Introduction 15 .1D simulations of two-phase flow fast transients Simulation 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. 50 1.Pressure [MPa] Introduction 16 .00 0 1 2 3 4 5 Time [sec] 6 7 8 9 10 UMSICHT WAHA RELAP5 P03 .00 2.00 1.Water hammer simulation of UMSICHT experiment Pressure near the valve 5.00 4.00 3.00 0.50 3.50 2.50 0.50 4. 4 0.3 0.2 0.5 0.8 0.0 0.7 0.9 0.6 0.Water hammer simulation of UMSICHT experiment Vapour volume fraction near the valve GS .0 0 1 2 3 4 5 Time [sec] 6 7 8 9 10 FZR WAHA RELAP5 Introduction 17 .Vapor volume fraction 1.1 0. Water hammer-like transient follows. but are within the uncertainty of the measurements. negative pressures could appear briefly near the valve in UMSICHT experiment. IAHR congress. due to the delayed cavity growth. spring 18 . Negative pressures were measured in water hammer experiment by Bergant and Simpson (1999. i..Phys. Graz) and tube-arrest experiment (designed specially for that purpose by Williams & Williams. Proc. D. 2222-2230) How to model transients with negative pressures? Introduction Tube-arrest experiment: Tube half-filled with purified water is accelerated upward and stopped suddenly.e. 35. Small negative pressures are actually measured in a few cases.Fluid dynamics at negative pressures Under special conditions. cold and purified liquid. J. 2002. 7-Equation Two-Fluid Model Alternative approach to 6-eq. Additional equation for volume fraction completes the system of equations: ∂α ∂α + vm = µ ( p g − pl ) ∂t ∂x 7-eq. while the pressure of the vapor phase remains positive. Very similar equations like 6-eq. two-fluid model: • 7-equation "two-pressure" two-fluid model (Saurel. several drawbacks… • 7-equation model allows simulations of liquid phase at negative pressure. Abgrall. Introduction 19 . 6-eq. : several advantages. model but with two separate phasic pressures. vs. 1999). 1D Simulation of tube-arrest experiment Introduction 20 . Two-phase flow modelling: Interface tracking algorithms • Rising bubble in the viscous fluid flattens the circular shape and causes vorticity in and behind the bubble VOF method explicitly tracks the interface between fluids and enables the streamline location • Streamlines around the bubble experiment (left) simulation (right) Introduction 21 . Comput. 776) Introduction 22 .Phys 171.Coupling of interface tracking method (VOF) and two-fluid model Fluid dispersion and stratification during the Rayleigh-Taylor instability (Černe.Introduction . J. Petelin. 2001. Tiselj. Inviscid linear analysis: step velocity and step density profiles assumed z=H U2 z=0 U1 z=-H Immiscible fluids ρ2 fluid 2 ρ1 fluid 1 Velocity and density profiles for linear inviscid U analysis ρ Results: Critical relative velocity Critical wave number Critical wave length ∆U 2 > 2 2 ρ1 + ρ2 ∆ρgσ ρ1ρ2 ∆ρ = ρ1 − ρ2 k* = ∆ρg / σ λ* = 2π / k* Introduction 23 .Kelvin-Helmholtz instability . A. Thorpe. 39.04 N/m µ1 = 0. Journal of Fluid Mechanics. Experiments on the instability of stratified shear flows: immiscible fluids. 25-48 Introduction 24 .2) m H=0.83 (0.03 m ρ2 = 780 kg/m3 Initial conditions h2 H h1 ρ2 ρ1 ρ1 = 1000 kg/m3 u2 x z=0 z Tube tilted for a small angle g =10 m/s2 σ = 0.0015 Pa ⋅ s S. 1969. 1969) L=1.Tilted tube experiment (Thorpe.001 Pa ⋅ s g u1 γ µ2 = 0. Introduction 25 . surface tension neglected in particular simulation.8 s. K-H instability in experiment is observed in the middle section of the tube after ~1.CFX simulation complete tube length simulated Temporal development of the interface predicted by CFX. Viscosity not neglected.K-H instability . (computational domain =20cmx3cm. Introduction 26 . time~2s).CFX simulation of Kelvin-Helmholtz instability Volume fraction of lighter fluid. 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.CFX model without surface tension is more stable than predicted by the linear inviscid analysis and experiment. . (No reasonable results on unstructured grid) -Surface tension terms in CFX destabilize the surface contrary to the actual physics of the surface tension force.“Structured” grid was used and quasi-2D simulations performed. which plays a stabilizing role in the K-H instability development. . . Introduction 27 .Never trust “beautiful” pictures produced by CFD codes. 28 Introduction .DNS of turbulent heat transfer with isoflux BC ⇒Boundary conditions: Computational domain and boundary conditions.CONST. L3 Free surface FREE SURFACE vnormalfreesurface= 0 . POWER DENSITY Y X θ + ( y = 1) = 0 or θ + ( y = 1) = 0 Outer wall boundary is adiabatic. FLOW dθ + ( y = − 1) = 0 dy -h L2=2h L1 Solid – fluid interface dθ + ( y = 1) = 0 dy ISOTHERMAL and ISOFLUX 0 Z h HEATED WALL . – wall of negligible thermal capacity and negligible thickness).DNS of turbulent heat transfer with isoflux BC Instantaneous dimensionless temperature field on the heated wall with isoflux BC (i.e. Introduction 29 . Technical University of Catalonia and Heat 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.consrv eqs 1 . Slovenia 1D 2-fluid models . Ljubljana. conservation equations 4) 1D two-fluid models . 2006 Table of contents INTRODUCTION Lectures 1-2 TWO-FLUID MODELS 3) 1D two-fluid models .Two-phase flow modelling.flow regime maps and closure equations 5) Characteristic upwind schemes for two-fluid models 6) 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 . seminar at UPC.consrv eqs 2 . Introduction .conservation equations Contents .classification of two-fluid models . .Hyperbolicity .Drift-flux model.6-equation two-fluid models. . .Interfacial area transport equation. .1D two-fluid models .Homogeneous equilibrium model.Two-pressure two-fluid models.consrv eqs 3 . 1D 2-fluid models . S.com/RELAP5/manuals/index.htm . Corradini. Yadigaroglu.consrv eqs 4 . 2006. Hibiki.conservation equations Selected references . G.RELAP5 computer code manuals: http://www.G. G. annual 1-week seminar at ETH Zurich. Springer. Ishii.Materials of the "Short Courses on Multiphase Flow nad Heat Transfer".Hewitt. T.L. Thermo-fluid dynamics of two-phase flows. One-dimensional two-phase flowm McGraw-Hill. Wallis. (Lead lecturers: S. Tryggvason.B. Zaleski) 1D 2-fluid models .M. 1969 . G.F. M. Hetsroni.1D two-fluid models .edasolutions. G. Banerjee. . consrv eqs 5 . ψ A B r P C r vector of n independent variables n*n matrix of terms with time derivatives n*n matrix of terms with spatial derivatives source term vector .Classification of two-fluid models General form of the two-fluid model equations: r ∂ψ ∂ψ A + B = P ∂t ∂x r r r ∂ψ ∂ψ +C = S ∂t ∂x r r ⋅ A−1 "Standard" two-fluid models do not contain terms with second order derivatives.closure relations without derivatives n*n matrix (preferably with n real eigenvalues and n linearly independent eigenvectors) 1D 2-fluid models .Introduction . Introduction . (Diffusive terms can be found in two-fluid models of CFD codes Their accuracy is questionable. but they certainly have a positive influence on the stability of the numerical schemes.) 1D 2-fluid models . Their inclusion would not improve the accuracy of these models.consrv eqs 6 . Viscous stresses and heat conduction are described with constitutive equations that do not contain derivatives.Classification of two-fluid models "Standard" two-fluid model equations: r r r ∂ψ ∂ψ +C = S ∂t ∂x do not contain terms with second order derivatives. Introduction . additional equation for interfacial area concentration) 8+ . example: different types of bubbles modelled with separate balance equations) 1D 2-fluid models .Classification of two-fluid models Classification according to the number of equations .consrv eqs 7 r r r ∂ψ ∂ψ +C = S ∂t ∂x .equation models (multi-field models. CATHARE) 7-equation models (two-pressure models.dimension of the r vector ψ : 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. v g .consrv eqs r inhomogeneous model without heat transfer 8 . pm ) r ψ = (α . Other possibilities exist for 3-equation two-fluid model: ψ = ( ρ m . v f ) 1D 2-fluid models . Important from the theoretical point of view .Homogeneous equilibrium model (3-equation model) n=3 (HEM model should not be called two-fluid model) conservative variables or basic 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 . ρ m um ) r ψ = ( ρ m . ρ m v m .represents a limit of higher two-fluid models. v m . Very strong interaction between both phases assures equal phasic velocities and equal phasic temperatures. 2D. gravity + F f .wall + qwall ∂t ∂x ∂x 1D 2-fluid models .wall ∂t ∂x ∂x •Momentum balance •Energy balance ∂ ρ m um ∂ ρ m um vm ∂ vm + +p = vm Fm.consrv eqs 9 . 3D).Homogeneous-equilibrium model The simplest averaged model of two-phase flow (works in 1D. Such approximation is seldom acceptable. •Mass balance for mixture: ∂ ρm ∂ ρ m vm + =0 ∂t ∂x ρm ∂ vm ∂v ∂p + ρ m vm m + = F f . Homogeneous-equilibrium model • Equation of state (probably the most complicated part of the HEM model). • Complicated calculation from equations of state: α m = α m (ρ m − saturation . u m − saturation ) Sonic velocity exhibits strong discontinuity between the single-phase and two-phase flow. p m = p m (ρ m − saturation . u m − saturation • • ) Closure relations needed for wall friction and wall heat flux.consrv eqs 10 . No special model needed for single-to-two-phase flow transition. 1D 2-fluid models . Very popular model in the early days of nuclear thermal-hydraulics. ) 1D 2-fluid models . (Other types of 4-equation two-fluid models can be constructed.Drift flux model (4-equation model) n=4 (drift flux model .again not called two-fluid model) ψ = ( ρ m . (see Ishii. Mixture velocity obtained from the balance equations. but not from differential equation but from the empirical correlations. ρ g .gas) Drift flux model or 4-equation two fluid model: one phase in saturation conditions (usually vapor).consrv eqs 11 . ρ m v m .mixture. Hibiki for details). g . ρ m um ) r (m . relative velocity also available. other phase not necessarily in saturation. 40-years old model . Findlay. J. Heat Transfer 87) • • Mixture mass balance: Gas-phase mass balance ∂α ρg ∂t + ∂α ρ g v g ∂x ⎞ ρ ρ ∂ ⎛ ⎜ α (1 − α ) f g vr ⎟ = Γg − ⎟ ∂x ⎜ ρf ⎝ ⎠ 12 ∂ ρm ∂ ρ m vm + =0 ∂t ∂x 1D 2-fluid models .consrv eqs .Drift-flux model Drift flux model takes into account the relative velocity of two phases: vr = v g − v f The relative velocity depends on the type of the two-phase flow (flow regime) and must be supplied with appropriate correlations). 1965.still useful in engineering applications (Zuber. consrv eqs 13 . wall heat flux Fg .Drift-flux model • Mixture momentum balance 2 ρ ρ 2⎞ ∂ ρ m vm ∂ ρ m vm ∂ ⎛ ⎜ α (1 − α ) f g v r ⎟ + ∂ pm = Fg . viscous stress tensor in 2D. 3D versions (not written in balance equations) 1D 2-fluid models . gravity + Fg .wall + qwall ∂t ∂x ∂x • Closure relations: correlation for relative velocity vr correlation for inter-phase mass transfer Γg equation of state wall friction.wall + + ⎟ ρf ∂t ∂x ∂x ⎜ ∂x ⎝ ⎠ • Mixture energy (phases in thermal equilibrium): ∂ ρ m um ∂ ρ m um vm ∂v + + p m = vm Fm.wall qwall conductive heat flux. the other one in non-equilibrium. This type of two-fluid model was built into the computer code RELAP5/MOD1. mechanical non-equilibrium possible.5-equation two-fluid models n=5 a) ψ = ( ρ g . ρ f v f . ρ g u g ) r Thermal non-equilibrium between both phases possible. ρ m um ) r One phase in saturation conditions. ρ g v g .consrv eqs 14 . 1D 2-fluid models . ρ f . ρ f . Version of the computer code for nuclear thermal-hydraulics analyses from ~1985. ρ m vm . ρ f u f . mechanical equilibrium .homogeneous flow (not very realistic and not used in practise) b) ψ = ( ρ g . ρ g vg .6-equation two-fluid models n=6 ψ = (ρ g . 1D 2-fluid models . TRAC.France. TRACE (RELAP5 and TRAC merged 2-3 years ago) . Nuclear Engineering and Design 124 (3). ρ f . ρ f v f . Both pressures equal. The physical closure laws in the CATHARE code. 1990. This type of two-fluid model is built into the nuclear thermal-hydraulics computer codes that are still in use today and RELAP5. CATHARE code . ρ f u f ) Both phases can exhibit departure from saturation conditions. ρ g ug .D.all codes made in USA.consrv eqs 15 r . Mechanical non-equilibrium possible.manuals of the RELAP5 computer code (available online on internet) .Bestion. References: . more experiments needed.α ) ρ f ∂t ∂A α ρ g ∂t + ∂A (1 . Closure relations are mainly based on empirical approach.α ) ρ f v f + = − A Γg ∂x ∂A α ρ g v g ∂x = A Γg Γg vapor mass generation per unit volume A (x ) pipe cross-section 1D 2-fluid models . Thus.6-Equation Two-Fluid Model • Requires even more closure relations than the drift flux model. Mass balances: ∂A (1 .consrv eqs (streamwise variations allowed) 16 . gravity + F f .consrv eqs 17 .6-Equation Two-Fluid Model • Momentum balances ∂ vf ∂ v2 1 ∂p f + (1 .wall 2 ∂t ∂x ∂x (1 . wall + α ρg ∂t ∂x ∂x 2 CVM Virtual mass term.α ) ρ f α ρg ∂ vg ∂ v2 ∂p 1 g +α + CVM = −Ci | vr | vr + Γg (vi − v g ) + Fg .α ) ρ f + (1 . gravity + Fg . contains derivatives! Interface friction coefficient Interface velocity Ci vi 1D 2-fluid models .α ) − CVM = Ci | vr | vr − Γg (vi − v f ) + F f . α ) v f ∂Aα * +p = A Qif − Γg h f + v f F f .consrv eqs 18 .wall ∂t ∂x ( ) Qig Qif * hg h* f gas-interface and liquid-interface heat fluxes per unit volume specific gas and liquid enthalpies at the interface (usually saturation enthalpies) 1D 2-fluid models .α ) ρ f u f v f ∂t + ∂x • −p ∂A(1 .6-Equation Two-Fluid Model Energy balances: ∂A(1 − α ) ρ f u f ∂A(1 .wall ∂t ∂x ( ) ∂ Aα ρ g u g ∂t + ∂Aα ρ g u g v g ∂x +p ∂ Aα ∂A α v g * +p = A Qig + Γg hg + v g Fg . . others . ⎜∂ ⎟ ⎜ ∂p ⎟ ⎝ uk ⎠ p ⎝ ⎠u k • • • • • Correlations for inter-phase momentum transfer. Wall friction correlations. 1D 2-fluid models . Correlations for wall-to-fluid heat transfer ..consrv eqs Qig Qif Γg 19 .6-Equation Two-Fluid Model Closure relations: • Two additional equations of state for each phase k are: ⎛∂ρ ⎞ ⎛∂ρ ⎞ d ρ k = ⎜ k ⎟ d p + ⎜ k ⎟ d uk ... CVM Ci vi Correlations for inter-phase heat and mass transfer. .no derivatives .interface pressure term .virtual mass term .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..unsteady wall friction terms (in single-phase 1D flows).contribute to vector P .in dispersed flows (motion of the bubble/droplet causes motion of the neighbouring mass of the opposite phase ) .6-Equation Two-Fluid Model r ∂ψ ∂ψ A + B = P ∂t ∂x r r Closure relations: r • non-diferential closures .consrv eqs 20 . .stratified flows in 1D approximation .contain temporal and/or spatial derivatives of the variables contribute to matrices. examples: A. B . • differential closure equations . 1D 2-fluid models . • The same physical phenomena can be sometimes described with differential or non-differential model • closure equations with second-order derivatives . consrv eqs 21 . In practice such diffusion terms are not explicitly added. Differential terms (virtual mass. 1D 2-fluid models . interface pressure) may be used to improve hyperbolicity (interface pressure term added into CATHARE code two-fluid model without physical background. Standard 6-equation two-fluid model is non-hyperbolic (ill-posed. Even a small term with second-order derivatives removes ill-posedness of the two-fluid equations. has "slightly" complex −1 eigenvalues of the matrix C = A B ). B and mathematical character of the equations.6-Equation Two-Fluid Model r ∂ψ ∂ψ A + B = P ∂t ∂x r r Closure equations with first order derivatives influence the matrices and A.e. with purpose to remove non-hyperbolicity). i. but come in the form of the numerical diffusion of the first-order accurate schemes. .interfacial area concentration is a basis for all the closure laws describing inter-phase heat.. ρ g v g . ρ g u g .consrv eqs 22 . .7 th variable ) Possibilities for 7th variable: .transport equation for interfacial area concentration ..vapor volume fraction α model assumes phasic pressure nonequilibrium (two-pressure two-fluid model). Hibiki). ρ f . r 1D 2-fluid models .7-equation two-fluid models n=7 ψ = ( ρ g .. ρ f u f . mass and momentum transfer (Ishii. ρ f v f ..concentration of non-condensable gas (RELAP5) . 1999).α ) v f ( ρ f E f + p f ) ∂x + pi vi ∂(1 . model but with two separate phasic pressures.7-equation two-fluid model Two-pressure two-fluid model Alternative approach to 6-eq. J. Comput. two-fluid model: • 7-equation "two-pressure" two-fluid model (Saurel.wall ∂t ∂x ∂x ( ) 1D 2-fluid models . ∂(1 − α ) ρ f E f ∂t + ∂(1 . Abgrall. Physics 150 (2). Additional equation for volume fraction completes the system of equations: ∂α ∂α + vm = µ ( p g − pl ) ∂t ∂x • New terms in total energy equations.wall f ∂x ( ) ∂α ρ g E g ∂αv g ( g g E g + pg) ∂α * + + pi vi = µpi ( p g − pl ) + Qig + Γg hg + v g Fg.consrv eqs 23 . Very similar equations like 6-eq.α ) = −µpi ( p g − pl ) + Qif − Γg h* + v f F f . .Less problems with numerics (allows calculations of extremely large pressure and volume fraction gradients without oscillations) .Much simpler eigenstructure of the equations (simple analytical expressions for eigenvalues and eigenvectors) .Unknown relaxation time for the pressure non-equilibrium.consrv eqs 24 .The "two-pressure" model can be used as a single pressure model if instantaneous pressure relaxation is assumed ( µ = ∞ ). Problems: . 1D 2-fluid models . model comparing to standard 6-eq.No problems with hyperbolicity (no need for virtual mass or empirical interfacial pressure term) .7-equation two-fluid model Two-pressure two-fluid model Advantages of the 7-eq.Pressure relaxation term is very stiff (very short relaxation time). model: . for example liquid in annular flow. is modelled with a separate conservation equation for the liquid film at the wall and a separate equation for the droplets in the vapor code of the flow.8+ -equations two-fluid models n=8 and more . ETH Zurich.) 1D 2-fluid models .multi-group models: for bubbly flows: bubble size spectra divided into various classes. . Each class of bubbles treated with a separate balance equation (see publications by U.multi-field models (see lecture notes of S. Banerjee at Modelling and Computation of Multiphase Flows.consrv eqs 25 . Forschungszentrum Rossendorf and CFX5 code manual. Rohde. annual seminars) The same phase. mass and momentum exchange in two-phase flows.Interfacial area transport equation . .ai is flow regime dependent . Advantage of the transport equation over the "standard" (non-differential) closures for ai is more continuous transition between the correlations of different flow regimes.Interfacial area ai is the most important parameter that governs the inter-phase heat. .consrv eqs 26 .Advantage of the transport equation for ai ∂ ai ∂a + v i = SOURCES + SINKS ∂t ∂x Advantage of additional equation .Like all other variables .it can actually serves as a quantity describing the flow regime.more accurate closure relations in transients that change the flow regimes. Hibiki) 1D 2-fluid models . (reference: Ishii. Technical University of Catalonia and Heat and Mass Transfer Technological Center. Ljubljana. 