💧Multiphase Flow Modeling Unit 6 – Numerical Methods & CFD in Multiphase Flow

Key Concepts and Fundamentals

• Multiphase flow involves the simultaneous presence of two or more phases (gas, liquid, or solid) in a system
• Interactions between phases lead to complex flow behavior and heat/mass transfer phenomena
• Interfacial forces (surface tension, drag) play a crucial role in determining the flow characteristics
• Volume fraction represents the proportion of each phase in a given control volume
  • Denoted by $\alpha_k$ for phase $k$
  • Satisfies the constraint $\sum_{k=1}^{n} \alpha_k = 1$, where $n$ is the number of phases
• Slip velocity refers to the relative velocity between different phases
  • Influences the mixing and transport processes in multiphase systems
• Interfacial area concentration quantifies the amount of interfacial area per unit volume
  • Affects the rates of heat, mass, and momentum transfer between phases
• Flow regimes describe the spatial distribution of phases (bubbly flow, slug flow, annular flow)
  • Determined by factors such as phase velocities, fluid properties, and geometry

Governing Equations

• Conservation of mass (continuity equation) for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \mathbf{v}_k) = 0$
  • $\rho_k$ is the density and $\mathbf{v}_k$ is the velocity of phase $k$
• Conservation of momentum for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k \mathbf{v}_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \mathbf{v}_k \mathbf{v}_k) = -\alpha_k \nabla p + \nabla \cdot (\alpha_k \mathbf{\tau}_k) + \alpha_k \rho_k \mathbf{g} + \mathbf{M}_k$
  • $p$ is the pressure, $\mathbf{\tau}_k$ is the stress tensor, $\mathbf{g}$ is the gravitational acceleration, and $\mathbf{M}_k$ represents the interfacial forces
• Conservation of energy for each phase $k$:
  • $\frac{\partial (\alpha_k \rho_k h_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k h_k \mathbf{v}_k) = -\nabla \cdot (\alpha_k \mathbf{q}_k) + Q_k$
  • $h_k$ is the specific enthalpy, $\mathbf{q}_k$ is the heat flux, and $Q_k$ represents the interfacial heat transfer
• Closure models are required to describe the interfacial forces, heat transfer, and turbulence
  • Drag force, lift force, virtual mass force, and turbulent dispersion force are commonly considered
• Equation of state relates the pressure, density, and temperature of each phase
  • Ideal gas law for gases and incompressible fluid assumption for liquids are often used

Discretization Techniques

• Finite difference method (FDM) approximates derivatives using Taylor series expansions
  • Suitable for structured grids and simple geometries
  • Explicit and implicit schemes are available
• Finite volume method (FVM) divides the domain into control volumes and applies conservation principles
  • Handles complex geometries and unstructured grids
  • Ensures conservation of quantities at the discrete level
• Finite element method (FEM) uses a variational formulation and shape functions to approximate the solution
  • Provides high-order accuracy and flexibility in handling irregular geometries
  • Computationally more expensive compared to FDM and FVM
• Spectral methods represent the solution using a linear combination of basis functions (Fourier series, Chebyshev polynomials)
  • Offers high accuracy for smooth solutions and periodic boundary conditions
• Temporal discretization schemes include explicit (forward Euler), implicit (backward Euler), and semi-implicit methods (Crank-Nicolson)
  • Explicit schemes are simple but have stability limitations
  • Implicit schemes are stable but require the solution of a system of equations at each time step
• Spatial discretization schemes include upwind, central, and high-resolution schemes (QUICK, TVD)
  • Upwind schemes are stable but introduce numerical diffusion
  • Central schemes have lower numerical diffusion but may lead to oscillations
  • High-resolution schemes combine the advantages of upwind and central schemes

