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**Multiphase and Reactive Flow Modelling**

BMEGEÁTMW07 K. G. Szabó Dept. of Hydraulic and Water Management Engineering, Faculty of Civil Engineering Spring semester, 2012

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**Introduction Physical phenomena**

Major concepts, definitions, notions and terminology Equilibrium vs. non-equilibrium models Lagrangean vs. Eulerian description Dimensionless numbers Modelling strategies Update me!

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**Basic notions and terminology**

Ordinary phases: Solid Liquid Gaseous preserve volume Condensed phases expands preserves shape Fluid phases deform There also exist extraordinary phases, like plastics and similarly complex materials The property of fluidity serves in the definition of fluids

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**Properties and models of solids**

Properties of solids: Mass (inertia), position, translation Extension (density, volume), rotation, inertial momentum Elastic deformations (small, reversible and linear), deformation and stress fields Inelastic deformations (large, irreversible and nonlinear), dislocations, failure etc. Modelled features: Mechanics Statics: mechanical equilibrium is necessary Dynamics: governed by deviation from mechanical equilibrium Thermodynamics of solids Mass point model Rigid body model The simplest continuum model Even more complex models

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**Models and properties of fluids**

Key properties of fluids: Large, irreversible deformations Density, pressure, viscosity, thermal conductivity, etc. (are these properties or states?) Features to be modelled: Statics Hydrostatics: definition of fluid (inhomogeneous [pressure and density]) Thermostatics: thermal equilibrium (homogenous state) Dynamics Mechanical dynamics: motion governed by deviation from equilibrium of forces Thermodynamics of fluids: Deviation from global thermodynamic equilibrium often governs processes multiphase, multi-component systems Local thermodynamic equilibrium is (almost always) maintained Only continuum models are appropriate!

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**Modelling Simple Fluids**

Inside the fluid: Transport equations Mass, momentum and energy balances 5 PDE’s for Constitutive equations Algebraic equations for Boundary conditions On explicitly or implicitly specified surfaces Initial conditions Primary (direct) field variables (p,T) represaentattion Secondary (indirect) field variables

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**Thermodynamical representations**

Representation (independent variables) TD potential enthropy and volume (s,1/ρ) internal energy temperature and volume (T,1/ρ) free energy enthropy and pressure (s,p) enthalpy temperature and pressure (T,p) free enthalpy All of these are equivalent: can be transformed to each other by appropriate formulæ Use the one which is most practicable: e.g., (s,p) in acoustics: s = const ρ(s,p) ρ(p). We prefer (T,p) Note

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**Some models of fluids Stoksean fluid compressible**

In both of these, the heat transport problem can be solved separately (one-way coupling): Mutually coupled thermo-hydraulic equations: Non-Newtonian behaviour etc. compressible (or barotropic) fluid fluid dynamical equations heat transport equation (1 PDE) general simple fluid fluid dynamical equations heat transport equation models for complex fluids

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**Phase transitions Evaporation, incl. Condensation Freezing Melting**

Boiling Cavitation Condensation Freezing Melting Solidification Sublimation All phase transitions involve latent heat deposition or release

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**Typical phase diagrams of a pure material:**

(crystal structures) may exist Several solid phases TODO: 1/rho=spec volume, viscosity, chemical potential. Write equations of equilibrium. In equilibrium 1, 2 or 3 phases can exist together Complete mechanical and thermal equilibrium

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**Conditions of local phase equilibrium in a contact point in case of a pure material**

2 phases: T(1)=T(2)=:T p(1)=p(2)=:p μ(1)(T,p)= μ(2)(T,p) Locus of solution: a line Ts(p) or ps(T), the saturation temperature or pressure (e.g. ‘boiling point´). 3 phases: T(1)=T(2) =T(3)=:T p(1)=p(2)=p(3) =:p μ(1)(T,p)= μ(2)(T,p) = μ(3)(T,p) Locus of solution: a point (Tt,pt), the triple point. This should be recast to the multicomponent case

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Multiple components Almost all systems have more than 1 chemical components Phases are typically multi-component mixtures Concentration(s): measure(s) of composition There are lot of practical concentrations in use, e.g. Mass fraction (we prefer this!) Volume fraction (good only if volume is conserved upon mixing!) Concentration fields appear as new primary field variables in the equation: One of them (usually that of the solvent) is redundant, not used. TODO: concentrations