2006 Seminar on Two-phase flow modelling 4) 1D two-fluid models flow regime maps and closure equations by Iztok Tiselj "Jožef Stefan“ Institute.closures 1 . Slovenia 1D 2-fluid models . conservation equations 4) 1D two-fluid models .flow regime maps and closure equations 5) Characteristic upwind schemes for two-fluid models 6) 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 . seminar at UPC.Two-phase flow modelling. 2006 Table of contents INTRODUCTION Lectures 1-2 TWO-FLUID MODELS 3) 1D two-fluid models .closures 2 . wall friction .Differential closure equations .wall-fluid heat transfer .closures 3 .vertical flow regimes .Non-differential closure equations .virtual mass .Contents .inter-phase heat and mass transfer .flow regime maps and closure eqautions .horizontal flow regimes .correlations for flow regime transitions .Flow regime maps .inter-phase friction .unsteady wall friction 1D 2-fluid models .1D two-fluid models .interface pressure . a complete set of 1D flow regimes and closure laws applied in one of the leading codes for analyses in nuclear thermal-hydraulics.RELAP5 manual .closures 4 .Reference .1D two-fluid models .flow regime maps and closure equtions . 1D 2-fluid models . 2*Bubbly flow . 1982): Flow regimes from left to right: .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.closures 5 .Slug or plug flow . Stromung und Warmeubergang in Gas-FlussigkeitsGemischen.Annular-whisp 1D 2-fluid models . Springer-Verlag.Annular . Closure laws are developed separately for each flow regime. which is based on geometry of the flow.not directly applicable in 2D. which determines borders between different flow regimes. 1D 2-fluid models .. Flow regime maps . 3D cannot base on "integral quantity". mass and momentum transfer and wall-to-fluid transfer strongly depend on the flow regime. 3D two-phase flow modelling: local closure laws in 2D. Thus .the first step in development of the closure laws for 1D twofluid models is to draw an accurate flow regime map.closures 6 .Flow regime maps Flow regime is an integral "quantity". Inter-phase heat.. closures 7 . Multiphase Flow 1.Example of horizontal flow regime map Flow regime map for horizontal flow. Int. 1974. J. j g = αv g jl = (1 − α )vl ( j f = jl ) 1D 2-fluid models . From Mandhane et al. 3 manual) 1D 2-fluid models .closures 8 .Horizontal flow regime map in RELAP5 code (drawing from RELAP5/mod3. j g = αv g jl = (1 − α )vl ( j f = jl ) 1D 2-fluid models .closures 9 . 1969. From Hewit. Roberts.Example of vertical flow regime map Flow regime map for cocurrent vertical upward flow. closures 10 .3 manual) 1D 2-fluid models .Vertical flow regime map in RELAP5 code (drawing from RELAP5/mod3. pipe diameter. Typical simulation of the transient in the nuclear power plant coolant loop. pipe inclination. They are based on a wide range of experiments but are are limited to the measurements and experimental conditions (type of fluid..).. 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 .Correlations for flow regime transitions Various flow regime maps exist. Flow regime maps are believed (I. Tiselj) to be the major source of uncertainty in the computer codes based on two-fluid models. Flow regime maps in the computer codes must operate in much wider range of parameters. pressure. temperature.closures 11 . Can take into account history or spatial distribution of the variables. . .closures 12 .no derivatives .Validity in transient conditions questionable. .contain temporal and/or spatial derivatives of the variables.Derived from steady-state experiments. .Influence the mathematical character of the equations and the speed of sound in the two-phase flow.Differential and non-differential closure laws • Non-diferential closures .Difficult to develop (experiments in transient conditions needed). . Can be obtained with theoretical approach. 1D 2-fluid models . • Differential closure equations .Easier to develop from the experimental data. wall Examples of Ci : . Standard laws for friction near the flat wall are applied.wall Differenti al terms = −Ci | v r | v r + Γg ( vi − v g ) + Fg . 1D 2-fluid models . bubble stable at the local relative velocity v r = v g − v f . gravity + Fg . gravity + F f . . bubble diameter=half of the max.stress terms due to the relative motion of both phases: Liquid and gas phase momentum equations: Differential terms= Ci | vr | vr − Γg ( vi − v f ) + F f . Assumption: interface is a flat plate.closures 13 .Horizontally stratified flow (RELAP5).Inter-phase friction friction Non-differential closure equations Physical background .Bubbly flow (RELAP5). Assumptions: all bubbles of the same size. 6α bub / d0 Re = (We ⋅ σ ) (1 − α ) ( ) µ f v2 fg (We ⋅ σ ) = max(5 ⋅ σ .75 ) / Re bubble .005α bub ) ⎠ ⎝ 2 fg d0 = (We ⋅ σ ) 1D 2-fluid models .5 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: agf = 3. 10 −10 ) ⎞ ⎛ 2 (We ⋅ σ ) ⎟ ⎜ vr . 0.1Re 0.1⎟ ⎝8 ⎠ C D = min 24(1 + 0. 0. v = max 1/ 3 ⎟ ⎜ ρ f min( D . 0.bubbly flow inter-phase friction (RELAP5): Drag coefficient of the bubble: ⎛1 ⎞ Ci = max⎜ ρ g CD agf .Example .closures ρ f v2 fg 14 . Example .Force of g on f: F f = Fg = C i v 2 r 1 1 f f ρ f (v f − v i ) 2 = f g ρ g (v g − v i ) 2 8 8 Friction factors near the flat wall f f = (0.stratified flow friction (WAHA code): Force of f on g = .79 ln(Re g ) − 1. v g − v i Ag ρ g ⎟ Re g = max⎜ ⎟ µg ⎟ ⎜ ⎠ ⎝ Inter-phase friction coefficient: ( v g − vi ) 2 1 a Ci = ρ g f g 2 gf 8 (v g − v f ) ( v f − vi ) 2 1 a ) (or Ci = ρ f f f 2 gf 8 (v g − v f ) Approximate interfacial area concentration in 2 min(α .64) −2 ⎞ ⎛ ⎜1000 .64) −2 ⎞ ⎛ ⎜1000 .5(v f + v g ) 15 .79 ln(Re f ) − 1. v f − v i A f ρ f ⎟ Re f = max⎜ ⎟ µf ⎟ ⎜ ⎠ ⎝ f g = (0. (1 − α )) a gf = the circular pipe A 1D 2-fluid models .closures Iterative procedure starts with initial guess vi = 0. α ) ρ f v f + ∂A α ρ g v g ∂x ∂x = − A Γg ∂A α ρ g ∂t Γg vapor mass generation per unit volume = A Γg Differential terms= A Qif − Γg h* + v f F f .α ) ρ f ∂t + ∂A (1 .closures 16 .Inter-phase heat and mass transfer Non-differential closure equations Physical background: ∂A (1 .wall f * + Γg hg + v g Fg .wall ig ( Differential terms = A(Q ) ) Qig Qif * hg h* f gas-interface and liquid-interface heat fluxes per unit volume specific gas and liquid enthalpies at the interface (usually saturation enthalpies) 1D 2-fluid models . Fluxes . Details .elsewhere (RELAP5). are flow regime dependent TS (interfacial area dependent). hg = hg − saturation * .Inter-phase heat and mass transfer The vapor generation rate is calculated from known heat fluxes as: Γg = − Qif + Qig h −h * g * f h* = h f f * . hg = hg if if Γg > 0 Γg < 0 h* = h f − saturation f h = u + p/ ρ The liquid-to-interface and gas-to-interface volumetric heat fluxes Interface temperature is assumed to be a saturation temperature at the local pressure.closures 17 . Q Q ig Qif = H if (TS − T f ) Qig = H ig (TS − T g ) if 1D 2-fluid models . White correlation (for single phase flow): Laminar flow: 64 fw = 1 fw Re ⎛ 2.wall Differenti al terms = −Ci | v r | v r + Γg ( vi − v g ) + Fg .wall = f wf ρ f v f v f (1 − α ) ρ f 2D ρm Fg .27 ⎟ ⎜ Re f D⎟ w ⎝ ⎠ 1D 2-fluid models .Wall friction Non-differential closure equations Simple model .wall Darcy equations modified for the two-phase flow: F f . gravity + F f .wall = f wg ρ g v g v g αρ g 2D ρm Colebrook.closures Turbulent flow‫׃‬ Differential correlations to take into account transient effects. gravity + Fg .. 18 .51 k⎞ ⎜ = −2 log + 0.calculate single phase friction for two-phase mixture and split the friction between both phases: Differenti al terms= Ci | vr | vr − Γg ( vi − v f ) + F f .. wall + Qwg 1D 2-fluid models .closures 19 ( ) ( ) . f .energy eq.Wall-to-fluid heat transfer Non-differential closures Physical background: wallto-fluid heat transfer important in the flow around the fuel elements of the nuclear power plant.energy eq. differential terms= A Qif − Γg h* + v f F f . differential terms = * A Qig + Γg hg + v g Fg .wall + Qwf f g . closures ⎧ ⎪ 1 1 + 2α ⋅ ⎪ 2 1− a ⎪ ⎪ = ρ mα (1 − α ) ⎨ ⎪ ⎪ ⎛ 3 − 2α ⎞ 2 (1 − α )( 2α − 1) ⎪ ⎜ ⎟ + ⎪ ⎝ 2α ⎠ (1 − α + αρ g / ρ f ) 2 ⎩ α ≤ 0. This can be taken into account with a new term in momentum equation: α ρg ∂ vg ∂ v2 ∂p 1 g +α + CVM = non − differenti al terms + α ρg ∂t ∂x ∂x 2 Simplified term for the 1D two-fluid models (one of the possibilities): ∂ vg ∂ v f ∂vf ⎞ ⎛ ∂ vg CVM = CVM ⎜ ⎜ ∂t + v f ∂x .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.∂t .v g ∂x ⎟ ⎟ ⎝ ⎠ CVM 1D 2-fluid models .4 α > 0.6 20 . Moreover.Clearly and accurately defined only for spherical particles.. Bubbles/droplets are often non-spherical. Virtual mass term can make equations of the 6-equation two-fluid model hyperbolic.Even less than in the bubbly and droplet flow regimes is known about the virtual mass term in other flow regimes. . 1D 2-fluid models . size of the bubbles is not known.closures 21 .. .Historical reason for inclusion of the VM term: more stable numerics.Virtual mass term Differential closure equation Problem of the virtual mass term: . Interface pressure term Differential closure equation Physical background: interface pressure term allows simulations of the horizontally stratified flows with 1D two-fluid model .appears in momentum equations: α ρg (1 .α ) ρ f ∂ vg ∂ v2 1 ∂p ∂α g + α ρg +α + Pi = non − differenti al terms 2 ∂t ∂x ∂x ∂x ∂ v2 1 ∂p ∂α f + (1 .α ) − Pi = non − differential terms 2 ∂t ∂x ∂x ∂x ∂vf Interface pressure must be: Pi = α (1 − α )( ρ f − ρ g ) gD D pipe diameter to obtain solutions that behave like solutions of the shallow water equation 1D 2-fluid models .α ) ρ f + (1 .closures 22 . interface pressure term can make the two-fluid model hyperbolic. CATHARE code is using interface pressure term in stratified flow: this term is sufficient to make equations hyperbolic in Pi = α (1 − α )( ρ f − ρ g ) gD horizontally stratified flows and in all other flow regimes an expression which makes equations hyperbolic (almost hyperbolic): hyperbolicity can be lost 2 ρ g ρ f vr when relative velocity Pi = becomes comparable with αρ f + (1 − α ) ρ g the speed of sound in the two-phase mixture 1D 2-fluid models .Interface pressure term Differential closure equation Mathematical background: like virtual mass term.closures 23 . Such correlations are insufficient for some of the fast transients with pressure waves in the piping systems.Unsteady wall friction Differential closure equation Physical background: Standard wall friction correlations are developed from the steady-state measurements.closures 24 r r r r . Simplified single-phase momentum equation: r 4 rv ∂p 1 =− τ + + ρ ρ D v ∂x ∂x 2 Unsteady ∂ t fricton equation: wall ∂v ∂ v2 τ s steady state wall friction τ s − τ Dτ = θ relaxation time correlation D t θ More details in lectures on 1D simulations of fast transients. 1D 2-fluid models . Flow regime is integral "quantity". must be tested with "integral experiments".closures 25 . but. mass and momentum transfer and wall-to-fluid transfer depend on the flow regime. Applicability of a specific two-fluid model with a given set of closure equations for the particular transient in the nuclear power plant.Closure equations . Application of "integral quantity" on the local scale of partial differential equations is questionable. Results are especially questionable in simulations of the transients with flow regime transition. It "works" in 1D. 1D 2-fluid models . how to transport the flow regime information to 2D. 3D ? • Closure relations are the main source of uncertainty in the two-fluid models.conclusions • Closure equations describing inter-phase heat. Slovenia characteristic-upwind schemes 1 . 2006 Seminar on Two-phase flow modelling 5) Characteristic upwind schemes for two-fluid models by Iztok Tiselj "Jožef Stefan“ Institute.Technical University of Catalonia and Heat and Mass Transfer Technological Center. Ljubljana. seminar at UPC.Two-phase flow modelling.flow regime maps and closure equations 5) Characteristic upwind schemes for two-fluid models 6) 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 2 .conservation equations 4) 1D two-fluid models . 2006 Table of contents INTRODUCTION Lectures 1-2 TWO-FLUID MODELS 3) 1D two-fluid models . Introduction to high resolution shock capturing schemes for Euler equations of single-phase compressible flows.Integration of the stiff source terms.second-order accurate solutions . yes or no? characteristic-upwind schemes 3 .Integration of the geometric source terms.Characteristic-upwind schemes for two-fluid models: .Two-fluid models: conservative or non-conservative form? . eigenvectors of the two-fluid model equations.Characteristic-upwind schemes for two-fluid models .Characteristic-upwind schemes for two-fluid models. .Riemann solvers . .Pressure-based and characteristic upwind schemes.Eigenvalues.Contents . . . . Anderson. Physics 136 (2) 503-521. Vol. I. Lectures in Mathematics. Pember. No. 5.Characteristic upwind schemes for two-fluid models . Computational Fluid Dynamics. Math. Physics 150. S. J. ETH. R. Numerical Methods for Conservation Laws.Selected references Books: C. 1997. SIAM J. (1992). Saurel. Zurich. 53. Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation I. Papers: R. Comp. 425-467. LeVeque. Spurious Solutions". John Wiley & Sons. Hirsch. J. McGraw-Hill. 1-2. 1999. A Multiphase Godunov method for compressible multifluid and multiphase flows. Numerical computation of internal and external flow. (1995). (1988). Modelling of two-phase flow with second-order accurate scheme. B. Comp. 1293 (1993) characteristic-upwind schemes 4 . D. R. J. R. Petelin. Abgrall. New York. J. Tiselj. Appl. Pressure-based and characteristic upwind schemes Pressure-based schemes: pressure is a "privileged" variable comparing to density. . density. 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. Characteristic upwind schemes: pressure treated like all other variables (velocity..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) . Suitable for incompressible flows..suitable for Euler equations of compressible flows. (argument for pressurebased schemes) characteristic-upwind schemes 5 .Pressure based schemes are not limited only to incompressible but can usually handle "slightly" compressible flows. temperature) . longer history .Characteristic upwind scheme can be easily upgraded into secondorder accurate scheme. Their weak side is numerical dissipation. which means reduced numerical diffusion.Advantage of characteristic upwind approach: for fast transients with pressure waves.New pressure-based schemes are improved also for slightly compressible flows.Pressure-based and characteristic upwind schemes Characteristic upwind approach vs. . 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. pressure-based methods: . which tends to smear discontinuities on coarse grids. . . second-order accurate versions available (CFD codes). characteristic-upwind schemes 6 . robust and efficient.older versions were firstorder accurate in time and space.Pressure-based methods . A ρ e] Equation of state (ideal gas): r p 1 2 E= + ρv γ −1 2 γ= cp cv 7 characteristic-upwind schemes . A ρ v .High resolution shock capturing schemes for Euler equations Euler equations of single-phase compressible quasi-1D flow of ideal gas: ⎡ Aρ ⎤ ⎡ Aρv ⎤ ⎡0⎤ ⎢ Aρv ⎥ + ⎢ A( ρv 2 + p )⎥ = ⎢ p ⎥ dA Conservative form ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ AE ⎥ t ⎣ ⎦ ⎢ Av ( E + p ) ⎥ x ⎣ ⎦ Non-conservative vectorial form: r ∂ψ ∂ψ +C = S ∂t ∂x 1 ⎞ ⎛ E = ρ e = ρ u + ρ v2 ⎟ ⎜ 2 ⎠ ⎝ r r ⎢0⎥ ⎣ ⎦ dx Conservative variables are used in vector ψ = [ A ρ . High resolution shock capturing schemes for Euler equations Jacobian matrix: 0 1 0 ⎤ ⎡ (3 − γ )v γ − 1⎥ C = ⎢ (γ − 3)v 2 / 2 ⎥ ⎢ 3 2 ⎢(γ − 1)v / 2 − vh h − (γ − 1)v γv ⎥ ⎦ ⎣ Diagonalized: C = L⋅ Λ⋅ L Eigenvalues: ⎡v + c −1 ⎛ 2 γ p⎞ ⎟ ⎜c = ⎜ ρ ⎟ ⎠ ⎝ h = e+ p/ρ Λ=⎢ 0 ⎢ ⎢ 0 ⎣ 0 0⎤ v − c 0⎥ ⎥ v⎥ 0 ⎦ Eigenvectors ⎡ 1 1 ⎤ L = ⎢ v +c v −c v ⎥ ⎥ ⎢ 2 ⎢h + cv h − cv v / 2⎥ ⎦ ⎣ 1 8 characteristic-upwind schemes High resolution shock capturing schemes for Euler equations Equation: r ∂ψ ∂ψ +C =S ∂t ∂x r ∂A ∂ψ −1 ∂ψ + L⋅Λ⋅ L + R =0 ∂x ∂x ∂t r r r −1 ∂ψ −1 ∂ψ −1 −1 ∂A L + Λ⋅L + Λ⋅ Λ ⋅L R = 0 ∂t ∂x ∂x r r r r rewritten: Modified characteristic variables introduced: δξ = L δψ + Λ ⋅ L −1 r −1 −1 r R δA CHARACTERISTIC FORM OF EQUATIONS: ∂ξ ∂ξ +Λ =0 ∂t ∂x characteristic-upwind schemes 9 r r High resolution shock capturing schemes for Euler equations - discrete form Vectorial equations r ∂ψ ∂ψ +C =S ∂t ∂x ψ nj - ψ n-1 j r r r r are numerically solved with explicit time integration (n - time, j - space): ψ n+1 - ψ nj j r r ∆t ∆x ∆x r ++ n r -- n A j - A j -1 A j+1 - A j + ( R ) j+1/2 =0 + ( R ) j -1/2 ∆x ∆x + C ++ ( ) n j -1/2 + C −− ( ) n j+1/2 ψ nj+1 - ψ n j r r ( ) (C ) ++ n C j −1 / 2 −− n j +1 / 2 ( = (L = L ⋅Λ ++ −1 n ⋅L j −1 / 2 n j +1 / 2 ⋅ Λ − − ⋅ L −1 ) ) r n r -- n −− −1 ( R ) j +1 / 2 = ( L ⋅ F ⋅ L R ) j +1 / 2 r n r ++ n ++ −1 ( R ) j +1 / 2 = ( L ⋅ F ⋅ L R ) j +1 / 2 CFL limit on time step: ∆t < ∆x / max(v − c , v + c ) characteristic-upwind schemes 10 High resolution shock capturing schemes for Euler equations - discrete form Matrices Λ + + , Λ −− , F + + , F −− : + λk + = λk ⋅ f k+ + k = 1,3 − λk − = λk ⋅ f k− − k = 1,3 ⎛ ∆t ⎞ λ ⎞ φ ⎛ - 1⎟ f ++ = max⎜ 0 , k ⎟ + k ⎜ λ k k ⎜ | |⎟ 2 ⎝ ∆x ⎠ λk ⎠ ⎝ f -k ⎛ ∆t ⎞ λ ⎞ φ ⎛ = min⎜ 0 , k ⎟ - k ⎜ λ k - 1⎟ ⎜ | |⎟ 2 ⎝ ∆x ⎠ λk ⎠ ⎝ SECOND-ORDER CORRECTIONS Flux (slope) limiters: ξ k, j+1-m - ξ k, j -m ∆ξ k, j+1 / 2-m λ k, j+1/2 = , m= θ k, j+1/2 = MINMOD | λ k, j+1/2 | ξ k, j+1 - ξ k, j ∆ξ k, j+1 / 2 φ k = max(0 , min(1 ,θ k )) r r r −1 −1 −1 ∆ξ j+1/ 2 = L ∆ψ + Λ ⋅ L RA ∆A j+1/ 2 Van Leer φ k = ( θ k + θ k ) /( θ k + 1) Superbee φk = 0 1st-order upwind φ k = max( 0, min( 2θ k ,1), min(θ k ,2)) φk = 1 2nd-order Lax-Wendroff ( ) characteristic-upwind schemes 11 High resolution shock capturing schemes for Euler equations - discrete form Jacobian matrix averaging (Roe's approximate Riemann solver): C j +1 / 2 0 ⎡ 2 (γ − 3)vave / 2 =⎢ ⎢ 3 ⎢(γ − 1)vave / 2 − vave have ⎣ A j ρ j v j + A j +1ρ j +1 v j +1 A j ρ j + A j +1ρ j +1 1 (3 − γ )vave 2 have − (γ − 1)vave 0 ⎤ γ − 1⎥ ⎥ γvave ⎥ ⎦ A j ρ j h j + A j +1ρ j +1 h j +1 A j ρ j + A j +1ρ j +1 ( vave ) j +1 / 2 = ( have ) j +1 / 2 = ( ρ ave ) j +1 / 2 = ρ j ρ j +1 ( Aave ) j +1 / 2 = A j A j +1 characteristic-upwind schemes 12 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: ω∆ ( Aρ ) + ∆ ( Aρv ) = 0 ω∆ ( Aρv ) + ∆A ( ρv 2 + p ) = p (dA / dx )ω ω∆ ( AE ) + ∆ ( Av ( E + p ) ) = 0 ( ) ω ∆ propagation velocity of the shock wave difference between the quantities ahead and behind the shock cross-section derivative in point of the discontinuity (dA / dx )ω (entropy fix procedure - see LeVeque for details - must be added to remove the discontinuities that violate entropy law - rarefaction shock waves.) characteristic-upwind schemes 13 High resolution shock capturing schemes for Euler equations - solutions (Sod's shock-tube) p ρ v length (m) 1- shock wave, 2- rarefaction wave, 3 - contact discontinuity Sod, JCP 27, 1978 characteristic-upwind schemes 14 High resolution shock capturing schemes for Euler equations .Lax Wendroff fails for Sod's case due to the very large discontinuity.shock-tube solutions (100 grid points) velocity (m/s) upwind 1st-order Lax-Wendroff 2nd-order length (m) analytical high resolution 2nd-order (Not Sod's shock tube ..) characteristic-upwind schemes 15 .. diagonalization can be performed numerically.High resolution shock capturing schemes for Euler equations .what is applicable for two-fluid models? Problems of two-fluid models: . 