6.2 Finite volume method

The finite volume method is a powerful numerical technique for solving complex fluid flow problems. It discretizes the domain into control volumes, applying conservation laws to each. This approach is particularly useful for multiphase flows, where it can handle intricate geometries and nonlinear phenomena.

Key aspects include integral formulation of equations, discretization techniques, and flux evaluation. The method also addresses mesh generation, boundary conditions, and solution algorithms. For multiphase flows, extensions like VOF and Eulerian-Eulerian approaches are crucial for capturing phase interactions and interfaces.

Finite volume method overview

• Fundamental numerical approach for solving partial differential equations (PDEs) that govern fluid flow and heat transfer
• Discretizes the computational domain into a finite number of control volumes, over which conservation equations are integrated
• Suitable for complex geometries and nonlinear problems encountered in multiphase flow modeling

Integral form of conservation equations

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• Conservation equations for mass, momentum, and energy are expressed in integral form over a control volume
• Integral formulation ensures conservation of quantities at a discrete level, even with coarse meshes
• Gauss's divergence theorem is applied to convert volume integrals of divergence terms into surface integrals

Control volume approach

• Computational domain is divided into a finite number of non-overlapping control volumes
• Conservation equations are applied to each control volume, with fluxes evaluated at the control volume faces
• Provides a natural framework for enforcing conservation and handling discontinuities in material properties

Discretization techniques

• Process of converting continuous PDEs into a system of algebraic equations that can be solved numerically
• Involves approximating spatial and temporal derivatives using finite differences or interpolation schemes
• Choice of discretization scheme affects accuracy, stability, and computational cost of the solution

Spatial discretization

• Approximation of spatial derivatives in the conservation equations using finite differences or interpolation
• Common schemes include central differencing, upwind differencing, and higher-order methods (QUICK, MUSCL)
• Selection of spatial discretization scheme depends on the desired accuracy, stability, and computational efficiency

Temporal discretization

• Approximation of temporal derivatives in the conservation equations using finite differences
• Explicit schemes (Forward Euler) evaluate derivatives using known values from the previous time step, while implicit schemes (Backward Euler, Crank-Nicolson) involve unknown values at the current time step
• Implicit schemes require the solution of a system of equations at each time step but allow for larger time steps and improved stability

Explicit vs implicit schemes

• Explicit schemes are computationally inexpensive but have strict stability limits on the time step size (CFL condition)
• Implicit schemes are more computationally demanding but allow for larger time steps and unconditional stability
• Choice between explicit and implicit schemes depends on the problem's stiffness, desired accuracy, and available computational resources

Mesh generation

• Process of discretizing the computational domain into a set of control volumes or elements
• Quality of the mesh directly impacts the accuracy and stability of the numerical solution
• Mesh generation techniques aim to create a grid that accurately represents the geometry and provides a suitable framework for discretization

Structured vs unstructured grids

• Structured grids have a regular connectivity pattern, with control volumes arranged in a logical i-j-k indexing system (Cartesian, cylindrical, curvilinear)
• Unstructured grids have an irregular connectivity pattern, with control volumes of arbitrary shapes (triangles, tetrahedra, polyhedra)
• Structured grids are simpler to generate and lead to more efficient solvers, while unstructured grids offer greater flexibility for complex geometries

Grid quality considerations

• Aspects of grid quality that affect the accuracy and stability of the numerical solution
• Skewness: measures the deviation of control volume faces from orthogonality, with high skewness leading to numerical errors
• Aspect ratio: ratio of the longest to shortest edge of a control volume, with high aspect ratios causing numerical instabilities
• Smoothness: gradual variation in control volume size and shape, with abrupt changes leading to discretization errors

Boundary-fitted coordinates

• Mesh generation technique that maps the physical domain onto a computational domain with a regular structure
• Allows for accurate representation of complex geometries while maintaining a structured grid topology
• Governing equations are transformed into the computational domain, with additional terms arising from the coordinate transformation

Flux evaluation

• Computation of the fluxes of conserved quantities (mass, momentum, energy) across control volume faces
• Accurate flux evaluation is crucial for maintaining conservation and capturing the correct physics
• Various schemes are available for approximating the face values and gradients required for flux calculation

Upwind vs central differencing

• Upwind schemes consider the direction of the flow when approximating face values, ensuring numerical stability
• Central differencing schemes use a symmetric stencil around the face, providing second-order accuracy but potentially leading to oscillations
• Hybrid schemes combine upwind and central differencing based on a local Peclet number to balance stability and accuracy

Higher-order schemes

• Flux approximation methods that use a larger stencil to achieve higher-order accuracy
• Examples include Quadratic Upstream Interpolation for Convective Kinematics (QUICK) and Monotonic Upstream-centered Scheme for Conservation Laws (MUSCL)
• Higher-order schemes reduce numerical diffusion but may introduce overshoots and undershoots near discontinuities

Flux limiters

• Methods for modifying higher-order schemes to enforce monotonicity and prevent spurious oscillations
• Flux limiters (Total Variation Diminishing, Flux Corrected Transport) constrain the gradients used in the flux approximation to ensure physical bounds are respected
• Flux limiting maintains higher-order accuracy in smooth regions while providing stability near discontinuities

Source term treatment

• Incorporation of source terms (body forces, chemical reactions, phase change) into the discretized conservation equations
• Accurate treatment of source terms is essential for maintaining conservation and capturing the correct physics
• Various techniques are employed to approximate the source term integral and ensure numerical stability