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**Notations to be used (or at least attempted)**

Phase index (upper): (p) or (s), (l), (g), (v), (f) for solid, liquid, gas, fluid, vapour Component index (lower): k Coordinate index (lower): i, j or t Examples: Partial differentiation: Note

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**Material properties in multicomponent mixtures**

One needs constitutional equations for each phase These algebraic equations depend also on the concentrations For each phase (p) one needs to know: the equation of state the viscosity the thermal conductivity the diffusion coefficients

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**Conditions of local phase equilibrium in a contact point in case of multiple components**

Suppose N phases and K components: Thermal and mechanical equilibrium on the interfaces: T(1)=T(2) =T(3)=:T p(1)=p(2)=p(3)=:p Mass balance for each component among all phases: (N-1)K equations for 2+N(K-1) unknowns This should be recast to the multicomponent case

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**Phase equilibrium in a multi-component mixture**

Gibbs’ Rule of Phases, in equilibrium: If there is no (global) TD equilibrium: additional phases may also exist in transient metastable state or spatially separated, in distant points TD limit on the # of phases

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Miscibility The number of phases in a given system is also influenced by the miscibility of the components: Gases always mix → Typically there is at most 1 contiguous gas phase Liquids maybe miscible or immiscible → Liquids may separate into more than 1 phases (e.g. polar water + apolar oil) Surface tension (gas-liquid interface) Interfacial tension (liquid-liquid interface) (In general: Interfacial tension on fluid-liquid interfaces) Solids typically remain granular

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**Topology of phases and interfaces**

A phase may be Contiguous (more than 1 contiguous phases can coexist) Dispersed: solid particles, droplets or bubbles of small size usually surrounded by a contiguous phase Compound Interfaces are 2D interface surfaces separating 2 phases gas-liquid: surface liquid-liquid: interface solid-fluid: wall 1D contact lines separating 3 phases and 3 interfaces 0D contact points with 4 phases, 6 interfaces and 4 contact lines Topological limit on the # of phases (always local)

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**Special Features to Be Modelled**

Multiple components → chemical reactions molecular diffusion of constituents Multiple phases → inter-phase processes momentum transport, mass transport and energy (heat) transfer across interfaces. (Local deviation from total TD equilibrium is typical)

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**Class 3 outline Balance equations**

Mass balance — equation of continuity Component balance Advection Molecular diffusion Chemical reactions

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extensive quantity: F density: φ=F/V=ρ∙f specific value f=F/m molar value f=F/n molecular value F*=F/N

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**Differential forms of balance equations**

Conservation of F: equations for the density general only convective flux equation for the specific value These forms describe passive advection of F

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**Incomplete without class notes**

! Diffusion — continued further diffusion models the advection—diffusion equations Chemical reactions the advection—diffusion—reaction equations stochiometric equation reaction heat chemical equilibrium reaction kinetics frozen and fast reactions

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**Further diffusion models**

Thermodiffusion and/or barodiffusion Occur(s) at high concentrations high T and/or p gradients For a binary mixture: coefficient of thermodiffusion coefficient of barodiffusion Analogous cross effects appear in the heat conduction equation

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**Further diffusion models**

Nonlinear diffusion model Cross effect among species’ diffusion Valid at high concentrations more than 2 components low T and/or p gradients (For a binary mixture it falls back to Fick’s law.) Make D caligraphic. C.f. flug 8.9

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**The advection–diffusion equations**

local rate of change advective flux diffusive flux The concentrations are conserved but not passive quantities

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**The advection–diffusion–reaction equations**

local rate of change advective flux diffusive flux reactive source terms The concentrations are not conserved quantities

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**Incomplete without class notes**

! Mathematical description of interfaces implicit description parametric description (homework) normal, tangent, curvature interface motion Transport through interfaces continuity and jump conditions mass balance heat balance force balance

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**Interfaces and their motion**

Description of interface surfaces: parametrically by implicit function (the explicit description is the common case of the two) Moving phase interface: (only!) the normal velocity component makes sense New primary(?) field variables Incomplete without class notes !