1997.Diagonalization of the Jacobian matrix of 6-equation two-fluid model is a difficult task: . JCP 136. . (Details: Tiselj. . 2004) characteristic-upwind schemes 16 . Petelin.diagonalization can be performed with analytical approximations. WAHA code manual.Equations are "Euler-like" but not necessarily hyperbolic. i. Fluids 8 (2). (See example of shock wave in bubbly mixture.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).e.shocks in two-phase flow are not discontinuities. 1996) experiment pressure (bar) analitical solution of hypothetical two-fluid model time (ms) characteristic-upwind schemes 17 . Moreover ..High resolution shock capturing schemes for Euler equations . Phys. Rankine-Hugoniot conditions are unknown... Kameda. Matsumoto. which represent external forces (gravity. 2) Source terms describing interphase mass.not stiff (probably).. and energy transfer. momentum.e.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 . SPECIAL TREATMENT REQUIRED. wall friction) and wall heat transfer . which tend to establish mechanical and thermal equilibrium – i.High resolution shock capturing schemes for Euler equations . characteristic-upwind schemes 18 .can be treated with characteristic upwind in the convection part of equations. RELAXATION source terms. 3) Other source terms. These source terms are STIFF (their time scale can be much shorter than the time scale of the sonic waves). wall friction and volumetric forces are solved in the first sub step with upwind discretisation: r r r ∂ψ ∂ψ A +B = SNON _ RELAXATION . Operator splitting: 1) Convection and non-relaxation source terms .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. ∂t ∂x 2) Relaxation (inter-phase exchange) source terms: r r dψ A = SRELAXATION dt characteristic-upwind schemes 19 .source terms due to the smooth area change. characteristic-upwind schemes 20 . ∂t ∂x ∂x This part of the scheme is the same as for the Euler equations of the single-phase compressible flow. r r r ∂A r ∂ψ ∂ψ −1 + L⋅Λ⋅ L + RA + RF = 0 . ∂t ∂x C = L ⋅ Λ ⋅ L−1 r r Eigenvalues and eigenvectors of Jacobian matrix are found: Source terms are rewritten: r RA contains source terms due to the variable pipe cross-section r RF contains wall friction and volumetric forces (no derivatives).1st substep of operator splitting: convection terms with non-relaxation source terms Equation solved: r ∂ψ ∂ψ −1 +C = A ⋅ S N −R . ) characteristic-upwind schemes 21 r .1st substep of operator splitting: basic variables Basic variables are ~ primitive variables. u f . (1 .α ) ρ f v f . No conservation of momentum: equations of two-fluid model cannot be written in conservative form. virtual mass terms.vg .α ) ρ f .. ug ) ( ρ f ..α ) ρ f e f .α .(1 . (Conservation of momentum is less important than conservation of mass/energy.α ρ g . and possibly other correlations that contain derivatives.α ρ g eg] Conservative equations + and -: 1)+ Numerical conservation of mass and energy can be assured with conservative variables. interfacial pressure terms. due to the pressure gradient terms. v f . ρ g replaced with u f .α ρ g vg . ψ = ( p. u g ) The preferred set of variables would be conservative variables: r ϕ = [(1 . α ) ρ f u f . v g ."Non-standard" water property subroutines are required that calculate two-phase properties ( p . (1 − α ) ρ f .αρ g e g thanαthef primitive variables: ψ = ( p. (1 − α ) oscillations . (1 .αρ g v g . 4)+/-The conservative quantities as components of vector. u f . u g ) characteristic-upwind schemes 22 r r . α ρ g .α ) ρ f .Primitive variables are very convenient for evaluation of eigenvalues and eigenvectors. (1 . α ρ g u g 3). (1 − ) ρ e f ) are more( sensitive to the numerical ρ f v f . ρ ) from the conservative f g variables ( ).1st substep of operator splitting: basic variables Conservative variables + and -: 2). ψ = αρ g .α . ρ .α . v f . v f . (1 − α ) ρ f e f ) characteristic-upwind schemes 23 r .Specific numerical oscillations are induced near the property discontinuities (Karni. Abgrall. +/. (1 − α ) ρ f .αρ g e g .The optimal set of variables might be a mixture of conservative and nonconservative variables: ψ = ( αρ g . 1994. + Non-conservation of mass and energy can also cause numerical oscillations near the strong pressure and volume fraction discontinuities. 1996) when conservative variables are used.1st substep of operator splitting: basic variables Conservative variables + and -: 4) CONTINUED . Petelin. JCP 136. 1997 Influence of the basic variables on the solution of the Toumi's shock tube problem for the 6-equation two-fluid model.1st substep of operator splitting: basic variables see Tiselj. characteristic-upwind schemes 24 . αRIGHT=0. JCP 136. Initial vapor volume fraction discontinuity: αLEFT=0.1st substep of operator splitting: basic variables .25.examples see Tiselj. 1997 Influence of the basic variables on the solution of the Toumi's shock tube problem for the 6-equation two-fluid model. Petelin.1 characteristic-upwind schemes 25 . 1st substep of operator splitting: basic variables . αRIGHT=0.examples see Tiselj. Petelin. JCP 136.1 characteristic-upwind schemes 26 . Initial vapor volume fraction discontinuity: αLEFT=0.9. 1997 Influence of the basic variables on the solution of the Tiselj's shock tube problem for the 6-equation two-fluid model. 7) equations in two-phase volume. 6 (5. Problem: transition from single-phase to two-phase flow.1st substep of operator splitting: basic variables ..conclusions Optimal scheme for the convective part of equations remains to be found. 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). Implicit time schemes might be preferred. 3 equations in single-phase volume. characteristic-upwind schemes 27 .. i.2nd substep of operator splitting: integration of stiff relaxation source terms r r dψ A = S RELAXATION dt Relaxation source terms: inter-phase heat. characteristic-upwind schemes 28 . Integration of the relaxation sources within the operatorsplitting scheme is performed with variable time steps. their characteristic time scales can be much shorter that the time scales of the hyperbolic part of the equations. which depend on the stiffness of the source terms. Upwinding is not used (difficult to use) for calculation of the relaxation source terms. mass and momentum exchange terms are stiff.e.. or to prevent the change of sign of phasic temperature differences. The maximal relative change of the basic variables in one step of the integration is limited to 0. Time step is further reduced when it is necessary to prevent the change of relative velocity direction.01 to obtain results that are "numerics" independent.2nd substep of operator splitting: integration of stiff relaxation source terms Second equation of the operator splitting scheme ψ r m+1 rm r rm = ψ + A (ψ )S (ψ )∆tS rm −1 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. characteristic-upwind schemes 29 . Probably the best solution: implicit integration of relaxation sources. em ρm . It is in principle possible to choose a set of basic variables: ψ M = ( ρm .Tg ) that enables simplified integration of the relaxation source terms.vm ρm . and mixture total energy should remain unchanged after the integration of the relaxation source terms. It is difficult to calculate the state of the fluid from the variables that are result of such relaxation. Tf .vg − v f . mixture momentum. Only a system of three differential equations is solved instead of the system of six. r 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. .Numerical schemes for hyperbolic equation with stiff source terms .Stiff source terms are integrated with variable time step depending on the stiffness.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 nonphysical 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. energy and momentum exchange in two-fluid models do not produce spurious solutions and do not modify the propagation velocity of the discontinuities. two-fluid model confirmed the results of Pember: the stiff sources describing inter-phase mass. characteristic-upwind schemes 31 .Numerical tests with the 6-eq. Also very though test for numerics. • 2) Simple water hammer experiments (Simpson. Brookhaven Nat. al. 1989). 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. characteristic-upwind schemes 32 . • 3) Two-phase flow in the nozzle. Lab.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). 1981.). Hig) 100 characteristic-upwind schemes 33 . heat and mass transfer (Hif.45 40 35 30 25 20 15 10 5 0 0 20 Vhem Vf Ci=10 Hif=Hig=10^3 Vg Propagation velocities of shock and rarefaction waves in two-fluid models 640 Them Tf Tg Ci=10 Hif=Hig=10^3 40 60 80 635 100 630 625 620 615 610 0 20 40 60 80 Shock waves of two fluid model with various interphase momentum (Ci). 45 40 35 30 25 20 15 10 5 0 0 20 40 60 80 Ci=10^3 Hif=Hig=10^6 640 635 100 630 625 620 615 610 0 20 40 60 80 Ci=10^3 Hif=Hig=10^6 100 characteristic-upwind schemes 34 . 45 40 35 30 25 20 15 10 5 0 0 20 40 60 80 Ci=10^4 Hif=Hig=10^7 640 635 100 630 625 620 615 610 0 20 40 60 80 Ci=10^4 Hif=Hig=10^7 100 characteristic-upwind schemes 35 . 45 40 35 30 25 20 15 10 5 0 0 20 40 60 80 Ci=10^5 Hif=Hig=10^9 640 635 100 630 625 620 615 610 0 20 40 60 80 Ci=10^5 Hif=Hig=10^9 100 characteristic-upwind schemes 36 . 45 40 35 30 25 20 15 10 5 0 0 20 40 60 80 Ci=10^6 Hif=Hig=10^11 640 635 630 625 620 615 610 0 20 40 60 80 100 Ci=10^6 Hif =Hig=10^11 100 characteristic-upwind schemes 37 . characteristic-upwind schemes 38 • • . which states that the stiff relaxation source terms do not produce spurious solutions.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. 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. Results with the two-fluid model confirm the Pember's conjecture from (1993). 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. Universite Catholique de Louvain. Grenoble. Slovenia. characteristic-upwind schemes 39 .conclusions Is it reasonable to develop new codes based on characteristic upwind schemes? New code for simulation of water hammer transients . France. Belgium. Comissariat a l'Energie Atomique.WAHA .Characteristic upwind schemes for two-fluid models . Authors: Jozef Stefan Institute.has been developed using characteristic upwind scheme within the WAHALoads project financed by EU's 5th research program. Slovenia pressure-based schemes 1 .Technical University of Catalonia and Heat and Mass Transfer Technological Center. Ljubljana. 2006 Seminar on Two-phase flow modelling 6) Pressure-based solvers for two-fluid models by Iztok Tiselj "Jožef Stefan“ Institute. 2006 Table of contents INTRODUCTION Lectures 1-2 TWO-FLUID MODELS 3) 1D two-fluid models .conservation equations 4) 1D two-fluid models . seminar at UPC.flow regime maps and closure equations 5) Characteristic upwind schemes for two-fluid models 6) 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 2 .Two-phase flow modelling. Numerical diffusion.Numerical scheme of RELAP5 .Pressure-based solvers for two-fluid models Contents .Introduction . accuracy pressure-based schemes 3 . Pressure-based solvers for two-fluid models Selected references Book: Ferziger.com/Wiki/Numerical_methods RELAP5. Peric. 1997. NEPTUNE manuals pressure-based schemes 4 . Computational methods for fluid dynamics. Fluent. Springer. Internet: http://www.cfd-online. CFX. pressure-based methods Pressure equation arises from the requirement that the solution of the momentum equation also satisfies continuity. (CFX is known to have problems with inviscid flows) pressure-based schemes 5 . Such discretisation is often unstable . Often in the conservative form.Introduction .especially if diffusive terms (second-order derivatives) are absent. "Standard" two-fluid model equations: r r r ∂ψ ∂ψ +C = S ∂t ∂x Equations are discretised "directly". Schemes developed for conservation laws in single-phase flow are usually applied also for two-phase flows . Number of conservation laws not important. Pressure-velocity coupling: .Rhie-Chow type of velocity interpolation on coincident grids (used in general-purpouse CFD codes) pressure-based schemes 6 . .use staggered grid. Stability comes from the numerical diffusion of first-order accurate discretisation and artificial viscosity term.Introduction ..30 years old numerical scheme .no second-order terms in RELAP5 two-fluid model.especially in 2D.avoid checker-board of pressure-velocity field: .pressure-based methods RELAP5 .. 3D CFD codes. w) 2) Solve pressure correction equation (SIMPLE..p) 2) Solve transport equations for other scalars pressure-based schemes 7 .w.for twophase flows) 1) Solve Momentum equations (u.. NEPTUNE. Fluent .) – Correct fluxes and velocities 3) Solve transport equations for other scalars Coupled Solver (CFX.Introduction .v.Pressure equation system in one go (u.for single-phase flows) 1) Solve the Momentum equations. Fluent . CFX.v.pressure-based methods Segregated Solver (RELAP5. pressure-based methods Overview of the segregated solver (from Fluent manual): pressure-based schemes 8 .Introduction . pressure-based methods Overview of the coupled solver (from Fluent manual): pressure-based schemes 9 .Introduction . velocities calculated at the boundaries of the control volumes.Acoustic terms: ∂ρ ∂v +ρ =0 ∂t ∂x ρ ∂v ∂p + =0 ∂t ∂x . explicit for non-acoustic terms (semiimplicit scheme) .Staggered grid .Implicit for the acoustic terms.RELAP5 numerical scheme (simplified) RELAP5 continuity and momentum equation for single-phase flow: ∂ρ ∂ρv =0 + ∂x ∂t ∂v ρ ∂v 2 ∂p =0 ρ + + ∂t 2 ∂x ∂x RELAP5 code discretisation properties: . .Artificial viscosity term added for stability in the momentum equation. pressure-based schemes 10 . vg momentum control volume pressure-based schemes 11 .RELAP5 numerical scheme (simplified) Staggered grid in RELAP5: mass.α.g.uf. energy scalar node control volume p.g vf j vg j+1 j+1/2 velocity node vf.ρf. density for example: ρi +1 / 2 = ⎨ ⎧ρ i +1 vi +1 / 2 < 0 ⎪ ⎪ ρ i vi +1 / 2 > 0 ⎩ the same for velocity Difference equations obtained for the positive velocities in the grid points i and i+1/2: ρ n +1 i ρ −ρ + ∆t n i n n +1 i v i +1 / 2 −ρ ∆x n vin−+1/ 2 i −1 1 =0 1st-order accurate difference ⎞ ∆x ⎤ ⎟ ⎥ ⎟ 2 ⎥ ⎠ ⎦ vin++1/ 2 − vin+1 / 2 ρ in ⎡ ( v 2 ) in+1 − ( v 2 ) in ⎛ ( v 2 ) in+3 / 2 − 2(v 2 ) in+1 / 2 + ( v 2 ) in−1 / 2 1 + −⎜ ⎢ ⎜ 2 ⎢ ∆t ∆x ∆x 2 ⎝ ⎣ artificial viscosity term pin++1 − pin +1 1 + =0 ∆x 2nd-order accurate difference ρ in+1 / 2 pressure-based schemes 12 .RELAP5 numerical scheme (simplified) Donor-cell discretisation of the convective terms. other sources . the velocity field is updated.with explicit integration. Other variables . pressure-based schemes 13 . Inter-phase exchange terms are also calculated implicitly.calculated in two steps . Velocity is eliminated and a linear system of N-equations is solved with unknown pressure pn+1.mainly due to the stiff interphase exchange source terms. (N number of volumes) After calculation of the pressure field.RELAP5 numerical scheme (simplified) ρ in +1 − ρ in ∆t ρ in+1 / 2 + ρ in vin++1/ 2 − ρ in−1vin−+1/ 2 1 1 ∆x =0 ⎞ ∆x ⎤ ⎟ ⎥ ⎟ 2 ⎥ ⎠ ⎦ vin++1/ 2 − vin+1 / 2 ρ in ⎡ ( v 2 ) in+1 − ( v 2 ) in ⎛ ( v 2 ) in+3 / 2 − 2(v 2 ) in+1 / 2 + ( v 2 ) in−1 / 2 1 + −⎜ ⎢ ⎜ 2 ⎢ ∆t ∆x ∆x 2 ⎝ ⎣ pin++1 − pin +1 1 + =0 ∆x Two-equations written in each point. RELAP5 and other codes in nuclear thermalhydraulics TRAC. More implicit approach allows use of longer time steps . CFX) .however. CATHARE .even more implicit treatment of equations. More implicit approach means more stability. but not more accuracy (stability is a result of numerical diffusion of the implicit schemes). CATHARE . time step longer than the characteristic time of the physical phenomena means non-accurate simulation of the phenomena..fully implicit: ψ n +1 −ψ n ∆t r r + r r r n +1 ∆f (ψ n +1 ) = S (ψ ) ∆x Multi-dimensional codes (NEPTUNE..fully implicit. pressure-based schemes 14 . 15 0.05 0.002 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/c RELAP5 2 ∆t=0.01∆x/c 2F . 15 0.WAHA ∆t=∆x/c adaptive time step for relaxation source terms. fraction 0.1 0.001 0 0 0.25 tim e (s ) pressure-based schemes ."Water hammer due to the valve closure" simulation RELAP5 1 2 F 2 nd -o rd e r RELAP5 2 Vapor vol.2 0. 18 16 Pressure (MPa) 14 12 10 8 0 2 4 Length (m) 6 8 10 This is a consequence of the 2nd-order central differencing of the pressure gradient: v n +1 − v n p n +1 − p n +1 ρ in+1 / 2 i +1 / 2 ∆t i +1 / 2 + .RELAP5 at very small time steps RELAP5 1st RELAP5 2nd 2nd-order scheme Quasi second-order pressure waves are predicted by the RELAP5 when a very small time step is used. numerical oscillations appear near the shock wave.. The resolution of the steep gradients is improved. however..CONVECTION + i +1 i ∆x =0 pressure-based schemes 16 . Advantages of the characteristic based schemes seem to be insufficient to justify development of the codes based on the characteristic upwind schemes.conclusions Work fairly well. pressure-based schemes 17 .the main problem of the two-phase flows is not numerics and numerical errors but physics and physical models. although the various numerical artifacts are less controlled than in the characteristic upwind schemes.it is not very important .Pressure-based methods . Characteristic upwind or pressure-based schemes . Slovenia 7-3D-two-phase-flows 1 . Ljubljana.Technical University of Catalonia and Heat 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. Two-phase flow modelling. 2006 Table of contents INTRODUCTION TWO-FLUID MODELS Lecture 1-2 Lectures 3-6 INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows . seminar at UPC.mathematical background 8) Interface tracking models 9) Coupling of two-fluid models and VOF method 10) Simulations of Kelvin-Helmholtz instability ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME 7-3D-two-phase-flows Lectures 11-14 Lectures 15-18 2 . computer codes CFX. . Fluent. NEPTUNE (new 3D code for nuclear thermal hydraulics).Introduction. NEPTUNE . Fluent.Turbulence in two-fluid model codes. Fluent.3D closure laws in CFX.3D two-fluid models in CFX.3D two-phase flows .mathematical background .Contents . NEPTUNE . 7-3D-two-phase-flows 3 . 2006) .References . 30:139– 65 7-3D-two-phase-flows 4 .Tsai & Yue Annu. Rev. McFadden. Wheeler. Hibiki (book.Detailed surface modelling (non-zero thickness of the interface. Fluent manuals Additional: .Ishii.3D two-phase flows . Fluid.. 1998. Mech..mathematical background . Fluid Mech.about freesurface flows in oceanography .NEPTUNE.28:249-78 . Annu. Rev. CFX.): Anderson. 1996. Conclusions: "Two-phase CFD is much less mature than single phase CFD. Thus it is clear that the physical modelling will have to be improved over a long time period. ECORA strongly recommends further investigations on this topic.ECORA .. The flows are much more complex and myriads of basic phenomena may take place at various scales.. . 