Numerical Schemes for Multiphase Flow

• Two-fluid model treats each phase as a separate fluid with its own set of conservation equations
  • Coupling between phases is achieved through interfacial terms
  • Suitable for dispersed and separated flows
• Mixture model considers the mixture of phases as a single fluid with averaged properties
  • Solves a single set of conservation equations for the mixture
  • Requires additional equations for the volume fraction and relative velocities
• Volume of Fluid (VOF) method tracks the interface between immiscible fluids using a color function
  • Solves a single set of equations for the mixture while advecting the color function
  • Captures sharp interfaces but may suffer from numerical diffusion
• Level-set method represents the interface as a zero level-set of a higher-dimensional function
  • Advects the level-set function and reconstructs the interface
  • Maintains a smooth interface but requires reinitialization to preserve the signed distance property
• Lagrangian particle tracking follows the motion of individual particles or bubbles
  • Coupled with the continuous phase equations through source terms
  • Suitable for dilute dispersed flows and particle-laden flows
• Eulerian-Lagrangian methods combine Eulerian description of the continuous phase with Lagrangian tracking of dispersed phase
  • Particle-in-Cell (PIC) and Discrete Element Method (DEM) are examples
  • Captures the detailed motion and interactions of particles or bubbles

CFD Software and Tools

• Commercial CFD packages (ANSYS Fluent, STAR-CCM+, COMSOL Multiphysics) provide comprehensive multiphase flow modeling capabilities
  • User-friendly interfaces and extensive documentation
  • Offer a wide range of physical models and numerical schemes
• Open-source CFD software (OpenFOAM, MFIX, OpenFVM) allows customization and development of new models
  • Require more programming expertise but provide flexibility and cost-effectiveness
  • Active user communities and increasing availability of tutorials and resources
• Meshing tools (ANSYS Meshing, Pointwise, Gmsh) generate computational grids for complex geometries
  • Structured, unstructured, and hybrid mesh options
  • Mesh quality assessment and refinement capabilities
• Post-processing and visualization software (ParaView, Tecplot, VisIt) enable analysis and interpretation of simulation results
  • Interactive visualization of flow fields, contours, and streamlines
  • Quantitative analysis and data extraction features
• High-performance computing (HPC) resources are essential for large-scale multiphase flow simulations
  • Parallel computing using Message Passing Interface (MPI) or OpenMP
  • GPU acceleration for computationally intensive tasks
• Coupling with other physics solvers (structural mechanics, electromagnetics) allows multiphysics simulations
  • Fluid-structure interaction (FSI) for deformable structures
  • Magnetohydrodynamics (MHD) for electrically conducting fluids

Simulation Setup and Boundary Conditions

• Geometry and computational domain definition based on the physical problem
  • Simplifications and assumptions to balance accuracy and computational cost
  • Consideration of symmetry, periodicity, and far-field boundaries
• Mesh generation and refinement to capture relevant flow features
  • Sufficient resolution near interfaces, boundaries, and regions of high gradients
  • Mesh independence study to ensure solution accuracy
• Initial conditions specify the state of the system at the start of the simulation
  • Phase distribution, velocity, pressure, and temperature fields
  • Consistent with the physical problem and boundary conditions
• Boundary conditions define the behavior at the domain boundaries
  • Inlet: specified velocity, volume fraction, or mass flow rate
  • Outlet: prescribed pressure, outflow, or convective conditions
  • Walls: no-slip, free-slip, or partial-slip conditions
  • Symmetry: zero normal gradients and zero fluxes across the boundary
• Material properties and constitutive relations for each phase
  • Density, viscosity, surface tension, and other relevant properties
  • Models for interfacial forces, drag, lift, and turbulence
• Numerical schemes and solution parameters
  • Choice of discretization methods, time integration schemes, and solver settings
  • Convergence criteria and tolerance for iterative solvers
• Monitoring and data extraction during the simulation
  • Residuals, force balances, and integral quantities of interest
  • Transient data recording and averaging for unsteady flows