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**Description of an interface by an implicit function**

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**Equation of motion of an interface given by implicit function**

Equation of interface Path of the point that remains on the interface (but not necessarily a fluid particle) Differentiate For any such point the normal velocity component must be the same Propagation speed and velocity of the interface Only the normal component makes sense

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**Parametric description of an interface and its motion**

Homework: Try to set it up analogously

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**Mass balance through an interface**

Steps of the derivation: describe in a reference frame that moves with the interface (e.g. keep the position of the origin on the interface) velocities inside the phases in the moving frame mass fluxes in the moving frame flatten the control volume onto the interface Incomplete without class notes !

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**The kinematical boundary conditions**

Incomplete without class notes ! This condition does not follow from mass conservation The tangential eqs do not follow from the principle of conservation of mass!! continuity of velocity conservation of interface

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**Diffusion through an interface**

Mass flux of component k in the co-moving reference frame: Case of conservation of component mass: on a pure interface (no surface phase, no surfactants) without surface reactions (not a reaction front) The component flux through the interface: The tangential eqs do not follow from the principle of conservation of mass!!

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Examples Impermeability condition Surface reaction

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**Momentum balance through an interface**

Effects due to surface tension surface viscosity surface compressibility mass transfer

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**Incomplete without class notes**

Surface tension The origin and interpretation of surface tension Incomplete without class notes !

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**Dynamical boundary conditions with surface/interfacial tension**

Fluids in rest normal component: Moving fluids without interfacial mass transfer tangential components: The viscous stress tensor: Modifies the thermodynamic phase equilibrium conditions Incomplete without class notes !

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**The heat conduction equation**

The equation Fourier’s formula (thermodiffusion not included!) Volumetric heat sources: viscous dissipation direct heating chemical reaction heat Boundary conditions Thermal equilibrium Heat flux: continuity (simplest) latent heat (phase transition of pure substance) Even more complex cases: chemical component diffusion chemical reactions on surface direct heating of surface Jump conditions

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**Boundary conditions on moving interfaces**

Physical balance equations imply conditions on the interface elements: continuity conditions jump conditions Other conditions prescribed to obtain a well set mathematical model With and without mass transfer Incomplete without class notes !

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**Approaches of fine models**

Phase-by-phase Separate sets of governing equations for each phase Each phase is treated as a simple fluid Describing/capturing moving interfaces Prescribing jump conditions at the interfaces One-fluid A single set of governing equation for all phases Complicated constitutional equations Describing/capturing moving interfaces Jumps on the interfaces are described as singular source terms in the governing equations 1-fluid = embedded interface

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**Phase-by-phase mathematical models**

A separate phase domain for each phase A separate set of balance equations for each phase domain, for the primary field variables introduced for the single phase problems, supplemented by the constitutional relations describing the material properties of the given phase The sub-model for the motion of phase domains and phase boundaries (further primary model variables) Prescribing the moving boundary conditions: coupling among the field variables of the neighbouring phase domains and the interface variables II/13Bp6f1

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**The one-fluid mathematical model**

A single fluid domain Characteristic function for each phase Material properties expressed by the properties of individual phases and the characteristic functions A single set of balance equations for the primary field variables introduced for the single phase problems, supplemented by discrete source terms describing interface processes The sub-model for the motion of phase domains and phase boundaries (further primary model variables) II/13Bp6f1

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**The sub-models of phase motion (interface sub-models)**

The choice of the mathematical level sub-model is influenced by the available effective numerical methods.

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**Numerical implementations of interface sub-models**

Main categories Grid manipulation Front capturing: implicit interface representation Front-tracking: parametric interface representation Full Lagrangian Specific methods MAC: (Marker-And-Cell) VOF: (Volume-of-Fluid) level-set phase-field CIP E.g. SPH

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**Front tracking methods**

on a fixed grid by connected marker points (Suits the parametric mathematical description) In 3D: triangulated unstructured grid represents the surface Tasks to solve: Advecting the front Interaction with the grid (efficient data structures are needed!) Merging and splitting (hard!) II/13Bp11f7

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**MAC (Marker-And-Cell method)**

An interface reconstruction — front capturing — model (the primary variable is the characteristic function of the phase domain, the interface is reconstructed from this information) The naive numerical implementation of the mathematical transport equation : 1st (later 2nd) order upwind differential scheme Errors (characteristic to other methods as well!): numerical diffusion in the 1st order numerical oscillation in higher orders Incomplete without class notes ! Due to the discontinuities of the function

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**Incomplete without class notes**

MAC Incomplete without class notes !