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.project of 5th research program of EU ECORA document: Recommendation on use of CFD codes for nuclear reactor safety analyses .." 7-3D-two-phase-flows 5 .. This makes that the separation between physics and numerics is not always well defined. NURESIM . Key objectives of NURESIM: (i) integration of advanced physical models in a shared and open software platform. thermal-hydraulics.project of 6th research program of EU The European Platform for NUclear REactor SIMulations. and coupled (multi-) physics modeling. verification and benchmarking. 7-3D-two-phase-flows 6 . (iv) training. NURESIM is planned to become common European standard software platform for modeling. best practice and quality assurance. (iii) progress assessment by using deterministic and statistical sensitivity and uncertainty analyses. recording. dissemination. (ii) promoting and incorporating the latest advances in reactor and core physics. and recovering computer data for nuclear reactors simulations. NURESIM . standardization and robustness of the envisaged NURESIM European Platform would address current and future needs of industry.NEPTUNE code. Thermal hydraulics . academic. government. core physics. and private institutions. and multi-physics) of the present and future reactors. 7-3D-two-phase-flows 7 .project of 6th research program of EU The specific objectives of NURESIM are to initiate the development of the next-generation of experimentally validated. reactor safety organizations. The improved prediction capabilities. “best-estimate” tools for modeling (thermal-hydraulics. Future ??? Both codes have a strong two-phase flow modules. Fluent .Computer codes CFX. Neptune . Especially useful for particles (bubbly flows).major players on the market of CFD codes. CFX and Fluent used to be competitors.commercial CFD codes .academic licenses ~1000 EU per CPU . Future dissemination ??? 7-3D-two-phase-flows 8 .in development. but have recently got the same owner (ANSYS).nuclear thermal-hydraulics oriented code . t )) ∂t r f s (r . t ) equation of interface 7-3D-two-phase-flows 9 .Navier-Stokes equations.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 r ∇⋅v = 0 Equation for interface tracking (form continuity eq. whole-domain formulation Approach available in Fluent with VOF technique .) Momentum equation δ Dirac delta function r ∂ρ r + v ∇ρ = 0 ∂t r rr r ∂ρv + ∇ ⋅ (ρv v ) = ρF − ∇ ⋅ ( p I + τ ) + σκδ ( f s ( r . two-fluid model) 7-3D-two-phase-flows .Liquid Volume conservation α L + αG = 1 1 momentum equation r rr r r ∂ ρU + ∇ ⋅ ρUU − µ ∇U + (∇U )T = −∇p + ρg + S M ∂t ρ = α L ρ L + (1 − α L )ρG µ = α L µ L + (1 − α L )µG ( ) ( ( )) Density Viscosity User specified momentum source 10 Model available in CFX5 and Fluent CFD codes (3.4.3D two-fluid models .Gas User specified mass source Interphase mass transfer r ∂ (α L ρ L ) + ∇ ⋅ α L ρ LU = SML + ΓLG ∂t ( ) ( ) L .homogeneous (equal velocity) model 2 Continuity equations r ∂ (αG ρ G ) + ∇ ⋅ αG ρ GU = ΓGL ∂t G . or 5 eqs. 3D two-fluid models . Fluent User specified and Neptune CFD codes (4.5.6 momentum source eqs.inhomogeneous model (different velocities) 2 Countinity equations r ∂ (α L ρ L ) + ∇ ⋅ ρ LU L = SML + ΓLG ∂t ( ) r ∂ (α G ρG ) + ∇ ⋅ ρGU G = SMG + ΓGL ∂t ( ) User specified mass source Inter-phase mass transfer 2 momentum equations r r r r r r r r r ∂ ⎛ ⎛ ∇U + ∇U T ⎞ ⎞ + Γ U − Γ U + S + M α L ρ LU L + ∇ ⋅ α L ρ LU L ⊗ U L = −α L∇p + ∇ ⋅ ⎜ α L µ L ⎜ L L ⎟⎟ LG G GL L ML L ⎝ ⎠⎠ ∂t ⎝ r r r r r r r r r ∂ ⎛ ⎛ ∇U + ∇U T ⎞ ⎞ + Γ U − Γ U + S + M α G ρGU G + ∇ ⋅ α G ρGU G ⊗ U G = −α G ∇p + ∇ ⋅ ⎜ α G µG ⎜ G G ⎟⎟ GL L LG G MG G ⎝ ⎠⎠ ∂t ⎝ ( ( ) ( ( )) ( ) ( ) ) ( ( )) ( ) ( ) Model available in CFX5. two-fluid model) 7-3D-two-phase-flows Interfacial forces acting on phase L due to presence of other phase 11 . interfacial area ai is supposed to be a part of solution and not a user defined parameter.. 7-3D-two-phase-flows 12 . Adroplet = πD / 4 1 ρL U L − UG A 2 ( D ) Mixture model (for droplets) C LG r r CD = ai ρ LG U G − U G 8 ρ LG = α L ρ L + α G ρG Interfacial area per unit volume ai = α Lα G d LG Mixing length scale . Fluent.inter-phase momentum transfer in dispersed flows Drag Force r r r M L = CLG U G − U L ( ) Similar model found in CFX.. Neptune Dimensionless drag force coefficient CD = 2 r r 2 .3D two-fluid models .user specified . . CFX and Fluent offer drag forces for non-spherical bubbles.15 Re0. Fluent and Neptune can take into account also the following inter-phase momentum transfer in dispersed flows: lift. Approach probably useful for particle flows (and allows numerous user defined parameters to fit the experiments. virtual mass.. but should be switched on by user.687 Re ( ) Transitional area at medium Reynolds numbers. (How do one knows that bubbles changed their shape?) CFX...inter-phase momentum transfer in dispersed flows Dimensionless drag force coefficient for spherical particles (bubbles. turbulent dispersion force.3D two-fluid models . droplets) Low Reynolds Re<<1 CD = 24 Re High Re: Schiller-Naumann drag model CD = 24 1 + 0.) 7-3D-two-phase-flows 13 . . Fluent ..3D two-fluid models .no correlations 3D two-phase flows.inter-phase momentum transfer in stratified flows Neptune ..inter-phase momentum transfer in dispersed-to-stratified flows ??? (user defined.separate correlations CFX.) 7-3D-two-phase-flows 14 . 3D two-fluid models energy equations 2 Total Energy equations.tot + QL + S L Heat transfer induced by Interphase heat interphase mass transfer transfer 1 r r Total enthalpy htot = hstat + U ⋅ U 2 External heat source Static enthalpy ( ) 15 7-3D-two-phase-flows .tot − α L λL∇TL − ∇ ⋅ ⎜ α L ρ L hL .tot − α G λG ∇TG − ∇ ⋅ ⎜ α G µG ⎜ ∇U + ∇U α G ρG hG .tot − α L ∂t ∂t ⎝ ⎝ ( ) ( ) ( ) 2 r ⎞r − ∇U δ ⎟U 3 ⎠ ΓLG hG .tot + QG + SG r r ⎛ ∂ ∂p ⎛ r ⎜ α L µ L ⎜ ∇U + ∇U + ∇ ⋅ α L ρ LU hL .tot − α G ⎜ ∂t ∂t ⎝ ⎝ ( ) ( ) ( ) T T 2 r ⎞r − ∇U δ ⎟U 3 ⎠ ⎞ ⎟= ⎟ ⎠ ⎞ ⎟= ⎟ ⎠ ΓGL hL .tot − ΓGL hL. 1 Momentum equation r r ⎛ ∂ ∂p ⎛ r + ∇ ⋅ α G ρGU hG .tot − ΓLG hG . 3D two-fluid models energy equations 2 Thermal Energy equations. 2 Momentum equations r ∂ (α L ρ L hL ) + ∇ ⋅ α L ρ LU L hL − α LλL∇TL = ΓLG hG .tot − ΓGL hL.tot + QL + S L ∂t Heat transfer induced by interphase mass transfer External heat source r ∂ (α G ρG hG ) + ∇ ⋅ α G ρGU G hG − α G λG∇TG = ΓGL hL − ΓLG hG + QG + SG ∂t ( ) ( ) Interphase heat transfer 7-3D-two-phase-flows 16 . H LS = H L hG . SAT . hL Heat transfer coefficients 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 .3D two-fluid models inter-phase heat & mass transfer Interfacial heat transfer – Thermal phase change model & ΓGL = mGL AGL Interfacial area density Interfacial mass flux & mGL = qLG + qGL H GS − H LS Heat flux from phase G to L Heat flux from phase L to G qGL = hL (TSAT − TL ) qLG = hG (TSAT − TG ) & mGL > 0 → H GS = H G . SAT & mGL < 0 → H GS = H G . H LS = H L . nucleate boiling correlations (important for nuclear simulations) .. Acceptable if the wall-fluid area known for each phase . Neptune: .. again part of the solution is expected as a user defined parameter.3D two-fluid models wall-to-fluid heat transfer Single phase type of heat transfer assumed in CFX and Fluent..flashing flow model (flashing delay possible in Neptune) 7-3D-two-phase-flows 18 .. 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-phase flows .turbulence Characteristic length scales of the interface motion can be much larger than the characteristic scales of turbulent flows. example: turbulent flume. 7-3D-two-phase-flows 19 . 0 σ ε = 1.44 Turbulent kinetic energy k ( ) ) ( Turbulence production r r rT 2 r r Pk = µt ∇U ⋅ ∇U + ∇U − ∇ ⋅ U 3µt ∇ ⋅ U + ρk 3 ( ) ( ) Cµ = 0.turbulence Turbulence k-ε.92 Cε 1 = 1.3D two-fluid models .09 Effective Viscosity µ eff = µ + µ t 2 ρk 3 µt = Cµ ρ k2 Modified pressure p′ = p + ε Turbulent viscosity 7-3D-two-phase-flows 20 .3 Cε 2 = 1. for one phase or both phases r ⎛⎛ ⎞ ⎞ ∂ ⎜ ⎜ µ + µ t ⎟∇k ⎟ + Pk − ρε (ρk ) + ∇ ⋅ ρUk = ∇ ⋅ ⎜ ⎜ ∂t σk ⎟ ⎟ ⎠ ⎠ ⎝⎝ r ⎛⎛ ⎞ ⎞ ∂ Turbulent eddy ⎜ ⎜ µ + µ t ⎟∇ε ⎟ + ε (Cε 1 Pk − Cε 2 ρε ) (ρε ) + ∇ ⋅ ρUε = ∇ ⋅ ⎜ ⎜ dissipation ε ∂t σε ⎟ ⎟ k ⎝ ⎠ ⎠ ⎝ σ k = 1. 3D two-fluid models - turbulence Turbulence (NEPTUNE) Model of dispersed phase kinetic energy transport and fluid/particle fluctuating movement covariance Model of dispersed phase kinetic stress and fluctuating movement covariance Fluent, CFX: user can apply various turbulence models in every phase that he/she wants... 7-3D-two-phase-flows 21 Technical University of Catalonia and Heat 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 1 Two-phase flow modelling, seminar at UPC, 2006 Table of contents INTRODUCTION TWO-FLUID MODELS Lecture 1-2 Lectures 3-6 INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows - mathematical background 8) Interface tracking models 9) Coupling of two-fluid models and VOF method 10) Simulations of Kelvin-Helmholtz instability ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME 8-interface-tracking Lectures 11-14 Lectures 15-18 2 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 3 Interface tracking methods- References 1 Lagrangian 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 4 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 Flow Lemonnier , Jamet, Lebaigue, Begell House, 2005. (test cases for interface tracking methods) 8-interface-tracking 5 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 6 Marker-And-Cell (MAC) . Level-set: Each fluid is treated with function tracing the amount of each phase in the given point. Volume-Of-Fluid (VOF). allow very accurate representation of the surface (accurate surface tension calculations).Embedded interface methods All use surface markers. 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 7 .Eulerian interface tracking methods Marker methods: . .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. pressure correction schemes . NEPTUNE. CFX4).segregated solvers (Fluent..coupled solvers .The most efficient single-phase schemes are not necessarily successful in two-phase flow. Useful schemes: . .available for two-phase flow in CFX5 (not in CFX4) 8-interface-tracking 8 . 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. t )) ∂t –Interface is being tracked with the surface markers connected into the surface..closer relation with twofluid models.. r Front-tracking methods . 8-interface-tracking 10 . Volume-tracking preferred .Embedded interface methods (Tryggvason) –Fixed Eulerian grid –Whole-domain formulation r rr r ∂ρv + ∇ ⋅ (ρv v ) = ρ k F − ∇ ⋅ ( pk I + τ k ) + σκδ ( f s ( r .not further discussed in this seminar. 8-interface-tracking 11 .Volume of fluid (VOF) – To compute time evolution of free surface continuity equation for void fraction is solved r ∂ (α ) + ∇ αU = 0 ∂t ( ) – 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. 95 0. Position of the interface in the Eulerian grid and void fractions..0 0.2 1. all based on geometry 8-interface-tracking 12 .Volume of fluid (VOF) Interface reconstruction – Reconstructs surface from volume fraction with geometrical elements.0 1.7 Many different reconstruction schemes.07 0.0 0.4 0..0 1. 0. Simple Line Interface Reconstruction with Calculation (SLIC) step function First-order reconstructions. 8-interface-tracking 13 .Volume of fluid (VOF) Interface reconstruction Different types of interface reconstruction: . Volume of fluid (VOF) Interface reconstruction Different types of interface reconstruction: .y (i.x 8-interface-tracking 14 .j-1) i.j) ∆y ∆x Second-order approaches but very complicated in 3D j.Least-squares Volume-of-Fluid Interface Reconstruction Algorithm (LVIRA) r n (i-1.j) (i.Flux Line-Segment for Advection and Interface Reconstruction (FLAIR) . which is positive in the space occupied by the first fluid.Level-Set Use of a continuous level-set function φ. xI ∈ Interface ( ) φ=2 φ =1 φ = −1 φ =0 Free surface position is defined with the zero value of level set function φ (distance function) 8-interface-tracking φ = −2 15 . negative in the space occupied by the second fluid. r Value of φ in a point x r is distance from point x to the surface r r r Φ = min x − xI . − ε ≤ Φ ≤ ε Φ >ε Φ < −ε –Where ε corresponds to the half of the interface thickness. Φ > 0 –To achieve numerical robustness a smeared out Heavy side function is often used ⎧0. Φ < 0 H (Φ ) = ⎨ ⎩1.Level-Set Temporal development equation ∂φ r + v ⋅ ∇φ = 0 ∂t –Heavy side function is used to represent density and viscosity over interface ⎧0. 8-interface-tracking 16 . ⎪ 1 Φ 1 ⎪ + sin H (Φ ) = ⎨ + 2 2ε 2π ⎪ ⎪ 1. ⎩ (πεΦ ). easier implementation of level-set.problems with φ near the steep gradients (bigger than in VOF). 3D . 8-interface-tracking 17 . Level-set . VOF more problematic.Level-Set vs. VOF Mass conserved in VOF but not in Level-Set (special additional algorithms needed). 2005 • After advective step . Phys. α does not measure distance from the surface but volume fraction.5 on the surface. 210.a different level-set function is defined: ∂α r + u ⋅ ∇α = 0 ∂t • α=0. We denote time variable by τ to stress that this is an artificial time. Artificial compression flux α(1-α) acts in the regions where 0<α<1. 8-interface-tracking 18 • v n is normal at the interface and is calculated only once at the . Equation which acts as artificial compression is solved until steady state is reached v ∇α ∂α v n= + ∇ ⋅ (α (1 − α )n ) = ε∆α ∇α ∂τ beginning of the second step. Comput.Conservative Level-Set Olsson & Kreiss. J. Small amount of “viscosity” ε∆α is added to smear discontinues. not equivalent to an actual time t. r r r r r r ∇p 1 ∂u + (u ⋅ ∇ )u = − + ∇ ⋅ µ ∇u + (∇u )T + g ∂t ρ ρ (( )) ρ = αρ1 + (1 − α )ρ 2 µ = αµ1 + (1 − α )µ 2 ∂α r + u ⋅ ∇α = 0 ∂t SIMPLE pressure correction procedure to get divergence free velocity field r • Solving momentum equation to obtain intermediate velocity u * ⎛ ∇p ′ ⎞ ⎛ 1 ⎞ r ⎟ = −⎜ ⎟∇ ⋅ u * ∇⎜ • Solving pressure correction equation ⎜ ⎟ ⎝ ∆t ⎠ ⎝ ρ ⎠ • Solving momentum equation only with the contribution of pressure r u n+1 part to get • Solving continuity equation for volume fraction to obtain α n+1 8-interface-tracking 19 .Conservative Level-Set .Our implementation System of Navier-Stokes eq. j-1/2 Gi.Conservative Level-Set Our implementation Staggered grid to avoid checkerboard distribution of the variables All equations discretized with fluxes to ensure conservation Gi. +1 j = α in j .j+1/2 i. . − ∆t Fi n1 / 2.j i.j p u v α in.j+1/2 Fi-1/2.j i.j-1/2 i+1/2. (5-diagonal matrix in 2D) 8-interface-tracking 20 .j i-1/2. second order accurate in space and time CGSTAB algorithm to solve pressure correction eq. j Gin j +1 / 2 − Gin j −1 / 2 + + . ∆x ∆y Second order discretization with Van Leer limiter (combination of upwind and Lax Wendroff scheme) -> decreased numerical diffusion and dispersion. j − Fi n1 / 2.j Fi+1/2. 14 m hl=0.81 m/s2 H=0. dam break on dry and wet surface Water-air system g=9.2 m 8-interface-tracking 21 .Dam break • • • • see: Validation of Advanced Computational Methods for Multiphase Flow for details of the benchmark Surface tension was neglected due to the scale of the problem Two problems.1 m hr=0.Conservative L-S .01 m L=1. Dam break • Mass conservation 1.1 0.2 0.0E-08 1.4 0.00E-03 0 1.0E-06 t 8-interface-tracking r max residual = ∇ ⋅ u 1.00E-06 1.00E-05 mass 1.0E-07 1.Conservative L-S .3 0.5 ∆mr = (m0 − m ) / mo 1.00E-07 1.00E-04 0.00E-08 22 . time step=1e-2 s.5 h @3. time step=1e-3 s. CPU time=1. CPU time=15 h @3.0 GHz Pentium 4 Wet ground – jet is formed • • Grid:1024x128.Conservative L-S .0 GHz Pentium 4 Most of the CPU time for pressure correction eq.Dam break • Dry ground • • Grid:512x64. 8-interface-tracking 23 . 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 24 . Interface sharpening in two-fluid models CFX The implementation of free surface flow involves some special discretisation options to keep the interface sharp. 8-interface-tracking 25 .not documented yet (similar mechanism as in CFX). These include: • A compressive differencing scheme for the advection of volume fractions in the volume fraction equations. Neptune .interface sharpening supposed to exist . • 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. Slovenia 9-VOF+two-fluid 1 .Technical University of Catalonia and Heat and Mass Transfer Technological Center. Ljubljana. 2006 Seminar on Two-phase flow modelling 9) Coupling of two-fluid models and VOF method by Iztok Tiselj "Jožef Stefan“ Institute. Two-phase flow modelling. 2006 Table of contents INTRODUCTION TWO-FLUID MODELS Lecture 1-2 Lectures 3-6 INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows .mathematical background 8) Interface tracking models 9) Coupling of two-fluid models and VOF method 10) Simulations of Kelvin-Helmholtz instability ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME 9-VOF+two-fluid Lectures 11-14 Lectures 15-18 2 . seminar at UPC. Coupling of the interface Tracking and the Two-Fluid Models . 171. Tiselj J. 38:329–350 Numerical errors of the VOF. Int. 9-VOF+two-fluid 3 . Comp. Numer. J. 776–804 (2001). Petelin... Phys... Fluids 2002.9) Coupling of two-fluid models and VOF method ...Contents -VOF method -“Two-fluid” model -Model Coupling -Simulation of the Rayleigh-Taylor instability References: Cerne. Meth. Description of the problem Various two-phase flow regimes separated flow . ∂α ∂t r + ∇ ⋅ αu = 0 r r r ∂uk α k ρk + α k ρk (uk ∇)uk = ∂t r r r α k ρk g − α k ∇p − Chk ur ur r ∂α k + ∇ ⋅ (α k u k ) = 0 ∂t ( ) ∑α k = 1 k interfacial drag: Chk =Ch1 =−Ch2 9-VOF+two-fluid 4 .model VOF dispersed flow two-fluid model r ∇⋅u = 0 r r r ∂u + ρ (u ⋅ ∇ )u = ρ ∂t r ρg − ∇p + ∇ ⋅ (µ D ). Description of the problem Flow regime change v v v 9-VOF+two-fluid 5 . j ⎩ fluid 1 fluid 2 both fluids r ∂α + ∇ ⋅ (uα ) = 0. i.I Whole domain formulation of basic equations (no surface tension term): r ∇⋅ u = 0 r r r r ∂u ρ + ρ (u ⋅ ∇)u = ρg − ∇p + ∇ ⋅ (µ D). j = ⎨0 . ⎪ αi. ∂t ⎧1 .VOF method . ⎪0 < α < 1. ∂t 9-VOF+two-fluid 6 . x 9-VOF+two-fluid 7 .07 0.0 → n (i-1.VOF method II simulated structures are larger than the grid distance 0.0 0.4 0.95 0.j) j.2 (i.0 1.y 1.7 (i.0 0.j-1) i.j) ∆y ∆x 1. I reconstruction error 9-VOF+two-fluid 8 .VOF errors . bubble on a coarse grid S d h 9-VOF+two-fluid 9 .VOF errors .I reconstruction error . 6 1 δ = N ( i . j (t )) (t ) 2 0.I reconstruction error 0.2 d bubble diameter h distance between the grid points ∆x = ∆y 0 0 2 4 6 d/h 8 10 9-VOF+two-fluid 10 . j (t ) − α RECONSTRUCTEDi. j )∈V ∑ (α ACTUALi.4 0.VOF errors . bubbles with d<2.VOF errors .II Advection error Initial state: .5h move faster .different bubbles flows together with the surrounding liquid in a constant velocity field Final state: .shapes of the bubbles are changed 9-VOF+two-fluid 11 . VOF errors III Numerical dispersion error 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) u d 9-VOF+two-fluid 12 . Several fluid chunks with the characteristic size h<d<3h are provided.VOF errors III Numerical dispersion error Numerical Dispersion When the strip width is close to the grid size. the fluid chunks are stable despite the shear velocity field. 