Solution Methods and Algorithms

• Pressure-velocity coupling algorithms ensure the satisfaction of continuity and momentum equations
  • SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) and its variants (SIMPLEC, PISO)
  • Fractional Step Method (FSM) for time-dependent flows
• Iterative solvers for the linearized system of equations
  • Jacobi, Gauss-Seidel, and Successive Over-Relaxation (SOR) methods
  • Krylov subspace methods (Conjugate Gradient, GMRES) for large and sparse systems
• Multigrid methods accelerate the convergence of iterative solvers
  • Coarse-grid correction and smoothing operations
  • Algebraic multigrid (AMG) for unstructured grids
• Preconditioning techniques improve the conditioning of the linear system and enhance solver performance
  • Diagonal scaling, incomplete LU factorization (ILU), and block preconditioners
• Time advancement schemes for transient simulations
  • Explicit schemes (forward Euler, Runge-Kutta) for time-accurate simulations
  • Implicit schemes (backward Euler, Crank-Nicolson) for stability and larger time steps
• Adaptive time-stepping adjusts the time step size based on the flow dynamics and numerical stability
  • Courant-Friedrichs-Lewy (CFL) condition for explicit schemes
  • Error estimation and control for implicit schemes
• Parallel computing strategies for efficient execution on multi-processor systems
  • Domain decomposition and load balancing techniques
  • Communication and synchronization between processors using MPI or OpenMP

Validation and Verification

• Verification assesses the correctness of the numerical implementation and solution
  • Code verification: comparison with analytical solutions, manufactured solutions, or benchmark problems
  • Solution verification: grid convergence studies, temporal convergence, and iterative convergence
• Validation evaluates the accuracy of the mathematical model and its ability to represent the physical phenomena
  • Comparison with experimental data or well-established correlations
  • Quantitative assessment using error norms, statistical measures, and uncertainty quantification
• Sensitivity analysis investigates the influence of input parameters and model assumptions on the simulation results
  • Identification of critical parameters and their impact on the quantities of interest
  • Design of experiments (DOE) and parameter space exploration techniques
• Uncertainty quantification (UQ) characterizes the propagation of input uncertainties to the simulation outputs
  • Probabilistic methods (Monte Carlo, polynomial chaos) for quantifying output uncertainties
  • Sensitivity indices and variance-based methods for ranking input parameters
• Code-to-code comparison involves comparing the results from different CFD codes or numerical implementations
  • Identification of discrepancies and their sources (numerical schemes, physical models)
  • Establishment of best practices and guidelines for consistent and reliable simulations
• Experimental validation requires careful design and execution of experiments
  • Selection of representative test cases and operating conditions
  • Measurement techniques (PIV, LDV, X-ray tomography) for detailed flow characterization
  • Quantification of experimental uncertainties and their impact on validation metrics

Practical Applications and Case Studies

• Oil and gas industry: multiphase flow in pipelines, separators, and wellbores
  • Prediction of flow patterns, pressure drop, and phase distribution
  • Design and optimization of production systems and flow assurance strategies
• Chemical and process engineering: reactors, mixers, and separation equipment
  • Modeling of gas-liquid, gas-solid, and liquid-liquid flows
  • Optimization of mixing, mass transfer, and reaction kinetics
• Nuclear engineering: two-phase flow in nuclear reactors and steam generators
  • Prediction of void fraction, critical heat flux, and flow instabilities
  • Safety analysis and accident scenario simulations
• Environmental engineering: sediment transport, pollutant dispersion, and multiphase flows in porous media
  • Modeling of erosion, deposition, and resuspension processes
  • Remediation strategies and contaminant fate and transport studies
• Biomedical engineering: blood flow, drug delivery, and microfluidic devices
  • Simulation of red blood cell and platelet transport in blood vessels
  • Design and optimization of drug delivery systems and lab-on-a-chip devices
• Aerospace engineering: fuel injection, spray combustion, and icing on aircraft wings
  • Modeling of atomization, droplet breakup, and coalescence processes
  • Prediction of ice accretion and its impact on aerodynamic performance
• Renewable energy: wind and tidal turbines, solar receivers, and fuel cells
  • Simulation of wind and tidal flows around turbine blades
  • Modeling of heat transfer and fluid flow in solar receivers and fuel cell channels