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**VOF (Volume-Of-Fluid method)**

1D version (1st order explicit in time): Gives a sharp interface, conserves mass Requires special algorithmic handling The scheme of evolution:

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**VOF in 2D and 3D SLIC: Simple Line Interface Construction PLIC:**

Piecewise Linear Interface Construction Hirt & Nichols

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**Numerical steps of VOF Interface reconstruction within the cell**

determine n several schemes position straight interface Interface advection several schemes exist, goals: conserve mass exactly avoid diffusion avoid oscillations Compute the surface tension force in the Navier–Stokes eqs. several schemes

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**Implementation of VOF in**

Any number of phases can be present The transport equation for is adapted to allow variable density of phases mass transport between phases Contact angle model at solid walls is coupled Special (`open channel´) boundary conditions for VOF Surface tension is implemented as a continuous surface force in the momentum equation For the flux calculations ANSYS FLUENT can use one of the following schemes: Geometric Reconstruction: PLIC, adapted to non-structured grids Donor-Acceptor: Hirt & Nichols, for quadrilateral or hexahedral grid only Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM): a general purpose sheme for sharp jumps (e.g. high ratios of viscosities) for arbitrary meshes Any of its standard schemes (probably diffuse and oscillate) References in FLUENT Users Guide: [Youngs [412]], [145], [Ubbink [377]]

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**The level set method [hu: nívófelület-módszer]**

the interface is implicit F is continuous standard advection schemes work fine the curvature can be obtained easily the effect of surface tension within a cell can be computed

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The level set method If then the computational demand can be substantially decreased

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**Signed distance function as an implicit level-set function**

What kind of function is it? Signed distance from the interface! Alas, is not conserved. Generating F: τ is pseudo-time (t is not changed) Apply alternatively! Unfortunately, mass is not conserved in the numeric implementation. A better numeric scheme

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**Only first order accurate in h**

Numerical implementation of the interfacial source terms in the transport equations Missing function graph! With ε = 1.5h, the interface forces are smeared out to a three-cell thick band Only first order accurate in h

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For example, the normal jump condition due to surface tension can be expressed as an embedded singular source term in the Navier–Stokes equation: contribution to a single cell in a finite volume model: Other source terms (latent heat, mass flux) in the transport equations can be treated analogously. C.f. VOF

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**Level set demo simulations**

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**Evaluation criteria for comparison**

Not only for VOF and Level Set Ability to conserve mass/volume exactly numerical stability keep interfaces sharp (avoid numerical diffusion and oscillation) Ability and complexity to model more than 2 phases phase transitions compressible fluid phases Demands on resources number of equations grid spacing grid structure time stepping differentiation schemes Limitations of applicability grid types differential schemes accuracy

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Recommended books Stanley Osher, Ronald Fedkiw: Level Set Methods and Dynamic Implicit Surfaces Applied Mathematical Sciences, Vol. 153 (Springer, 2003). ISBN Details on the level set method Grétar Tryggvason, Ruben Scardovelli, Stéphane Zaleski: Direct Numerical Simulations of Gas–Liquid Multiphase Flows (Cambridge, 2011). ISBN Modern solutions in VOF and front tracking ISBN:

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**SPH Smoothed Particle Hydrodynamics**

The other extreme — a meshless method: The fluid is entirely modelled by moving representative fluid particles — fully Lagrangian There are no mesh cells interfaces PDE field variables Everything is described via ODE’s

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**SPH simulation of hydraulic jump**

Fr1 = 1.15 Fr1 = 1.37 Fr1 = 1.88

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**SPH simulation of dam-break**

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**Liquid vs. liquid-gas simulation**

Air Entrapped air Vacuum Void bubble

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**Evaluation of SPH Advantages Conceptually easy Best suits problems**

in which inertia dominates (violent motion, transients, impacts) FSI modelling with free surface or liquid–gas interface Interface develops naturally Computationally fast Easy to parallelise Can be adapted to GPU’s Disadvantages High number of particles Hard to achieve incompressibility Some important boundary conditions are not realised so far

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