9-VOF+two-fluid 13 . the tension of the reconstruction algorithm to keep the fluid chunk as compact as possible results in dispersion. vortex shear flow with the zero velocity in the origin and boundaries and maximum velocity in the middle circle bubble is put on the position of the maximum velocity gradient (point(0.0.VOF errors III Numerical dispersion error black coloured spot in the prescribed prescribed velocity field .5.85)) bubble is deformed into the spiral whirling to infinity 9-VOF+two-fluid 14 . VOF errors III Numerical dispersion error Numerical dispersion: left . right numerical solution on coarse grid 9-VOF+two-fluid 15 .solution on finer grid. 7 fluid 1 fluid 2 0.2 0.6 0.“Two-fluid” model .8 0.3 0.I simulated structures are smaller than the grid distance fluid 2 fluid 1 0.9 9-VOF+two-fluid 16 .7 0.3 0.9 0. “Two-fluid” model .II r ∂αk + ∇ ⋅ (αk uk ) = 0 ∂t ∑α k k =1 r r r r r r ∂uk f k ρk + αk ρk (uk ∇)uk = αk ρk g − αk ∇p + Ck (u1 − u2 ) + αk ∇ ⋅ (µk Dk ) ∂t interfacial drag r r 1 C1 = −C2 = cd ρc ai v1 − v2 8 C1 = −C2 = cd ρ α1α2 9-VOF+two-fluid 17 . I VOF two-fluid VOF 9-VOF+two-fluid 18 .Model coupling . j = 0 stratified fluids γ i. j (i +k . j +l (ξi . j V 9-VOF+two-fluid ∑H i +k .j) (i+1. j 1 = Vi. j ) − f i +k .II definition of the "dispersion" (i. j +l .j) γ i . j > 0 mixed fluids practical implementation: measured on 3x3 number of cells 19 γ i. j = func (local distribution of α ) γ i. j +l )∈ 1i .Model Coupling . j) is reconstructed γ i. j < γ 0 the interface in the cell (i.j) are calculated with the "two-fluid" model two-fluid model γ0 = 0 • VOF model γ 0 = max γ γ 0 = 0.Model Coupling III Switch criteria between models • • γ i. the fluids in the cell (i. j > γ 0 .8 20 Tests on simple two-fluid states 9-VOF+two-fluid .3 − 0. j = 0 .Transition between VOF and two-fluid model Wrong reconstruction: γ i.8 9-VOF+two-fluid 21 . 0 10 t 1.at switch to two-fluid model the error is increased due to the numerical diffusion.2 0.6 0.3 0.5 15 2.1 0 0 0.in the moment of numerical dispersion the VOF model significantly increases the error .the switch to denser nodalization model may delay the error increase .Advantage of the coupled model -the distributions of the volume fraction are compared to the exact solution .5 5 1.4 0.0 20 VOF 28x28 coupled 28x28 VOF swithed to 56x56 9-VOF+two-fluid 22 .5 0. but long time its prediction of volume fraction distribution is better than at VOF model δ 0. 8 t=1.6 t=3.6 t=4.6 t=2.VOF simulation Rayleigh-Taylor instability t=0 t=0.8 t=7 9-VOF+two-fluid 23 .Result .4 t=0. j (t )) (t ) 2 1.Comparison of VOF results for different grid densities δ nod = 1 N (i .2 0 1 10 f = α t 100 9-VOF+two-fluid 24 . j (t ) − α Li. j )∈ V ∑(α Mi.6 0.8 0.2 nod f 6x30 -f 12x60 f 12X60 -f 24X120 f 24X120 -f 48X240 f 48X240 -f 96x480 1 0.4 0. 8 t=7 9-VOF+two-fluid 25 .6 t=3.Rayleigh-Taylor instability t=0 t=0.8 t=1.4 t=0.coupling of VOF and two-fluid models .Results .6 t=2.6 t=4. j (t ) − α Li.1 0 1 10 f = α t 100 9-VOF+two-fluid 26 . j )∈ V ∑(α Mi.3 0.4 0. j (t )) (t ) 2 0.Comparison of coupled VOF+two-fluid model results for different grid densities δ nod = 1 N (i .5 0.2 0.6 nod fcoupled 6x30-fcoupled 12x60 fcoupled 12x60-fcoupled 24x120 fcoupled 24x120-fcoupled 48x240 fcoupled 48x240-fVOF 48x240 0. •The study in this paper was performed with the VOF method and the LVIRA piecewise linear reconstruction algorithm. the second solution is better. advection error and numerical dispersion. when the physical dispersion of the fluids is very fine. 9-VOF+two-fluid 27 .Conclusions VOF-two-fluid coupling CONCLUSIONS •The grid cell limitation causes some errors in the VOF model. •The numerical dispersion can be avoided either by grid refinement of the mesh or switching to the two-fluid model during the simulation. when the characteristic size of the chunks does not change much during the transient. The first solution is effective. like reconstruction error. On the other hand. Such errors cannot be reduced by applying better and more accurate interface tracking algorithm. however the results can be applied also for the other VOF reconstruction algorithms. I Estimate for the accuracy of the interface reconstruction: γ i.Accuracy of the interface reconstruction . j ≈ 0 separated fluids 9-VOF+two-fluid 28 . j r n γ i . j = ∇αi. j r n r n 9-VOF+two-fluid 29 .Accuracy of the interface reconstruction .II γ >1 mixed fluids i. Ljubljana. Slovenia 10 .K-H instability 1 .Technical University of Catalonia and Heat and Mass Transfer Technological Center. 2006 Seminar on Two-phase flow modelling 10) Simulations of Kelvin-Helmholtz instability by Iztok Tiselj "Jožef Stefan“ Institute. Two-phase flow modelling.K-H instability Lectures 11-14 Lectures 15-18 2 .mathematical background 8) Interface tracking models 9) Coupling of two-fluid models and VOF method 10) Simulations of Kelvin-Helmholtz instability ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME 10 . 2006 Table of contents INTRODUCTION TWO-FLUID MODELS Lecture 1-2 Lectures 3-6 INTERFACE TRACKING IN 3D TWO-PHASE FLOWS 7) 3D two-phase flows . seminar at UPC. Simulations of Kelvin-Helmholtz instability Contents Same phenomena simulated with: CFX .VOF simulation and two-fluid model simulation Conservative level-set (home-made code) Additional simulation: Condensation induced water hammer in horizontal pipe 10 .two-fluid model with and without interface sharpening Fluent .K-H instability 3 . Lebaigue. Begell House. • 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.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. 10 . 2005.K-H instability 4 . 001 Pa·s Wall. u=v=0 h2=15 mm H=30 mm h1=15 mm µ2=0.04 N/m ρ2=780 kg/m3 2 ∆U cr ≥ 2 ρ1=1000 kg/m3 µ1=0. J. 1969) Wall. u=v=0 L=1830 (200) mm U2 z=0 z ρ1 + ρ 2 ∆ρgσ ρ1 ρ 2 ∆ρ = ρ1 − ρ 2 kcr = ∆ρg / σ 2 λcr = 2π / kcr x U1 g γ=4. Fluid Mech.13 ° 10 .0015 Pa·s σ=0.Thorpe’s experiment (Thorpe.K-H instability 5 . 39. K-H instability 6 .5 s). Thorpe’s experiment is in agreement with results of the inviscid linear analysis. analytical solutions • Undisturbed velocity field (far from closed ends. Linear analysis is appropriate due to the small density ratio.Thorpe’s experiment vs. linear inviscid theory is insufficient at higher density ratios. 10 . neglected viscosity): U2 = − (ρ1 − ρ 2 )gh1 sin γ t ρ1 h2 + ρ 2 h1 U1 = (ρ1 − ρ 2 )gh2 sin γ t ρ1h2 + ρ 2 h1 • • • • • Experimental onset of instability is 1.88 s (analytical 1. Linear analysis is valid until amplitude is small. Experimental critical wavelength is 25-45 mm (analytical 27 mm). Two simulations were done: – Simulation with explicit time scheme for volume fraction with geometric VOF surface reconstruction. 10 . – Simulation with implicit time scheme for volume fraction without surface reconstruction.Fluent simulation of K-H instability • Continuity equation: r ∂ (αρ1 ) + ∇ αρ1U = 0 ∂t ( ) • Momentum equation with volumetric surface force: r rr r r r ∂ ρU + ∇ ⋅ ρUU − µ ∇U + (∇U )T = −∇p − σ (∇n )n ∇α + ρg ∂t ( ) ( ( )) • • Implicit (first order accurate) time scheme was used to calculate velocity field and SIMPLE pressure correction.K-H instability 7 . 4GHz Opteron 10 . Explicit time scheme. Grid:29x196.Fluent simulation .0 s to 3. time step=1e-4 s. with geometric surface reconstruction used.VOF • • • • Volume fraction field from 0. Surface is always sharp. CPU time=39 h @ 2.K-H instability 8 .55 s. 0 1.0 Growth of instability on mesh with 29x196 volumes and double precision.5 1.VOF 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 -9 measured and analytical time for onset of instability dt=1e-3 dt=5e-3 dt=1e-4 3.5 5. explicit time scheme for volume fraction.K-H instability 9 .5 4.0 2.0 0.5 Time [s] 3.5 2.0 0.Fluent simulation . geometric surface reconstruction. 10 .0 4. Fluent simulation . time step=1e-4 s. Numerical diffusion of surface can be seen.0 s to 3.no surface reconstruction (4-equation two-fluid model) • • • • Volume fraction field from 2.0 s. Implicit time scheme. without surface reconstruction. Grid:29x196. CPU time=46 h @2.K-H instability 10 .4 GHz Opteron 10 . dt=1e-4 0. implicit time scheme for volume fraction. dt=1e-4 CFX.5 3.0 3.5 1.K-H instability 11 .Fluent simulation .5 Growth of instability on mesh with 29x196 volumes and double precision.5 2.0 0.0 Time [s] 2.0 1. without surface reconstruction. dt=1e-3 Fluent.no surface reconstruction 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 -9 Fluent. 10 . Fluent .K-H instability 12 • • • . but there is a significant diffusion of the surface. – With surface reconstruction.conclusions • With linearised Navier-Stokes equations one can analytically predict onset of K-H instability and the critical wavelength. Problem was simulated with Fluent CFD program. solving non-linear Navier-Stokes equations. but onset of instability cannot be predicted. VOF surface tracking in Fluent code was tested Fluent simulations: – Onset of instability can be predicted without surface reconstruction. surface is always sharp.VOF . 10 . 2D Continuity equation: r ∂ (ρ ) + ∇(ρU ) = 0 ρ = αρ1 + (1 − α )ρ 2 Momentum equation: r rr r r r ∂ (ρU )+ ∇ ⋅ (ρUU − µ (∇U + (∇U )T )) = −∇p − FSTF + ρg + FAD µ = αµ1 + (1 − α )µ 2 ∂t Volumetric surface tension force: r r r FSTF = σ (∇n )n ∇α Additional force as generator of the flow: r ρ + ρ2 ⎞ ρ ⎛ FAD = ⎜ ρ − 1 g sin γ ⎟ 2 ⎠ (ρ1 + ρ 2 ) / 2 ⎝ ∂t 10 . with surface sharpening Viscosity not neglected.CFX .K-H instability 13 .Kelvin-Helmholtz instability • • • • • • Homogeneous two-fluid model. 10 . Only a section of the tube was simulated with periodical boundary conditions. Space derivates are discretized with high resolution scheme (combination of first and second order accuracy).K-H instability • • • • 14 . 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. which reduces numerical diffusion and dispersion.Kelvin-Helmholtz instability • • Equations are solved with implicit second order accurate time scheme.CFX . 8 s. surface tension neglected in particular simulation. CPU time=20 h @2.K-H instability 15 . time step=1e-4 s. K-H instability in experiment is observed in the middle section of the tube after ~1.K-H instability . Viscosity not neglected.CFX simulation complete tube length simulated Temporal development of the interface predicted by CFX.4 GHz Opteron 10 . Grid:29x1790. •Analytically predicted λcr is 27 mm.0 GHz Pentium 10 . •Most unstable wavelength in simulation is 40 mm.Kelvin-Helmholtz instability with surface tension •Volume fraction field from 2.0 s to 3. CPU time=50 h @3. time step=1e-4 s. •Grid: 29x196.25 s. •In experiment λcr is 25-45 mm.K-H instability 16 . 0 1. double precision.7.0 0. Tough case for CFX-5.K-H instability 17 .5 3.0 3.5 1.5 Growth of instability on mesh with 29x196 volumes and max residual = 1e-5.5 2. very small timestep must be used 10 .0 Time [s] 2. different dt [s].Kelvin-Helmholtz instability with surface tension 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 analitical prediction dt=1e-4 dt=1e-3 Visible from volume fraction field 0. 5 4.5 Time [s] 3. double precision.0 Growth of instability on mesh with 29x196 volumes and max residual = 1e-5.0 1.K-H instability 18 .5 2.0 4. 10 .0 3. There is no need for small timestep in CFX-10.0 2.0 0.5 1.5 5.Kelvin-Helmholtz instability with surface tension 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 -9 dt=1e-3 dt=5e-3 0. 35 s.Kelvin-Helmholtz instability without surface tension •Volume fraction field from 2.K-H instability 19 .0 s to 3. •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 20 . different dt [s] 10 .Kelvin-Helmholtz instability without surface tension 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 analitical prediction dt=1e-1 dt=1e-2 dt=1e-3 dt=1e-4 0.5 2.0 3.0 Time [s] 2.5 3. double precision.0 0.5 1.0 1.5 Growth of instability on mesh with 29x196 volumes and max residual = 1e-5. 0 Growth of instability on mesh with 29x196 volumes.K-H instability 21 .0 2.5 2.5 3.5 4.Kelvin-Helmholtz instability without surface tension 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 double precision single precision 0.0 4.01 s and max residual = 1e-4 . 10 .0 0. dt = 0.0 Time [s] 3.0 1.5 5.5 1. double precision.Kelvin-Helmholtz instability without surface tension 1x10 1x10 1x10 Amplitude [m] 1x10 1x10 1x10 1x10 1x10 -1 -2 -3 -4 -5 -6 -7 -8 without surface tension with surface tension 0.0 0. 10 .0 3.0 1.0 Time [s] 2.5 Growth of instability on mesh with 29x196 volumes dt = 1e-4 and max residual = 1e-5.K-H instability 22 .5 3.5 1.5 2. – There is no need for such small timestep in CFX-10.0 10 .K-H instability 23 .7 but extremely (inconveniently) small time step must be used. Problem was simulated with CFD programs. Numerical diffusion of surface is relatively small.Kelvin-Helmholtz instability with CFX • • • With linearised Navier-Stokes equations we can analytically predict onset of K-H instability and critical wavelength. solving non-linear Navier-Stokes equations CFX simulations: – Onset of instability can be predicted with CFX-5. :1.K-H instability 24 . CPU time=2 h @3. anal. anal.35 s (exp.5 s) •Critical wavelength: 33 mm (exp.:1.0 GHz Pentium 4 Not real aspect ratio 10 . time step=1e-3 s.:25-45 mm.:27 mm) Grid:2440x40.Conservative Level-Set .9 s.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. only the 65 cm in the middle of the channel is shown Time [s] 2.28 3.98 3.43 10 .38 3.08 3.88 2.K-H instability 25 .18 3.Thorpe's K-H instability •Real aspect ratio.Conservative Level-Set . Direct Contact Condensation KFKI experiment done at PMK-2 facility in Hungary 2870 258 142 593 574 578 8.T3 8.K-H instability 26 .T2 1 8.T1 3 water 9 8.T4 vapour 2 7 10 1309 10 1150 10 5 4 11 6 10 . 5 bar TV=470 K d Steam.242 m/s TL=295 K 10 .5 bar v=0.87 m Pipe diameter d=73 mm Steam tank p=14.K-H instability 27 . TV=470 K d Cold water injection p=14.Direct Contact Condensation Simulation of the pipe in CFX Pipe length L=2. Direct Contact Condensation 2 continuity equations. k-ε turbulence model Thermal phase change model for interfacial heat transfer & ΓGL = mGL ai ai = ∇α & mGL = 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 10 .. sat − hL . 2 Energy eqs.K-H instability 28 HTCL (Tsat − TL ) hV . 1 Momentum eq. Direct Contact Condensation Heat transfer coefficient is calculated using surface renewal theory introduced by Hughes and Duffey 1991 ⎛a ⎞ HTC L = 2 ρ L c p .K-H instability 29 . L ⎜ L ⎟ ⎝π ⎠ 1/ 2 ⎛ ε ⎜ ⎜µ /ρ ⎝ L L ⎞ ⎟ ⎟ ⎠ 1/ 4 Thermal diffusivity aL = λL ρ L c p.L Turbulence eddy dissipation from k-ε turbulence model 10 . CPU time=9 h @3.K-H instability 30 .0 GHz Pentium 10 .03 s.Direct Contact Condensation 2D simulation Heat transfer coefficient Mass transfer rate Temperature of water Void fraction of water Not real aspect ratio Grid:10x400. time step=0. CPU time=7 h @3.Direct Contact Condensation 3D simulation Void fraction Grid:4000 volumes.03 s.K-H instability 31 . time step=0.0 GHz Pentium 10 . 03 0.01 0.035 0.67 CFL=0.03 0.03 0.02 0.Direct Contact Condensation Interfacial mass transfer rate vs.01 0.025 0.025 0.015 0. time 0.005 0 0 2 4 time [s] 6 8 10 ny=10 ny=20 ny=40 mc [kg/s] 0.02 0.015 0.03 0.015 0.005 0 dx/dy=1 dx/dy=2 dx/dy=4 dx/dy=8 10 .02 0.33 2 4 6 8 10 time [s] 0.025 mc [kg/s] 0.02 0.01 0.01 0.005 0 0 0.K-H instability 32 .025 mc [kg/s] 0.015 0.005 0 0 2 4 time [s] 6 8 10 0 2 4 6 time [s] 8 10 3D 2D mc [kg/s] CFL=1 CFL=0. 8 0. volume fraction T1 T2 T3 T4 0.7 0.1 0 0 5 10 time [s] 15 20 T1 T2 T3 T4 10 .9 0.7 0.6 0.5 0.8 0.K-H instability 33 .4 0.9 0.3 0.6 0.4 0.5 0.Direct Contact Condensation Temperature at the top of the pipe 200 180 temperature [°C] 160 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 20 T1 T2 T3 T4 Void fraction of steam 1 CFX volume fraction 0.2 0.3 0.2 0.1 0 0 5 10 time [s] 15 20 ? T1 T2 T3 T4 200 180 160 temperature [°C] 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 20 1 Exp. K-H instability 34 .Direct Contact Condensation Small increase of water temperature -> Small condensation rate Heat transfer coefficient was increased by factor 20 -> better agreement with experiment 10 . 1 0 0 5 10 time [s] 15 20 cfx exp temperature [°C] 200 180 160 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 cfx exp T1 T1 20 10 .2 0.3 0.4 0.K-H instability 35 .8 volume fraction 0.7 0.5 0.Direct Contact Condensation T1 measuring point 1 0.9 0.6 0. 3 0.4 0.1 0 0 5 10 time [s] 15 20 cfx exp temperature [°C] 200 180 160 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 20 cfx exp T2 T2 10 .5 0.K-H instability 36 .2 0.7 0.9 0.6 0.Direct Contact Condensation T2 measuring point 1 0.8 volume fraction 0. 2 0.3 0.1 0 0 5 10 time [s] 15 20 cfx exp temperature [°C] 200 180 160 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 20 cfx exp T3 T3 10 .4 0.K-H instability 37 .Direct Contact Condensation T3 measuring point 1 0.9 0.7 0.5 0.8 volume fraction 0.6 0. 1 0 0 5 10 time [s] 15 20 cfx exp temperature [°C] 200 180 160 140 120 100 80 60 40 20 0 0 5 10 time [s] 15 20 cfx exp T4 T4 10 .K-H instability 38 .Direct Contact Condensation T4 measuring point 1 0.9 0.6 0.8 volume fraction 0.4 0.3 0.2 0.7 0.5 0. 03 s.K-H instability 39 . time step=0.0 GHz Pentium 10 .Direct Contact Condensation Increased heat transfer coefficient by factor 20 Heat transfer coefficient Mass transfer rate Temperature of water Void fraction of water Not real aspect ratio Grid:10x400. CPU time=9 h @3. K-H instability 40 .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 . Slovenia WAHA-maths-numerics 1 . 2006 Seminar on Two-phase flow modelling 11) WAHA code .mathematical model and numerical scheme by Iztok Tiselj "Jožef Stefan“ Institute. Ljubljana.Technical University of Catalonia and Heat and Mass Transfer Technological Center. 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 2 .mathematical model and numerical scheme 12) WAHA code .Two-phase flow modelling.simulations 13) Hands on: simulation of two-phase water hammer transient and two-phase critical flow. 2006 Table of contents INTRODUCTION Lectures 1-2 3-6 7-10 TWO-FLUID MODELS Lecture INTERFACE TRACKING IN 3D TWO-PHASE FLOWS Lectures ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS 11) WAHA code . seminar at UPC. WAHA special models: pipe expansion.mathematical model and numerical scheme . contraction (abrupt area change).Two-fluid model of WAHA code .convective terms – 1st step .Water properties of the WAHA code WAHA-maths-numerics 3 . branch.source terms – 2nd step .Closure equations of WAHA code ."non-standard" terms in WAHA two-fluid model .WAHA code .operator splitting .Contents .WAHA code . forces .introduction .WAHA code numerical scheme . available on internet www2.WAHA code manual.ijs.si/~r4www/waha3_manual.reference .pdf WAHA-maths-numerics 4 .WAHA code .mathematical model and numerical scheme . α ) ρ f ( v f − w ) = − Γg − (1 . CATHARE.Two-fluid model of WAHA code Six-equation. wall ∂t ∂x ∂x ∂x ∂ (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x WAHA-maths-numerics 5 .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f K (1 . similar to codes like RELAP5. Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 .α ) ρ f ( v f − w ) + + (1 . etc.α )( v f − w ) K − (1 .α )w = + (1 .α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 . TRACE.α ) ρ f ( v f − w ) + (1 .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 . TRAC.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . two-fluid model. wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg .α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 . wall ∂x ∂x ∂x ∂t (1 − α ) ρ f ∂uf ∂uf ∂(1 . similar to codes like RELAP5. wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x LHS: cdifferential terms RHS: sources WAHA-maths-numerics 6 .α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 .α ) ρ f ( v f − w ) + (1 .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f K (1 .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . TRAC. CATHARE.α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 .α ) ρ f ( v f − w ) + + (1 .α )( v f − w ) K − (1 .α )w = + (1 . two-fluid model. etc.Two-fluid model of WAHA code Six-equation. wall ∂t ∂x ∂x ∂x ∂ (1 .α ) ρ f ( v f − w ) = − Γg − (1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg . TRACE. wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Pipe elasticity (Wylie. t )) WAHA-maths-numerics dAe D dp = = K dp A( x) d E 7 . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 . similar to codes like RELAP5.α )w = + (1 . t ) = A( x) + Ae ( p ( x.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg . TRAC. two-fluid model. wall ∂t ∂x ∂x ∂x ∂ (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α ) ρ f ( v f − w ) + (1 .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 .Two-fluid model of WAHA code Six-equation. etc.α )( v f − w ) K − (1 .α ) ρ f ( v f − w ) + + (1 .α ) ρ f ( v f − w ) = − Γg − (1 .α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 .α ) ρ f K (1 .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . CATHARE. TRACE.α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . Streeter): A( x.α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Additional closure relations: 1) Equations of state (more later): WAHA-maths-numerics ⎛∂ρ ⎞ ⎛ ∂ ρk ⎞ ⎜ ⎟ d p + ⎜ k ⎟ duk . TRAC.α )w = + (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . CATHARE.α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 .α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 .α ) ρ f ( v f − w ) + + (1 .Two-fluid model of WAHA code Six-equation.α ) ρ f ( v f − w ) + (1 .α ) ρ f K (1 .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . TRACE.α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg . d ρk = ⎜ ⎟ ⎜∂ ⎟ ⎝ ∂p ⎠uk ⎝ uk ⎠ p 8 .α )( v f − w ) K − (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . wall ∂t ∂x ∂x ∂x ∂ (1 . etc. two-fluid model. similar to codes like RELAP5.α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 .α ) ρ f ( v f − w ) = − Γg − (1 . CATHARE. etc.α )( v f − w ) K − (1 . similar to codes like RELAP5. wall ∂t ∂x ∂x ∂x ∂ (1 .α ) ρ f ( v f − w ) + (1 .α ) ρ f K (1 .α ) ρ m ⎜ +v f .vg ⎜ ∂t ∂x ∂t ∂x ⎟ ⎝ ⎠ WAHA-maths-numerics 9 . two-fluid model.α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 . TRACE. wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Additional closure relations: 2) Virtual mass term is used to obtain hyperbolicity of equations ∂ vg ∂ v f ∂vf ⎞ ⎛ ∂ vg ⎟ CVM = (1 − S )Cvm α (1.α ) ρ f ( v f − w ) + + (1 .α )w = + (1 .Two-fluid model of WAHA code Six-equation.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f ( v f − w ) = − Γg − (1 .α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg . TRAC. Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 . CATHARE.α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 .Two-fluid model of WAHA code Six-equation.α ) ρ f ( v f − w ) K ∂x A( x ) dx ∂t ∂t ∂x ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 .α ) ρ f K (1 . wall ∂t ∂x ∂x ∂x ∂ (1 .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . TRACE.α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α ) ρ f ( v f − w ) = − Γg − (1 . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Additional closure relations: i 3) Interfacial pressure term exists WAHA-maths-numerics only in stratified flow.α )( v f − w ) K − (1 . etc. two-fluid model. wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg .α ) ρ f ( v f − w ) + + (1 .α ) ρ f ( v f − w ) + (1 .α )w = + (1 . p = Sα (1 − α )( ρ f − ρ g ) gD 10 . TRAC.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . similar to codes like RELAP5.α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg .α ) ρ f ( v f − w ) + (1 .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 . similar to codes like RELAP5.α )w = + (1 .α )( v f − w ) ∂α ∂p ∂p ∂p −p + p (1 − α ) K +p + p(1 . etc.α ) ρ f ( v f − w ) ∂p 1 dA ( x ) + + (1 .1) Terms with Ci . TRAC.α ) ρ f ( v f − w ) K = − Γg − (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Additional closure relations: 4) Source terms are flow regime dependent.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . TRACE.α )( v f − w ) K − (1 .inter-phase drag WAHA-maths-numerics 11 . Continuity equations: Momentum equations: Internal energy equations: α ρg ∂p ∂ (1 .α ) ρ f ( v f − w ) ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . two-fluid model. wall ∂t ∂x ∂x ∂x ∂ (1 . CATHARE. Source terms are: 4.α ) ρ f K (1 .α ) ρ f ( v f − w ) ∂t ∂t ∂x ∂x A( x ) dx ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 .Two-fluid model of WAHA code Six-equation. α ) ρ f ( v f − w ) + (1 .Two-fluid model of WAHA code Six-equation.α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 . wall − α ( v g − w ) p − αw +p + pα (v g − w) K + pαK + α ρ g (v g − w) +p g A ( x ) dx ∂x ∂x ∂t ∂x ∂t ∂t ∂x Additional closure relations: 4. similar to codes like RELAP5. TRACE.α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . TRAC. etc. two-fluid model. wall ∂t ∂x ∂x ∂x ∂ (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 . CATHARE. wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂p ∂α = Q ig + Γg (u * − u g ) + v g Fg .α ) ρ f ( v f − w ) K = − Γg − (1 .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f .α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 .α )( v f − w ) K − (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α ) ρ f ( v f − w ) + + (1 .2a) Terms with inter-phase exchange of mass and energy with: Γg=-(Qif+Qig)/(hg-hf) .vapor generation term WAHA-maths-numerics 12 .α ) ρ f K (1 .α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 .α )( v f − w ) ∂α ∂p ∂p ∂p + (1 . wall − α ( v g − w ) p +p + pαK +p + α ρ g (v g − w) g A ( x ) dx ∂x ∂x ∂x ∂x ∂t ∂t ∂t Additional closure relations: 4. TRACE. CATHARE.2b) Terms with inter-phase exchange of mass and energy with: Qik=Hik (Ts-Tk) .α ) ρ f ∂ vf ∂ vf ∂ p ∂α + (1 .α ) ρ f ( v f − w ) + (1 .α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f ( v f − w ) K = − Γg − (1 .α ) ρ f ( v f − w ) + + (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . similar to codes like RELAP5.α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg 1 dA ( x ) ∂p ∂p ∂ α ρ g ( v g − w ) + + α ρ g (v g − w) K = Γg − α ρ g ( v g − w ) +α ρ g K ∂x ∂t ∂t ∂x A( x ) dx + (1 . etc. wall ∂t ∂x ∂x ∂x ∂ (1 .α )( v f − w ) K − (1 .α ) ρ f K (1 . TRAC. two-fluid model. wall − (1 − α )( v f − w ) p A( x ) dx ∂α ( v g − w ) ∂ ug ∂ ug 1 dA ( x ) ∂p ∂p ∂α ∂p + pα (v g − w) K − αw = Q ig + Γg (u * − u g ) + v g Fg .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f .Two-fluid model of WAHA code Six-equation.α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 . Continuity equations: Momentum equations: Internal energy equations: α ρg 1 dA ( x ) ∂p ∂p ∂ (1 .interface heat transfer terms WAHA-maths-numerics 13 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α )( v f − w ) ∂α ∂p ∂p ∂p + (1 . TRAC.α )( v f − w ) ∂α ∂p ∂p ∂p + (1 .α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 .α ) ρ f K ∂ vf ∂ vf ∂ p ∂α + (1 .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg . two-fluid model.α )( v f − w ) K − (1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂ ug ∂ ug ∂α ( v g − w ) 1 dA ( x ) ∂α ∂p ∂p ∂p + α ρ g (v g − w) +p + pαK +p + pα (v g − w) K − αw = Q ig + Γg (u * − u g ) + v g Fg . wall − α ( v g − w ) p g A ( x ) dx ∂t ∂x ∂t ∂t ∂x ∂x ∂x Additional closure relations: 4. CATHARE.α ) ρ f ( v f − w ) + (1 . wall ∂t ∂x ∂x ∂x (1 . etc. TRACE. wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α ) ρ f ( v f − w ) + + (1 .Two-fluid model of WAHA code Six-equation.3) Terms due to the variable pipe cross-section. similar to codes like RELAP5. WAHA-maths-numerics 14 .α ) ρ f 1 dA ( x ) ∂p ∂p ∂ (1 .α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg ∂p ∂ α ρ g ( v g − w ) ∂p 1 dA ( x ) + + α ρ g (v g − w) K +α ρ g K = Γg − α ρ g ( v g − w ) ∂t ∂t ∂x ∂x A( x ) dx + (1 .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . Continuity equations: Momentum equations: Internal energy equations: α ρg ∂ (1 .α ) ρ f ( v f − w ) K = − Γg − (1 . wall − (1 − α )( v f − w ) p A( x ) dx ∂ ug ∂ ug ∂α ( v g − w ) 1 dA ( x ) ∂α ∂p ∂p ∂p + α ρ g (v g − w) +p + pαK +p + pα (v g − w) K − αw = Q ig + Γg (u * − u g ) + v g Fg .wall .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α ) ρ f K ∂ vf ∂ vf ∂ p ∂α + (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α )( v f − w ) ∂α ∂p ∂p ∂p + (1 .Two-fluid model of WAHA code Six-equation.wall . similar to codes like RELAP5.α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg ∂p ∂ α ρ g ( v g − w ) ∂p 1 dA ( x ) + + α ρ g (v g − w) K +α ρ g K = Γg − α ρ g ( v g − w ) ∂t ∂t ∂x ∂x A( x ) dx + (1 . Continuity equations: Momentum equations: Internal energy equations: α ρg ∂ (1 . wall − α ( v g − w ) p g A ( x ) dx ∂t ∂x ∂t ∂t ∂x ∂x ∂x Additional closure relations: 4.α ) ρ f ( v f − w ) + + (1 . two-fluid model. etc. TRAC.4) Ff.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f .α )( v f − w ) K − (1 .α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 . WAHA-maths-numerics 15 .wall friction (Dynamical wall friction model available too).α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f 1 dA ( x ) ∂p ∂p ∂ (1 .α ) ρ f ( v f − w ) K = − Γg − (1 .α ) ρ f ( v f − w ) + (1 . CATHARE. Fg. TRACE. wall ∂t ∂x ∂x ∂x (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 . etc.α ) ρ f 1 dA ( x ) ∂p ∂p ∂ (1 .α )( v f − w ) K − (1 .α ) ρ f ( v f − w ) K = − Γg − (1 . wall − α ( v g − w ) p g A ( x ) dx ∂t ∂x ∂t ∂t ∂x ∂x ∂x Additional closure relations: 4. TRAC.α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 .α ) ρ f K ∂ vf ∂ vf ∂ p ∂α + (1 .α ) ρ f ( v f − w ) + + (1 .volumetric forces. WAHA-maths-numerics 16 .α )( v f − w ) ∂α ∂p ∂p ∂p + (1 . two-fluid model.α ) ρ f ( v f − w ) + (1 .α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f .α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg ∂p ∂ α ρ g ( v g − w ) ∂p 1 dA ( x ) + + α ρ g (v g − w) K +α ρ g K = Γg − α ρ g ( v g − w ) ∂t ∂t ∂x ∂x A( x ) dx + (1 . Continuity equations: Momentum equations: Internal energy equations: α ρg ∂ (1 .Two-fluid model of WAHA code Six-equation. similar to codes like RELAP5.5) Term with g cosθ .α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . wall − (1 − α )( v f − w ) p A( x ) dx ∂ ug ∂ ug ∂α ( v g − w ) 1 dA ( x ) ∂α ∂p ∂p ∂p + α ρ g (v g − w) +p + pαK +p + pα (v g − w) K − αw = Q ig + Γg (u * − u g ) + v g Fg . TRACE. wall ∂t ∂x ∂x ∂x (1 . CATHARE. 6) Terms for wall heat transfer are neglected in WAHA code.α ) ρ f ( v f − w ) K = − Γg − (1 . CATHARE.α ) − CVM − p i = C i | v r | v r − Γg ( v i − v f ) + (1 − α ) ρ f g cos θ − F f . wall ∂t ∂x ∂x ∂x (1 .α ) ρ f ( v f − w ) −p + p (1 − α ) K +p + p(1 .α )( v f − w ) ∂α ∂p ∂p ∂p + (1 . Continuity equations: Momentum equations: Internal energy equations: α ρg ∂ (1 .α ) ρ f 1 dA ( x ) ∂p ∂p ∂ (1 . TRAC. wall − α ( v g − w ) p g A ( x ) dx ∂t ∂x ∂t ∂t ∂x ∂x ∂x Additional closure relations: 4.α ) ρ f ( v f − w ) + (1 .α ) ρ f K ∂ vf ∂ vf ∂ p ∂α + (1 . wall ∂t ∂x ∂x ∂x (1 − α ) ρ f ∂uf ∂uf ∂(1 .α )w = ∂t ∂x ∂t ∂t ∂x ∂x ∂x 1 dA ( x ) Qif − Γg (u *f − u f ) + v f F f . similar to codes like RELAP5. two-fluid model. wall − (1 − α )( v f − w ) p A( x ) dx ∂ ug ∂ ug ∂α ( v g − w ) 1 dA ( x ) ∂α ∂p ∂p ∂p + α ρ g (v g − w) +p + pαK +p + pα (v g − w) K − αw = Q ig + Γg (u * − u g ) + v g Fg . etc.α ) ρ f α ρg ∂ vg ∂ vg ∂ p ∂α + α ρ g (v g − w) +α + CVM + p i = − C i | v r | v r + Γg ( v i − v g ) + αρ g g cos θ − Fg .α ) ρ f ( v f − w ) + + (1 .α )( v f − w ) K − (1 . TRACE.α ) ρ f ( v f − w ) ∂x ∂t ∂t ∂x A( x ) dx ∂α ρg ∂p ∂ α ρ g ( v g − w ) ∂p 1 dA ( x ) + + α ρ g (v g − w) K +α ρ g K = Γg − α ρ g ( v g − w ) ∂t ∂t ∂x ∂x A( x ) dx + (1 . WAHA-maths-numerics 17 .Two-fluid model of WAHA code Six-equation. 95 Droplet flow 0.95 > α > 0.5 vcrit vcrit ⎛ α (1 − α ) ⎞ ⎟ + vcrit = gD( ρ f − ρ g )⎜ ⎜ ρg ρf ⎟ ⎠ ⎝ vr Stratification factor S: S K −H ⎧ 1 ⎪⎛ vr ⎞ ⎪⎜ ⎟ = ⎨ L1 − ⎜ v critical ⎟ ⎪⎝ ⎠ ⎪ 0 ⎩ S = S K − H X inclination X ρv X v X α X 1−α v r < L 2 v critical L1 v critical (L1 − L2 ) L 2 v critical ≤ v r ≤ v r ≥ L1 v critical 1 ⎧ ⎪ X v = ⎨(100 − v m ) (100 − 25) ⎪ 0 ⎩ 1 ⎧ ⎪ −6 = ⎨ α − 5 ⋅10 10 −3 − 5 ⋅10 −6 ⎪ 0 ⎩ v m < 25 m / s 25 m / s ≤ v m ≤ 100 m / s v m ≥ 100 m / s α < 5 ⋅10 −6 5 ⋅10 −6 ≤ α ≤ 10 −3 α ≥ 10 −3 X ρv 1 ⎧ ⎪ ⎪ = ⎨(30000 − ρ m v m ) (30000 − 2500) ⎪ 0 ⎪ ⎩ ρ m v m < 2500 kg / m 2 s 2500 kg / m 2 s ≤ ρ m v m ≤ 30000 kg / m 2 s Xα ( )( ) ) ρ m v m ≥ 30000 kg / m 2 s θ < 30 0 X inclination 1 ⎧ ⎪ ⎪ 0 = ⎨ 60 − θ 60 0 − 30 0 ⎪ 0 ⎪ ⎩ ( )( ) 30 0 ≤ θ ≤ 60 0 X 1−α θ ≥ 60 0 1 ⎧ ⎪ = ⎨ (1 − α ) − 5 ⋅10 −6 10 3 − 5 ⋅10 −6 ⎪ 0 ⎩ ( )( WAHA-maths-numerics (1 − α ) < 5 ⋅10 −6 5 ⋅10 ≤ (1 − α ) ≤ 10 −3 (1 − α ) ≥ 10 −3 18 −6 .Two-fluid model of WAHA code Closure relations are flow regime dependent: WAHA flow regime map: Horizontally stratified flow S=1 Transitional area 1>S>0 Dispersed flow S=0 α > 0.5 Bubbly flow Critical velocity (Kelvin-Helmholtz instability) 0.5 Transitional flow α < 0. 6 (1 − α ) / d0 (1− q ) ⎛ 0.1Re0.5 ⎠ r Bubbly-to-droplet transition: Ci = Ci −bubbly Horizontally stratified flow: ( ) ⋅ (Ci −droplet ) q r = 0.95 − 0.5 Ci = Bubbly flow α > 0.1Re ) / Re ⎛ 24(1 + 0.75 ) ⎞ .1⎟ ⎝8 ⎠ 1 ρ f CD a gf 8 0.6α / d0 agf = 3.Inter-phase momentum transfer Dispersed flow: Vapor volume fraction: interfacial friction coefficient: drag coefficient: interfacial area concentration: LEGEND: .abub/ adrp is modified vapor/liquid volume fraction .95 Droplet flow ⎛1 ⎞ Ci = max⎜ ρ g CD a gf .95 − α ⎞ q=⎜ ⎟ ⎝ 0. 3 (vk − vi ) 2 1 Ci = ρ k fk a gf 8 (v g − v f ) 2 k = g.5⎟ CD = min⎜ ⎜ ⎟ Re ⎝ ⎠ agf = 3.Re is Reynolds number α < 0. 0. 0.75 CD = 24(1 + 0.d0 is average slug diameter . f Dispersed-to-horizontaly stratified: Ci = S Ci − stratified + (1 − S ) Ci − dispersed ( ) ( ) 19 WAHA-maths-numerics . Qik . f ⎛ ⎞ 1 ⎜ f k (Rek − 1000) Prk ⎟ ⎜ 4. 8 ⎟ Nuk = max ⎜ ⎟ f 1 + 12.7 k (Prk 0.specific enthalpies.Inter-phase heat&mass trans. Vapor generation rate Γg is calculated as: Γg = − Qif + Qig h −h * g * f The volumetric heat fluxes are calculated as: .67 − 1) ⎟ ⎜ 8 ⎝ ⎠ Dispersed-to-horizontaly stratified: interpolation H if = S H if − stratified + (1 − S ) H if − dispersed WAHA-maths-numerics ( ) ( ) 20 . hk* .liquid-to-interface and gas-to-interface heat fluxes Qik = H ik (TS − T f ) Horizontally stratified flow: Dittus-Boelter type of correlation: H ik = 2 Nuk akf kk α k = g. quality θ . Tk – phase temp.10 −9 ) 6 ρm – mixture density X .10 −5 ) = 10 ⋅ (1 + η ⋅ (100 + 25 ⋅ η )) max(α .relaxation time η . hk – phase enthalpy – fluid heat transfer coefficient Hif: (TS − T f ) – vapor or fluid volumetric heat flux Qik: H if = * − H ig TS − Tg − Γg hg − h* f ( ) ( ) Qik = H ik (TS − T f ) k = g or f WAHA-maths-numerics 21 .Inter-phase heat&mass trans.temperature relation TS – Saturation temp. Dispersed flow (Downar-Zapolski HRM model): – Homogeneous Relaxation Model (HRM) – vapor generation Γg: Γg = − ρ m X − X Saturation θ Legend: – vapor heat transfer coefficient Hig: H ig max(α . constant pressure (tank).Some other capabilities of WAHA code: Wall friction (steady): Fk . wall = f wk Minor loses at elbows: ρ k vk vk (1 − α ) ρ k 2D ρm f ml β D = 2π ∆x Unsteady wall friction: τ (t ) = τ s (t ) + τ un (t ) −∆t τ un (t ) = τ un (t − ∆t )e θ + kT ρ c∆v transient friction coefficient kT relaxation time Θ Instantaneous relaxation available for inter-phase heat. Tank allows modelling of critical flow at the boundary. mass. and constant mass flow rate (pump).such results are similar to results of HEM model. WAHA-maths-numerics 22 . and momentum transfer . Boundary conditions: closed end. Upwinding is not used for calculation of the relaxation source terms. mass and momentum exchange terms are stiff. wall friction and volumetric forces are solved in the first sub step with r r r upwind discretisation: A ∂ψ ∂ψ +B = S NON _ RELAXATION .e. which depend on the stiffness of the source terms.WAHA numerical scheme Numerical scheme is based on characteristic upwind methods and operator splitting. ∂t ∂x 2) Relaxation (inter-phase exchange) source terms: r r dψ A = S RELAXATION dt Relaxation source terms: inter-phase heat. Integration of the relaxation sources within the operator-splitting scheme is performed with variable time steps. i. WAHA-maths-numerics 23 . Operator splitting: 1) Convection and non-relaxation source terms .source terms due to the smooth area change. their characteristic time scales can be much shorter that the time scales of the hyperbolic part of the equations.. 1st substep of operator splitting: convection terms with non-relaxation source terms Equation solved: r ∂ψ ∂ψ −1 +C = A ⋅ SN − R . Source terms are rewritten: r r r ∂A r ∂ψ −1 ∂ψ + L⋅ Λ⋅L + RA + RF = 0 . Equation rewritten: L ∂t ∂x ∂x ∂x WAHA-maths-numerics 24 . ∂t ∂x ∂x r r r ∂A r ∂x ∂ψ ∂ψ −1 −1 −1 −1 −1 −1 + Λ⋅L + Λ ⋅ Λ ⋅ L RA + Λ ⋅ Λ ⋅ L RF =0 . ∂t ∂x r r Eigenvalues and eigenvectors of Jacobian matrix are found: C = L⋅ Λ⋅L −1 r RA contains source terms due to the variable pipe cross-section r RF contains wall friction and volumetric forces (no derivatives). ∂t ∂x ∂x ∂x −1 r r Modified characteristic variables are introduced as : −1 r r −1 −1 δξ = L δψ + Λ ⋅ L RA δA + Λ ⋅ L RF δx .1st substep of operator splitting: convection terms with non-relaxation source terms Equation rewritten: r ∂A r ∂x ∂ψ −1 ∂ψ −1 −1 −1 −1 L + Λ⋅L + Λ ⋅ Λ ⋅ L RA + Λ ⋅ Λ ⋅ L RF =0 . r −1 −1 characteristic-like form of Eqs: (allows 2nd order accurate discretisation with application of slope limiters) ∂ξ ∂ξ + Λ =0 . ∂t ∂x r r Slopes are not measured by “Modified characteristic variables” but rather r r −1 r with variables: −1 −1 δζ = Λ ⋅ δξ = Λ ⋅ L δψ + L RAδA + L RF δx WAHA-maths-numerics 25 . 1st substep of operator splitting: convection terms with non-relaxation source terms The combination of the first- and the second-order accurate discretisation is r n+1 r n rn rn rn rn (Godunov’s method): ξ j -ξ j ξ j - ξ j -1 ξ j+1 - ξ j -- n ++ n =0 + ( Λ ) j+1/2 + ( Λ ) j -1/2 ∆t ∆x ∆x where elements of diagonal matrices Λ + + , Λ − − are calculated as: λ ⎞ φ ⎛ ⎛ ∆t ⎞ ++ λ + + = λ k ⋅ f k+ + k = 1,6 - 1⎟ , k = 1,6 f k = max⎜ 0 , k ⎟ + k ⎜ λ k k ⎜ | |⎟ ∆x ⎠ 2 ⎝ and λk ⎠ ⎝ ⎛ ∆t ⎞ λ ⎞ φ ⎛ -λ − − = λ k ⋅ f k− − k = 1,6 k - 1⎟ , k = 1,6 f = min⎜ 0 , k ⎟ - k ⎜ λ k k ⎜ ⎝ | λk |⎟ ⎠ 2 ⎝ ∆x ⎠ Different slope limiters – second order correction: MINMOD φ p = max(0 , min(1 , θ p )) Van Leer φ p = ( θ p + θ p ) /( θ p + 1) Superbee φ p = max(0,min(2θ p ,1),min(θ p , 2)) Difference scheme (basic variables) used in the WAHA code for convective r r r part is: r ∆A j −1 / 2 + + ∆ψ j −1 / 2 − − ∆ψ j +1 / 2 −1 ∆ψ + + −1 − − −1 ++ −1 −1 L + + Λ Λ L RA + Λ L = Λ L ∆x ∆x ∆x ∆t r ∆A j +1 / 2 r r −− −1 −1 ++ −1 −1 −− −1 −1 + Λ Λ L RA + Λ Λ L R F j −1 / 2 + Λ Λ L R F j +1 / 2 ∆x ( ) ( ) WAHA-maths-numerics 26 1st substep of operator splitting: basic variables Basic variables are ~ primitive variables, ψ = ( p,α , v f ,vg , u f , ug ) r (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: ϕ = [(1 - α ) ρ f ,α ρ g ,(1 - α ) ρ f v f ,α ρ g vg , (1 - α ) ρ f e f ,α ρ g eg] 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 ( p ,α , ρ f , ρ g ) from the conservative variables ( (1- α ) ρ f ,α ρ g ,(1- α ) ρ f u f ,α ρ g u g ). WAHA-maths-numerics 27 r 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 r ψ m+1 = ψ m + A−1(ψ m )S (ψ m )∆tS r r r r 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 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 28 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: ψ M = ( ρm ,vm ρm ,em ρm ,vg − v f , Tf ,Tg ) 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. WAHA-maths-numerics 29 r WAHA special models: • 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 ∂ x [α k ρ k vk A] = 0 ∂ x [α k vk (ρ k wk + p ) A] = 0 2 ∂ x α k ρ k vk [ ( + p A = [α k p ]∂ x A )] i2 i2+1 i2+2 A1 i1-2 i1-1 i1 A2 k n WAHA-maths-numerics 30 WAHA special models: Abrupt area change two-phase test case [MPa] 16.0 15.0 14.0 13.0 12.0 11.0 10.0 9.0 0.0 1.0 2.0 3.0 4.0 0.7 0.6 WAHA, simplified WAHA, cons-char WAHA, conservative Relap5 mod3.2.2g 1.0 [α] WAHA, simplified WAHA, cons-char WAHA, conservative Relap5 mod3.2.2g Expansion at l = 3 m 0.8 0.9 pressure [m] 0.5 0.4 5.0 0.0 VVF 1.0 2.0 3.0 4.0 [m] 5.0 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 31 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. GEOMETRY (P1/P6/P2): Length l = 10/5/3 m Diameter d = 7.9/7.9/0.7 mm INITIAL CONDITIONS (P1/P6/P2): Temperature T = 293/293/293 K Vapor velocity v = 1/0.769/0.769 m/s Presure p = 80/80/80 bar Vapor volume fraction - pure liquid p= const. const. 1 1 2 3 4 2 .. Pipe 2.. 29 30 3 4 Closed end ... WAHA RELAP Pipe 1 ... 97 98 99 100 1 2 ...Pipe 6... p3 [MPa] 9.4 9.2 9.0 8.8 8.6 47 48 49 50 p1 [MPa] 9.2 9.0 8.8 8.6 8.4 8.2 8.0 7.8 0.000 0.002 0.004 0.006 0.008 p2 [MPa] 9.2 9.0 8.8 8.6 8.4 8.2 8.0 7.8 WAHA RELAP WAHA RELAP 8.4 8.2 8.0 7.8 0.002 0.004 0.006 0.008 [s] 0.010 0.000 [s] 0.010 0.000 0.002 0.004 0.006 0.008 [s] 0.010 WAHA-maths-numerics 32 Appendix A: Derivation ofv v v Ψ Ψ Ψ fluid force equations. 1 2 v Ψn n −1 . ... .2-1980. ANSI/ANS-58. “Design basis for protection of light water nuclear power plant against the effects of postulated pipe rupture”. ⎣ ⎦ ( ) WAHA-maths-numerics 33 .v. Revision of ANSI/ANS-58.WAHA special models: • Forces . ⎥ c.WAHA code can calculate forces on the 3D piping system.s .2-1988. Ain Aout Apipe c .. Pipe 1 .from American National Standard. ∆x i v Fi v ri ∆x v ri +1 i+1 v v v F0 F1 F2 v v v Fn −2 Fn−1 Fn Ai p i vi Ai+1 pi+1 vi+1 r F is dynamic fluid thrust force vector on pipe r ⎡ d ( ρ v ) dV r r r r r r r r ⎤ F = −⎢ ∫ + ∫ ρ v v ⋅ dA + ∫ pdAin + ∫ pdAout + ∫ pambient dApipe − ∫ ρ gdV ⎥ dt ⎢ c.Forces are calculated on the edges of the volumes.v. 005 0.015 total force A*p L = 4.WAHA special models: • Forces PIPE GS5 0 -5000 0 MEMBRANE GS1 force on the Edwards's pipe 0.6 0.097 m.4 0.force on the pipe 0 -5000 0 -10000 force (N) 0.2 0.2 10-2 m2 Edward'spipe .01 -15000 -20000 -25000 -30000 -35000 -40000 time (s) Total force pressure*crossection WAHA-maths-numerics 34 .8 CLOSED END GS7 -10000 force (N) -15000 -20000 -25000 -30000 -35000 -40000 time (s) 0. A = 4. Water properties of the WAHA code: EoS: ⎛ ∂ ρk ⎞ ⎛ ∂ ρk ⎞ ⎟ d p +⎜ d ρk = ⎜ ⎜ ∂ ⎟ du k ⎟ ⎜ ∂p ⎟ ⎝ uk ⎠ p ⎝ ⎠u k 250 Sa tura tion Va por s pinoda l (e xte nde d) Liquid s pinoda l (e xte nde d) Ne ga tive pre s s ure Va por s pinoda l Liquid s pinoda l • Thermodynamic properties of liquid and steam are based on NBS/NRC-84 formulation P res s ure [bar] 200 150 • Pre-tabulated and stored in ASCII file: – – 400 pressures (-95 – 1000 bar) 500 temperatures (273 – 1638 K) 100 50 • • Extended into negative pressure (up to –95 bar) Extended liquid and vapor spinodal lines 0 -50 Ne ga tive pre s s ure a re a in the Wa ha -100 250 300 350 400 450 500 550 600 650 Temperature [K] WAHA-maths-numerics 35 . 5 2 Single-phase liquid wave .60008ms 1500000 1300000 RELAP5 steam tables t=1.Water properties of the WAHA code: . 36 WAHA-maths-numerics . Single-phase vapor wave .pressure.0008ms 1100000 900000 WAHA steam tables t=1.Differences are more due to the slightly different time steps than due to the different water properties.0065ms 1.60514ms 1500000 1300000 1100000 900000 0 0. 2100000 1900000 1700000 p(Pa) 2100000 1900000 1700000 p(Pa) RELAP5 steam tables t=0.2 Gamma (internal version of WAHA code) .Propagation of pressure waves in single-phase liquid and in single-phase vapor shock tube .5 2 0 0.5 1 length (m) WAHA steam tables t=0.5 1 length (m) 1.pressure.2.Comparison: WAHA code with the WAHA steam tables and WAHA code with the steam tables of RELAP5/MOD3. E+06 p (Pa) 4.5 0.3 time (s) 0.E-01 6. 7.E+00 vapor volume fraction 8.E+06 2.2 0. WAHA-maths-numerics 37 .1 0.E-01 0.5 0.1 0.Comparison: WAHA code with the WAHA steam tables and WAHA code with the steam tables of RELAP5/MOD3.3 0.E+06 0.E-01 RELAP5 steam tables 4.Water properties of the WAHA code: .E-01 time (s) 0.4 0.2 0.E+06 6.4 0.E+00 0 -2.Calculations were performed with instantaneous relaxation of inter-phase heat. Edwards pipe problem .E-01 WAHA steam tables 2.E+06 1.vapor volume fraction.Edwards pipe problem .E+00 1.2 Gamma (internal version of WAHA code) .E+06 3. mass and momentum transfer.6 RELAP5 steam tables WAHA steam tables 1.6 Edwards pipe problem .E+00 0 0.2.pressure.E+06 5.rapid depressurization of the hot liquid in a horizontal pipe . simulations by Iztok Tiselj "Jožef Stefan“ Institute. 2006 Seminar on Two-phase flow modelling 12) WAHA code . Slovenia WAHA-simulations 1 . Ljubljana.Technical University of Catalonia and Heat and Mass Transfer Technological Center. seminar at UPC.Two-phase flow modelling. 14) Fluid-structure interaction in 1D piping systems DNS OF THE PASSIVE SCALAR TRANSFER IN THE CHANNEL AND FLUME Lectures 15-18 WAHA-simulations 2 .simulations 13) Hands on: simulation of two-phase water hammer transient and two-phase critical flow. 2006 Table of contents INTRODUCTION Lectures 1-2 3-6 7-10 TWO-FLUID MODELS Lecture INTERFACE TRACKING IN 3D TWO-PHASE FLOWS Lectures ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS 11) WAHA code .mathematical model and numerical scheme 12) WAHA code . R. Edwards.2 10-2 m2 Transient: .transition into the horizontally stratified flow WAHA-simulations A. P. 9.WAHA code .the pressure undershoot model .the two-phase critical flow .simulations Edwards pipe: CLOSED END GS7 GS5 PIPE MEMBRANE GS1 L = 4.propagation of the rapid depressurization wave .rapid depressurization of the hot liquid from the horizontal pipe Aim: . Studies of phenomena connected with the depressurization of water reactors. O'Brien.the flashing model . A = 4. 1970.097 m.to verify several WAHA code physical models like: .test case for codes used to simulate LOCA accidents in NPPs . 3 . Journal of the British Nuclear Energy Society.propagation of the void fraction wave . T. 125135. A = 4.097 m.left: closed end L = 4.WAHA code . WAHA-simulations 4 .right: constant pressure pT = 1 bar .5% smaller than cross-section of the pipe.cross-section of the break is 12.2 10-2 m2 .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. Initial conditions: -velocity: stagnant liquid -pressure p = 70 bar -temperature T = 515 K Conclusion: .simulations Edwards pipe: CLOSED END GS7 GS5 PIPE MEMBRANE GS1 Boundary conditions: . 3 0.WAHA code .6 Time [s ] WAHA-simulations 5 .5 0.1 0.4 0.simulations Pressure in GS1 [MPa] 7 Edwards pipe: 6 Expe rime nt WAHA3 P res s ure in GS 1 [MP a] 5 4 3 2 1 0 0 0.2 0. 2 0.simulations Pressure in GS7 [MPa] Edwards pipe: 7 Expe rime nt WAHA3 6 P res s ure in GS 7 [MP a] 5 4 3 2 1 0 0 0.6 Time [s ] WAHA-simulations 6 .5 0.WAHA code .1 0.3 0.4 0. 9 0.5 0.WAHA code .3 0.4 Expe rime nt WAHA3 0.8 0.1 0 0 0.7 0.4 0.simulations Vapor volume fraction in GS5 1 Edwards pipe: Vapor volume fraction in GS 5 [] 0.3 0.6 0.5 0.6 Time [s ] WAHA-simulations 7 .2 0.2 0.1 0. liquid WAHA3 .va por 500 Temperature in GS 5 [K] 450 400 350 300 250 0 0.2 0.4 0.5 0.1 0.simulations Temperature in GS5 [K] Edwards pipe: 550 Expe rime nt WAHA3 .3 0.6 Time [s ] WAHA-simulations 8 .WAHA code . inlet outlet “Transient”: .to verify conservation properties of the WAHA code in the variable crosssection geometry.steady state two-phase critical flashing flow in the convergent-divergent nozzle (Faucher).simulations Super Moby Dick exp. WAHA-simulations 9 .high pressure “Super Moby Dick” experiment performed at CEA in Grenoble in 80’s . Aim: .WAHA code . (2002). E.to verify the Homogeneous-Relaxation Model (Lemonnier) used in the WAHA code to model inter-phase heat and mass transfer in dispersed flow . Faucher. Doctorat de l’universe Paris Val de Marne. Simulation numerique des ecoulements unidimensionnels instationnaires avec autovaporisation. temperature T = 465.5% (strong phase changes).overall loss of mass flow along the nozzle is less than 0. inlet outlet Boundary conditions: -inlet: constant pressure pR = 80 bar.5 K Conclusion: .6 K (20/465.simulations Super Moby Dick exp. WAHA-simulations 10 .WAHA code .advantage of the WAHA code: critical flow is simulated with standard discretisation and boundary conditions .maximum non-conservation is less than ~1.7) -outlet: constant pressure pL = 47 bar.non-conservative numerical scheme: . temperature T = 549.7 % . 60 0.75 [m] 0. Why critical flow? c Vliq Vvap [m/s] 10000 1000 100 10 1 0.45 0.WAHA code .90 11 WAHA-simulations .15 0.simulations Super Moby Dick exp.30 0.00 0. 45 0.0 0.60 0.6 0.00 0.1 0.75 0.30 0.4 490 480 470 460 450 440 430 420 410 [K] T sat T liq VVF [VVF] 1.WAHA code .30 0.5 0.4 0.15 0.75 [m] 0.90 pinlet = 20 bar poutlet = 16 bar > pSAT = cca 13 bar WAHA-simulations pinlet = 20 bar poutlet = 4 bar < pSAT 12 . Why flashing flow? (VVF=α) [K] 490 485 480 475 470 465 460 455 0.0 [m] 0.0 0.00 0.45 0.2 0.90 0.simulations Super Moby Dick exp.15 0.3 0.60 0.2 T sat T liq VVF [VVF] 0.8 0. 0 0.0 6.30 0.00 0.90 13 WAHA-simulations .75 [m] 0.0 2.15 0.simulations Super Moby Dick exp.WAHA code .0 4.0 8.45 0.0 10.0 0.60 0. Pressure [bar] WAHA 1 EXP 1 WAHA 2 EXP 2 WAHA 3 EXP 3 [MPa] 12. 7 0.60 0.15 WAHA 1 EXP 1 Vapor volume fraction α WAHA 2 EXP 2 WAHA 3 EXP 3 0.00 0.8 0.2 0.90 14 WAHA-simulations .0 0.WAHA code .6 0.0 0.3 0.9 0.75 [m] 0.45 0.4 0. [α] 1.simulations Super Moby Dick exp.5 0.30 0.1 0. simulations Super Moby Dick exp.0 14.0 0.90 15 WAHA-simulations .WAHA code .00 0.0 16.45 0.75 [m] 0.60 0.15 WAHA 1 EXP 1 0.0 12.0 10. Mass flow rate [kg/s] WAHA 2 EXP 3 WAHA 3 EXP 2 [kg/s] 20.30 0.0 18. initial conditions and . The University of Michigan.6 mm. e = 1.column separation water hammer induced due by rapid valve closure.fundamental benchmark for two-phase computer codes because of the simple: . L = 36 m. Department of Civil Engineering. Ph.water hammer initiating mechanism. E = 120 GPa Aim: .geometry. A.85 10-4 m2.simulations Simpson’s pipe: TANK PIPE VALVE Measuring point Initial flow direction Transient: . Large water hammer pressures due to column separation in sloping pipes. . 1986. Simpson. R. A = 2.WAHA code . WAHA-simulations 16 .D thesis. 85 10-4 m2.4 m/s -pressure p = 3. WAHA-simulations 17 . E = 120 GPa Conclusion: At low temperatures flashing and condensation of the steam are not governed by the heat and mass transfer between both phases. A = 2.simulations Simpson’s pipe: TANK PIPE VALVE Measuring point Initial flow direction Boundary conditions: -right: closed end (valve) -left: constant pressure pT = 3. e = 1.6 mm. L = 36 m.3 K Effect of the elasticity taken into account.419 bar Initial conditions: -velocity v = 0.WAHA code .419 bar -temperature T = 296. but by the dynamics of the liquid column (energy equations are not needed). WAHA code .s tiff pipe WAHA3 .05 0.4 0.3 Time [s ] WAHA-simulations 18 .1 0.2 0 0 0.2 0.e la s tic pipe WAHA3 .uns te a dy friction 0.simulations Pressure near the valve [MPa] P res s ure his tory near the valve [MP a ] Simpson’s pipe: 1.8 0.2 1 Expe rime nt WAHA3 .6 0.15 0.25 0. simulations Simpson’s pipe: VVF his tory near the valve [] 0.1 0.2 0.004 0.WAHA code .25 0.05 0.01 0.006 0.3 Time [s ] WAHA-simulations 19 .012 Vapor volume fraction α near the valve WAHA3 .s tiff pipe WAHA3 .008 0.15 0.002 0 0 0.uns te a dy friction 0.e la s tic pipe WAHA3 . Energietechnik UMSICHT.simulations PPP pipeline (A. Fraunhofer Institut Umwelt-.database with over 400 experiments performed at UMSICHT’s test loop (total length ~ 200 m) . Siecherheits-.advanced measuring equipment (wire mesh sensor – void distribution) WAHA-simulations 20 . . Oberhausen).WAHA code . Dudlik. 5 m F 2 P turning point B2 -18.4 m -14.9 m F 3 P 137.9 m 67.WAHA code .2 m PIPELINE 81.7 m P09 P12 50.9 m closure valve 0m N=1 bridge P23 P01 F 1 P P02 P03 WM 0.4 m 142.0 m 90.7 m 75.5 m 60.6 m 139.0 m VALVE TANK 146.5 m P18 88.2 m P06 34.6 m 145.5 m WAHA-simulations 21 .8 m P15 67.simulations PPP pipeline: 44.5 m -8.7 m 77.4 m 84.8 m -0.2 m 149. 00m PPP pipeline: VALVE 10.50 m 3.simulations 34.1 K T = 419.7 K T = 392.92 bar.13 bar. v = 4.L = 149.50m PIPELINE 2.6 K 22 WAHA-simulations 1.00m 7.WAHA code .009 m/s. Boundary conditions: -left: closed end (valve) -right: constant pressure Initial conditions: -case 135: p = 1. -case 329: p = 10.00m 2.50m 6.column separation water hammer induced due by rapid valve closure.975 m/s.5 m (valve – tank) 4.00m . -case 307: p = 9. v = 3. T = 293. v = 3.50m 6.50 Transient: .50m 3. 00 TANK m 46.18 bar.50 Modelled section: .975 m/s. 5 1 0 5 10 15 20 Time [s ] WAHA-simulations 23 .5 Case 135 – Steady state – pressure in P03 WAHA3 .simulations 5.5 4 3.s te a dy s ta te PPP pipeline: 135: P res s ure his tory in P 03 [ba r] 5 4.5 3 2.5 2 1.WAHA code . simulations PPP pipeline: 135: P re s s ure his tory in P 03 [bar] 60 Case 135: Pressure near the valve [MPa] Expe rime nt WAHA3 50 40 30 20 10 0 0 1 2 3 4 5 6 7 Time [s ] WAHA-simulations 24 .WAHA code . WAHA code .simulations PPP pipeline: 307: P re s s ure his tory in P 03 [bar] 60 Case 307: Pressure near the valve [MPa] Expe rime nt WAHA3 50 40 30 20 10 0 0 1 2 3 4 5 6 7 Time [s ] WAHA-simulations 25 . simulations Case 329: Pressure near the valve [MPa] 55 PPP pipeline: 329: P res s ure his tory in P 03 [ba r] 50 45 40 35 30 25 20 15 10 5 0 0 2 4 6 8 Expe rime nt WAHA3 10 Time [s ] WAHA-simulations 26 .WAHA code . 8 0.WAHA code .2 0.simulations 1 Case 135: α near the valve [MPa] Expe rime nt WAHA3 135: Va por volume fraction in P 03 [bar] PPP pipeline: 0.6 0.9 0.3 0.1 0 0 1 2 3 4 5 6 7 Time [s ] WAHA-simulations 27 .7 0.5 0.4 0. 7 0.4 0.simulations 1 Case 307: α near the valve [MPa] Expe rime nt WAHA3 307: Vapor volume fraction in P 03 [bar] PPP pipeline: 0.8 0.9 0.2 0.5 0.WAHA code .6 0.3 0.1 0 0 1 2 3 4 5 6 7 Time [s ] WAHA-simulations 28 . 5 0.2 0.WAHA code .4 0.9 0.3 0.1 0 0 2 4 6 8 Expe rime nt WAHA3 10 Time [s ] WAHA-simulations 29 .6 0.8 0.7 0.simulations Case 329: α near the valve [MPa] 1 PPP pipeline: 329: Vapor volume fra ction in P 03 0. simulations Case 329: pressure near the valve [MPa] PPP pipeline: 329: P res s ure his tory in P 03 [ba r] 60 50 Expe rime nt Wa ha 3 Wa ha HEM Wa ha ne g.WAHA code . pre s s ure 40 Influence of different relaxation models 30 20 10 0 0 1 2 3 4 5 Time [s ] WAHA-simulations 30 . WAHA code .simulations 55 Case 329: pressure near the valve [MPa] Expe rime nt WAHA3 RELAP5 PPP pipeline: 329: P res s ure his tory in P 03 [ba r] 50 45 40 35 30 25 20 15 10 5 0 0 2 4 6 8 Comparison to RELAP5 code 10 Time [s ] WAHA-simulations 31 . simulations Case 329: α near the valve [MPa] PPP pipeline: 329: Vapor volume fraction in P 03 1 0.3 0.6 0.8 0.1 0 0 2 4 6 8 10 Expe rime nt WAHA3 RELAP5 Comparison to RELAP5 code Time [s ] WAHA-simulations 32 .4 0.9 0.7 0.WAHA code .2 0.5 0. WAHA code .simulations Case 329: pressure near the valve [MPa] 50 PPP pipeline: 329: P res s ure his tory in P 03 [ba r] 45 40 35 30 25 20 15 10 5 0 0 1 2 3 Ste a dy s ta te .0 K Influence of different initial temperature 4 5 Time [s ] WAHA-simulations 33 . T = 415. T = 419.6 K Ste a dy s ta te . T = 424.0 K Ste a dy s ta te . 5 m (N = 299) dx = 0.25 m (N = 598) dx = 1.simulations PPP pipeline: 329: P res s ure his tory in P 03 [bar] 50 45 40 35 30 25 20 15 10 5 0 0 1 2 3 4 5 Case 329: pressure near the valve [MPa] dx = 0.WAHA code .0 m (N = 150) Grid refinement Time [s ] WAHA-simulations 34 . CWHTF . Altstadt. R.overpressure accelerates a column of liquid water into vacuum at the closed vertical end of the pipe p1 Vapo r vo l.database with 20 experiments performed at FZR’s cold water hammer test facility (CWHTF) .two discontinuities initially present in the pipe that propagate with different velocity. Weiss. H. Carl.0 VALVE p2 evacuation pressure LV E.WAHA code .|Cold Water-Hammer Test Facility. α = 1. WAHA-simulations 35 .0 Wate r o nly α = 0. Forschungszentrum Rossendorf.simulations CWHTF: TANK CLOSED END LE Preferences: . Transient: . frac t. T ~ 295 K p1 CLOSED END TANK LE Vapo r vo l.left: constant pressure (tank) constant pressure (precise geometry .p1 = 1 bar . frac t.right: closed end .no FSI effects considered WAHA-simulations 36 . α = 1.0 Wate r o nly α = 0.0 VALVE p2 evacuation pressure LV Warnings: .p2 = 29 mbar .pipe) Initial conditions: .simulations CWHTF: Experiment labeled “150601”: Boundary conditions: .v = 0 m/s .WAHA code .absence of non-condensable gas model in the WAHA . WAHA code .2 0.15 0.25 0.simulations 50 Pressure near the closed end [bar] Expe rime nt Ta nk Pipe P res s ure his tory near the clos ed end [bar] CWHTF: 45 40 35 30 25 20 15 10 5 0 0 0.1 0.3 0.05 0.4 Time [s ] WAHA-simulations 37 .35 0. WAHA code .9 0.2 0.05 0.1 0.35 0.3 0.6 0.simulations 1 α near the closed end [bar] Ta nk Pipe VVF his tory near the clos ed end [bar] CWHTF: 0.5 0.1 0 0 0.7 0.2 0.25 0.4 Time [s ] WAHA-simulations 38 .4 0.15 0.3 0.8 0. 15 0.25 Expe rime nt Ta nk (ide a l ga s ) Pipe (ide a l ga s ) Liquid – ideal gas mixture 0.3 0.2 0.WAHA code .simulations Pressure near the closed end [bar] 50 P res s ure his tory ne ar the clos ed end [ba r] CWHTF: 45 40 35 30 25 20 15 10 5 0 0 0.05 0.35 0.4 Time [s ] WAHA-simulations 39 .1 0. 55 56 57 58 Cold water injection INITIAL CONDITIONS: steam temperature Ts = 470 K liquid temperature Tl = 295 K liquid velocity vl = 0.simulations KFKI exp.242 m/s pressure p = 14.very complicated thermally controled transient H.: Preferences: . Baranyai. Steam .. WAHALoads project deliverable D48.left: cold water intake (constant velocity) . G. WAHA-simulations 59 40 . G.right: steam tank . 2004..WAHA code .95 m pipe diameter d = 73 mm number of volumes N = 59 Steam tank 2 1 3 4 5 ...M. PMK-2 water hammer tests.condensation induced water hammer was observed in the steamline of the integral experimental device PMK-2 that is located at the Hungarian Atomic Energy Research Institute Experiment labeled “E22”: GEOMETRY: pipe length l = 2. Prasser. condensation caused by cold water injection into main steam-line of VVER.5 bar Boundary conditions: ...440-type PWR. Ezsol. 1 0 0 0.3 0.2 0.5 3 Vapor bubble Liquid slug Vapor WAHA-simulations Length [m] 41 .1 0 0 0.: Transient: .8 0.5 0.strong water hammer appears when the bubble is condensed and two liquid columns collide.WAHA code .6 0.7 0.6 0. 0.9 E05: Liquid volume fraction [ ] KFKI exp.amplitude of the waves increase until the liquid slug is formed that captures the vapor bubble .condensation rate increases and consequently increases relative vapor velocity over the liquid head .4 0.liquid-vapor surface becomes wavy .5 3 Vapor Liquid Length [m] Profile at t = 5.5 1 1.condensation of the entrapped vapor bubble accelerates columns of liquid on both sides of the bubble .5 2 2.5 0.5 2 2.3 0.7 0.5 1 1.simulations Profile at t = 3.liquid flows into the pipe (steam) .4 0.9 E05: Liquid volume fraction [ ] 0.8 0.08 s 1 0.2 0.75 s 1 0. 9 4.: 180 160 140 P res s ure [bar] 120 100 80 60 40 detail dispersed 20 0 4.WAHA code .95 Time [s ] WAHA-simulations 42 .85 4.75 4.8 4.simulations 200 Pressure near the water intake [bar] Expe rime nt WAHA3 KFKI exp. Slovenia WAHA-hands-on 1 . Ljubljana.Technical University of Catalonia and Heat 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. seminar at UPC.Two-phase flow modelling. 2006 Table of contents INTRODUCTION Lectures 1-2 3-6 7-10 TWO-FLUID MODELS Lecture INTERFACE TRACKING IN 3D TWO-PHASE FLOWS Lectures ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS 11) WAHA code .mathematical model and numerical scheme 12) WAHA code .simulations 13) 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-hands-on 2 . WAHA code .mathematical model and numerical scheme . available on internet WAHA-hands-on 3 .reference .WAHA code manual. 85e-4 0. 0.85e-4 0.85 10-4 m2.input file for Simpson's water hammer transient title simpson test .0 * from to comp001c0 002-99 000-00 *------------------------------------------------------* type name comp002ty tank tank_01 * length elast thick rough w.67e3 0. 0. 0.01 time01 0.0 0. A = 2.4 0.80 0.01 * fluid order abr_model eig_val_out extend_out maj_results switch 1 2 3 1 1 1 * ambient_press force_out force 1.0e0 0 0 0 * area incl azim f_coeff which comp002g1 2.f p.0 1.f h.6 mm.001 *-------------minor output -------------------------------TANK * pipe volume variable print00 1 100 1 print02 1 100 2 print03 1 100 3 print04 1 100 5 print05 1 100 7 print06 1 100 9 print07 1 20 1 print08 1 20 2 print09 1 20 3 *-------------pipes -------------------------------* type name comp001ty pipe cev_01 * length elast thick rough w.419e5 0..67e3 0.e+5 0.e-3 1.fr.0 0.0 * from to comp002c0 000-00 001-01 *************************************************************** end PIPE VALVE Measuring point Initial flow direction nods 100 L = 36 m.419e5 0.tr.e-3 0.f p.0 1.2 3.e-3 2. comp002g0 0.0 30 + 2. E = 120 GPa wch_nods 100 nods 0 wch_nods 0 WAHA-hands-on 4 .0 97.0 100 * type press alpha_g velf velg uf ug comp001s0 agpvu 3.6e-3 0.85e-4 0.0-0 1.0e-1 4.0 0.0 0.m.elastic pipe *--------time constants---------------------------------* beg end maj_out min_out diff restart time00 0 2. comp001g0 36.e-3 0.2e11 1.fr.0 0 * type press alpha_g velf velg uf ug comp002s0 agpvu 3.m.tr.fr.fr.0e-1 2.80 0.f h.0e0 0 0 0 * area incl azim f_coeff which comp001g1 2.0 97.0-0 0.0 1. e = 1.0 0. - 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 * 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. 4.266039E-3 0. type press wch_nods *comp001s0 agpvt 20.08e5 comp001s0 agpvt 80.00e5 *comp001s0 agpvt 120.06e5 * from comp001c0 002-99 003-01 ... slide 3/3 0.0 1.0 89 0.0 1.0 90 alpha_g velf 0.00 0.00 0.00 90 { velg tf tg 90 90 90 0.1e0 0.0 0.1e0 0.0 0.1e0 0.0 465.5 0.0 549.6 0.0 578.7 0.0 WAHA-hands-on 5 - input file for Moby-Dick two-phase critical flow transient 2/3 inlet comp001g1 + + + + + + + + + + + + + + + + + + + + + + + + + + 0.003494 0.0032 0.0029 0.0024 0.0020 0.0017 0.0013 0.0008 0.0005 0.000350 0.0003183 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1 2 3 4 5 6 7 8 9 10 46 47 40 45 + + + + + + + + + + + + + outlet 1.3091682E-3 0. 1.3884127E-3 0. 1.4699855E-3 0. 1.5538868E-3 0. 1.6401168E-3 0. 1.7286747E-3 0. 1.8195612E-3 0. 1.9127764E-3 0. 2.0083198E-3 0. 2.1061913E-3 0. 2.2063916E-3 0. 2.30892E-3 2.413777E-3 0. 0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 3.5791826E-4 0. 3.9990842E-4 0. 4.4422692E-4 0. 4.908738E-4 5.398491E-4 5.911529E-4 0. 0. 0. Nozzle cros s -s ection [cm ] 48 49 50 51 52 53 54 55 56 57 58 2 35 30 25 20 15 10 5 0 + + + + + + + + + + 2.5209623E-3 0. 2.630476E-3 2.742318E-3 0. 0. 2.8564883E-3 0. 2.9729874E-3 0. 3.0918144E-3 0. 3.2129711E-3 0. 3.336455E-3 0. 6.4478494E-4 0. 7.007455E-4 7.590344E-4 8.196517E-4 8.825974E-4 0. 0. 0. 0. 3.4622678E-3 0. 3.5904084E-3 0. 3.7208776E-3 0. 3.8536756E-3 0. 3.9888015E-3 0. 4.126256E-3 4.266039E-3 0. 0. + + 9.4787165E-4 0. 1.0154742E-3 0. 1.085405E-3 0. 59 60 61 62 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Length [m] 0.9 + + + 1.1576643E-3 0. 1.232252E-3 0. WAHA-hands-on 6 - 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 tg wch_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 tg wch_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 WAHA-hands-on 7 Technical University of Catalonia and Heat 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 1 Two-phase flow modelling, seminar at UPC, 2006 Table of contents INTRODUCTION Lectures 1-2 3-6 7-10 TWO-FLUID MODELS Lecture INTERFACE TRACKING IN 3D TWO-PHASE FLOWS Lectures ONE-DIMENSIONAL SIMULATIONS OF FAST TRANSIENTS 11) WAHA code - mathematical model and numerical scheme 12) WAHA code - simulations 13) 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 2 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 3 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 behaviour of hydraulic machinery under steady oscillatory conditions, 2001. D. C. Wiggert, Coupled transient flow and structural motion in liquid-filled piping systems 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 4 • • • • • • Fluid-structure interaction in 1D piping systems .Introduction 1D-piping-FSI 5 . 3494 injuries. 1D-piping-FSI 6 . vibration. 357 deaths. frequency change. • Statistical data USA (1986-2000) "Failed Pipe (Internal Force)“: 5979 accidents. costing over $1 billion. displacements and stresses (pipe) and extreme pressures (fluid). • With appropriate FSI analysis: reduction of the extreme pressures in the fluid and maximum stresses in the structure. energy transfer control and prevention of the failures.Fluid-structure interaction in 1D piping systems .Introduction • Fluid-Structure Interaction = FSI Moving fluid Pressure load Deformed structure Redistribution of the pressure load • Consequences: Noise. 5 -1 -1.2 • Time [s] Pressure near the valve (rapid valve closure transient.05 0.1 0. with FSI (s oft pipe) Cas e 5.5 1 } 50% higher maximal pressure! • Wylie about FSI: – 98% pipelines not subjected – no simple FSI inspection criterion – FSI analysis necessary for all pipelines! FSI analyses have been performed only for the most important pipelines P re s s ure [MP a ] 0.Fluid-structure interaction in 1D piping systems .Introduction • • 2 Cas e 4. no FSI (s tiff pipe) FSI during fast transients: accidental condition Conventional simulation of the fast transient: NO FSI (stiff and supported pipe) 1. free valve) 1D-piping-FSI 7 .5 0 -0. full axial coupling.15 0.5 0 0. 1D-piping-FSI 8 .Combinations of the above . T-junctions oxidation 5. The intensity will depend on the proximity of the frequency of vibration to a natural frequency of the pipeline.Common Sources of FSI 1.Vibrating machinery can induce vibrations in the pipeline.Fluid-structure interaction in 1D piping systems . . Long lengths of unsupported or poorly supported pipework Practice: 2.features numbered 1-5 are independently important while combinations are very important. Transient in the fluid (liquid density) .Introduction . Unsupported/unrestrained valves to cavitation or 4. Unsupported/unrestrained elbows vulnerable are parts submerged 3. only Poisson coupling – pipe breathing TANK PIPE Pipe properties: L = 20 m.Fluid-structure interaction in 1D piping systems .5 mm. ρf = 1000 kg/m3 Measuring point 1D-piping-FSI 9 . p = 0 Pa. ρs = 7900 kg/m3 Inital flow direction VALVE Initial conditions: v = 1 m/s.Introduction • Another example (with and without FSI) – Tank-straight pipe-valve system: Pressure near the rapidly closed valve (rapid valve closure transient. R = 398. e = 8 mm. ν = 0. valve is fixed no jucntion coupling effect.3. E = 210 GPa. Introduction • 2 Another example (with and without FSI) – Tank-straight pipe-valve system: P re s s ure his tory ne a r the va lve .5 0 0.5 0 -0.Fluid-structure interaction in 1D piping systems .2 Time [s ] -2.1 0.5 -1.5 -1 -1 -1.de ta il 1.5 0 0.5 no FSI with FSI Pre s s ure his tory ne a r the va lve 3 2.05 0.5 3 3.5 2 2.5 4 Time [s ] 1D-piping-FSI 10 .5 1 0 0.5 1 no FSI with FSI P res s ure [MP a] 2 P res s ure [MP a] 0.5 1.5 -0.15 -2 0.5 1 1. Poisson coupling: pressure waves in the fluid are coupled with axial waves in the structure and changes of the pipe cross-section. .flexural. 1D-piping-FSI 11 .). area changes. junctions. Figurativelly known as pipe breathing .Types of FSI -There are several types of waves that characterize FSI: . valves.Fluid-structure interaction in 1D piping systems . . etc.According to the interaction between these waves one can differentiate the following types of the coupling: .axial. .pressure waves in the fluid.torsional stress waves in the pipeline .Friction coupling: axial waves in the structure are initiated due to the difference between fluid and structure velocity – less important.radial and . .Junction coupling: different waves are appropriately coupled together at geometric changes (elbows. .rotational. It is convenient to classify the dynamic forces into two groups: .Poisson coupling leads to precursor waves . the classical waterhammer waves. .The interaction is always caused by dynamic forces which act simultaneously on fluid and pipe. which travel faster than and hence ahead of.Fluid-structure interaction in 1D piping systems .these are stress wave induced disturbances in the liquid.distributed forces (Poisson and friction coupling) .local forces (junction coupling) 1D-piping-FSI 12 .Types of FSI . 1D-piping-FSI 13 .most recent FSI methods.Two-way coupling . Most of the FSI analyses in the past in fluid-filled systems comprised two separate analyses undertaken sequentially (uncoupled calculation). AnsysCFX. The results are identical in both cases.Fluid-structure interaction in 1D piping systems .Types of FSI .Classification according to the fact whether the fluid knows for pipe deformations or not: . which are used as input to a structural dynamics code. that the fluid code takes into account also deformations of the structure (Abaqus-Fluent. It is also possible to couple codes in each calculation time step (one-way coupling).One-way coupling or uncoupled calculation (fluid transient is evaluated in undeformed structure). etc). Fluid-transient code is used to determine pressure and velocity histories in rigid and anchored structure. where FSI is defined with mathematical model or where two computer codes are coupled successively in such way. . 5 0 -0.de ta il 2 no FSI with FSI -Time-domain and frequency-domain analyses .15 0.5 0 0.5 -1 -1.05 0.Typical outcome of a frequencydomain analysis is a series of graphs highlighting the dominant frequencies in the response of various parameters.Mathematically.Typical outcome of a time-domain analysis is a series of graphs showing how parameters vary in time. time-domain and frequency-domain analyses contain the same information.2 Time [s ] 14 . 1D-piping-FSI 1.5 1 P res s ure [MP a] 0. .Fluid-structure interaction in 1D piping systems .1 0. It is possible. Inversely not always true. . to obtain frequency-domain results from a Fourier analysis of the output from a time-domain analysis. for example.Types of FSI Pre s s ure his tory ne a r the va lve . free structure.. A f p = At 1D-piping-FSI z .no . rod = At .Fluid-structure interaction in 1D piping systems – 1D models .no friction .no two-phase flow . M x =0 15 z v = u z .. u z = 0 v = u z . rigidly anchored structure: Closed pipe. A f p Y rod u z v 0. .no convective term . rod impact: p = const.Skalak’s basic 4 equation model – axial movement: • Set of four linear first-order PDEs: . Qy =0 . free structure: Closed pipe.no damping . fluid: ∂v ∂p ρf + =0 ∂t ∂s ∂v 2ν ∂N x ⎛ 1 2R ⎞ ∂p + 1 − ν2 ⎟ + = 0 ⎜ K Ed EAt ∂t ∂s ⎝ ⎠ ∂t ( ) pipe: Qy & ∂u x ∂N x = ρt At − Rp ∂t ∂s & νR ∂p ∂u x 1 ∂N x =0 EAt ∂t Ed ∂t ∂s Poisson coupling Junction coupling relations – pipe end Constant pressure (tank). 15 Time [s ] 16 0. R = 398.2 0. no FSI (s tiff pipe ) 1 1 P res s ure [MP a] 0. no FSI (s tiff pipe ) 1.Valve closure transient TANK PIPE Pipe properties: L = 20 m.3. E = 210 GPa.5 Ca s e 1.05 0.5 -1 -1 -1. e = 8 mm.5 0 0 -0.5 P res s ure [MP a] 0.1 0.05 0.Fluid-structure interaction in 1D piping systems – 1D models . p = 0 Pa.5 0 0.5 -0. ρs = 7900 kg/m3 Inital flow direction VALVE Initial conditions: v = 1 m/s. ρf = 1000 kg/m3 Pressure near the valve: Left: Right: 2 Measuring point valve fixed valve free 2 1.5 0 Time [s ] 1D-piping-FSI -1.5 mm. with FSI (s oft pipe ) Ca s e 5.5 Ca s e 4.15 0.1 0.2 . with FSI (s oft pipe ) Ca s e 5. ν = 0. 02 170.025 1D-piping-FSI 0 0.5 2 1.5 Expe rime nt Ca lcula tion Ga le Ca lcula tion Tijs s e ling 2. 2.Rod impact experiment .005 0.Fluid-structure interaction in 1D piping systems – 1D models .5 0 -0.5 Expe rime nt Ca lcula tion Ga le Ca lcula tion Tijs s e ling 2 1.5 -1 -1.no influence of supports .02 -2 0.005 0.5 P re s s ure right [MP a] 1 0.The experiment performed at University of Dundee -The experimental apparatus is relatively simple: .01 0.5 0 -0.015 0.no initial deformation .015 0.5 -1 -1.025 Time [s ] Time [s ] .no valve closing time effect.5 -2 0 P re s s ure le ft [MP a] 1 0.01 0.no initial flow influence .5 0. 1 = u x . 1 z .2 . 1 v1 u z .Fluid-structure interaction in 1D piping systems – 1D models .Valentin’s 8 equation model – axial. 2 A f .1 = x . 1 p1 At . 2 z . u z . 2 p2 At . 1 (straight sections) 1D-piping-FSI z . 1 = u y . 2 Junction coupling relations – elbow p1 = p2 . 2 v 2 u z .Skalak’s model + Timoshenko’s beam equations (from beam eq. M x . 2 Singular coupling! A f .) (ρ A +ρ t t f Af ) & ∂u y ∂t - ∂Qy ∂s =0 & 1 ∂Qy ∂u y & = . 1 = Q y .1 = M x .2 18 .ϕz 2 ∂s κ GAt ∂t & 1 ∂M z ∂ϕ z =0 EI t ∂t ∂s & ∂ϕ z ∂M z ρt I t ∂t ∂s = Qy A f . 2 . rotational and flexural movement – for plane pipelines with elbows . 1 = A f . 2 = Q y . uy . 02 1D-piping-FSI 19 .005 0.5e+006 -2e+006 1.Rod impact experiment .01 0.5e+006 1e+006 500000 Calculation Experiment 2e+006 1.005 0.no initial flow influence .02 0 0.015 0.015 0.no influence of supports .01 0.Fluid-structure interaction in 1D piping systems – 1D models .no valve closing time effect. 2.5e+006 2e+006 1.5e+006 1e+006 500000 0 -500000 -1e+006 Calculation Experiment 0 -500000 -1e+006 -1.no initial deformation .5e+006 -2e+006 0 0.The experiment performed at University of Dundee -The experimental apparatus is relatively simple : . Fluid-structure interaction in 1D piping systems – 1D models .ϕz 2 ∂s Rp κ GAt ∂t & 1 ∂M z ∂ϕ z =0 EI t ∂t ∂s (ρ A +ρ ρt I t Af ) & ∂u y ∂t - ∂Qy ∂s = Af p .Valentin’s 8 equation model – smoth model ∂v ∂p + =0 ρf ∂t ∂s ρt At t t 2 Rp 2ν ∂N x ⎛ 1 2R 2 ⎞ ∂p ⎜ K + Ed 1 − ν ⎟ ∂t .x .N x Rp & ∂ϕ z ∂M z ∂t ∂s = Qy Where Rp is curvature radius of the pipe Junction coupling relations – elbow 1D-piping-FSI 20 .EA ∂t + 2 2 R ⎝ ⎠ t ( ) Qy & ∂u x ∂N x = − ∂t ∂s Rp f ⎛ & (1 − 2ν ) u y R 2 ⎞ ∂v ⎜1 − 1 − ⎟ = 2 ⎜ Rp R p ⎟ ∂s ⎝ ⎠ & uy & 1 ∂N x νR ∂p ∂u x = EAt ∂t Ed ∂t ∂s Rp & & u 1 ∂Qy ∂u y & = . 02 .02 0 1D-piping-FSI 0.pressure Pre s s ure at impac t e nd 2500 2000 1500 2000 1500 1000 Pre s s ure at re mo te e nd P res s ure [kP a ] P res s ure [kP a ] 0 0.005 0.Rod impact experiment Comparison between singular and smoth coupling .01 0.005 0.01 0.015 1000 500 0 -500 -1000 -1500 -2000 500 0 -500 -1000 -1500 -2000 Time [s ] 0.015 Time [s ] 21 0.Fluid-structure interaction in 1D piping systems – 1D models . for deformations is 200) L1 = 5 m.3985 m. initial pressure in the pipe is zero. e . 00 E kg = 2 /m 3 10 GP a Valve is fixed (multiplication fact.Fluid-structure interaction in 1D piping systems – 1D models .3 .Valve closure – single elbow pipe Valve is free VALVE P1 (multiplication fact. e = 8 mm. for deformations is 50) PIPE 2 ELBOW TANK PIPE 1 =0 . E = 210 GPa ν = 0.ρ =8 t = 7 9 mm.3 9 ν = 85 m 0. ρt = 7900 kg/m3. fluid velocity v = 1 m/s 1D-piping-FSI L. 2 R Initial flow direction 22 . Valve closure.3. R = 0. 0000 s 0. fluid velocity v = 1 m/s 1D-piping-FSI 23 .2 -0.8 1 1.Fluid-structure interaction in 1D piping systems – 1D models . for deformations is 50) 0 -0.Valve closure – Tank-pipe-valve system.4 1.4 0.4 -0.2 Width [m] (multiplication fact.6 0.6 0 0.2 0.4 Valve is free 0.6 Initia l De forme d 0.6 Length [m] Valve closure.2 1. pipe is arbitrary De fo rmatio ns at t =0. initial pressure in the pipe is zero. Wiggert’s 14 equations model – 1D pipe in 3D space – full coupling: .additional eqs.Fluid-structure interaction in 1D piping systems – 1D models . for x-z plane .additional equations for torsional motion .radial deformations still not included (negligible) 1D-piping-FSI 24 . Fluid-structure interaction in 1D piping systems – numerical methods Vectorial form of the equations – valid for any system: r r ∂ψ ∂ψ +B =0 A ∂t ∂z C = A ⋅B -1 r r ∂ψ ∂ψ +C =0 ∂t ∂z The Jacobian matrix C has some very important properties: .the eigensystem is constant during 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 of Characteristics (MOC) MOC is most common method. 1D-piping-FSI Characteristic lines – processor demanding with increasing time 25 .it is analytically diagonalizable . and Godunov’s method (WAHA). component synthesis method. other methods are mixed MOCFEM procedure. Condensation .Godunov method: near future.Fluid-structure interaction in 1D piping systems – 1D models Two phase flow modelling (void generally reduces FSI effect): . first it has to cause the cavity to collapse.old = Vc.right – uf.new + Af (uf.predicts most of the cavitation situations (in cold water – inertially controlled cavitation) .simple model to implement .Coupling of two codes – using best “market” codes + coupling at fluidstructure interface (Newton’s law) 1D-piping-FSI 26 .when overpressure wave transverses a cavity.Cavitation starts when pressure falls below sat.left) ∆t .MOC: column separation concentrated cavity model (Bergant) . coupling of Valentine's 8 equation model with WAHA code – the result will be two-phase flow FSI coupling .pressure in the cavity fixed at saturation . pressure – the cavity volume Vc is evaluated using: Vc. The delay action associated with this behavior emulates the reduction of fluid wave speed and its dependency on the void fraction .