Point-wise approximation

• Simplest approach, where the source term is evaluated at the control volume center and multiplied by the cell volume
• Suitable for sources that vary smoothly across the domain, but may lead to inaccuracies for highly localized sources
• Can be improved by using higher-order interpolation or adaptive mesh refinement near strong sources

Linearization techniques

• Methods for linearizing nonlinear source terms to facilitate the solution of the discretized equations
• Explicit linearization evaluates the source term using values from the previous iteration, leading to a linear system of equations
• Implicit linearization approximates the source term using a Taylor series expansion, resulting in a more stable but computationally expensive scheme
• Partial linearization strikes a balance between explicit and implicit approaches, linearizing only the stiff components of the source term

Boundary condition implementation

• Specification of the conditions on the dependent variables or their fluxes at the boundaries of the computational domain
• Proper implementation of boundary conditions is crucial for obtaining a well-posed problem and a physically meaningful solution
• Different types of boundary conditions arise in multiphase flow problems, each requiring specific treatment

Dirichlet vs Neumann conditions

• Dirichlet (fixed value) boundary conditions specify the value of the dependent variable at the boundary
• Neumann (fixed gradient) boundary conditions specify the gradient of the dependent variable normal to the boundary
• Implementation involves modifying the discretized equations for control volumes adjacent to the boundary to incorporate the prescribed values or gradients

Inlet/outlet boundaries

• Inflow boundaries require specification of the velocity, pressure, and other scalar quantities entering the domain
• Outflow boundaries should allow the flow to exit the domain with minimal disturbance to the interior solution
• Techniques for handling inlet/outlet boundaries include prescribed profiles, extrapolation, and characteristic-based conditions

Wall treatment

• Solid walls require special treatment to accurately capture the effects of viscosity and heat transfer
• No-slip condition (zero velocity) is imposed at the wall, with appropriate modifications for moving or rotating walls
• Wall functions are often employed to bridge the near-wall region and avoid resolving the viscous sublayer, reducing computational cost

Solution algorithms

• Procedures for solving the system of discretized equations arising from the finite volume method
• Choice of solution algorithm depends on the problem characteristics, desired accuracy, and computational efficiency
• Multiphase flow problems often involve additional challenges, such as coupling between phases and disparate time scales

Pressure-velocity coupling

• In incompressible flows, pressure and velocity are coupled through the continuity equation, requiring special treatment
• Segregated algorithms (SIMPLE, PISO) solve for pressure and velocity separately, using a predictor-corrector approach
• Coupled algorithms solve for pressure and velocity simultaneously, providing better convergence but requiring more memory

SIMPLE vs PISO algorithms

• SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is a widely used segregated algorithm for steady-state problems
• PISO (Pressure Implicit with Splitting of Operators) extends SIMPLE for transient problems, using additional corrector steps
• PISO provides better accuracy and stability for time-dependent flows but requires more computational effort per time step

Multigrid methods

• Accelerate the convergence of iterative solvers by operating on a hierarchy of grids with different resolutions
• Smooth the error on each grid level using relaxation methods (Jacobi, Gauss-Seidel), and transfer corrections between grids
• Multigrid methods are particularly effective for elliptic equations (pressure Poisson equation) and can significantly reduce computational time

Convergence and stability

• Assessing the quality and reliability of the numerical solution obtained from the finite volume method
• Convergence refers to the reduction of the solution error as the mesh is refined or the time step is decreased
• Stability refers to the ability of the numerical scheme to produce bounded solutions without excessive oscillations or divergence

Courant–Friedrichs–Lewy (CFL) condition

• Stability criterion for explicit time-stepping schemes, relating the time step size to the mesh spacing and local velocity
• CFL number $C = \frac{u \Delta t}{\Delta x}$ should be less than a critical value (typically 1) for stability
• Implicit schemes are not subject to the CFL condition, allowing for larger time steps but requiring the solution of a system of equations

Under-relaxation factors

• Technique for improving the stability and convergence of iterative solvers by limiting the change in variables between iterations
• Under-relaxation factors (0 < α < 1) blend the old and new values of variables: $\phi^{new} = \alpha \phi^{calculated} + (1 - \alpha) \phi^{old}$
• Smaller under-relaxation factors improve stability but slow down convergence, requiring a trade-off between robustness and efficiency

Residual monitoring

• Tracking the convergence of the solution by monitoring the residuals (imbalance) of the discretized equations
• Residuals should decrease monotonically with iterations, indicating that the solution is approaching a steady state or satisfying the governing equations
• Convergence criteria are typically based on the normalized residuals falling below a specified tolerance (e.g., 10⁻⁶) or reaching a plateau

Multiphase flow extensions

• Adapting the finite volume method to handle the additional complexities introduced by the presence of multiple phases
• Multiphase flow models capture the interactions between phases, such as interfacial forces, mass, and heat transfer
• Different approaches are employed depending on the nature of the multiphase system (dispersed vs. separated) and the desired level of detail

Volume of fluid (VOF) method

• Interface-capturing approach for immiscible fluids, where a scalar function (volume fraction) represents the distribution of phases
• Volume fraction equation is solved alongside the conservation equations, with special techniques for reconstructing the interface geometry
• VOF method is suitable for flows with sharp interfaces and topological changes (bubbles, droplets, waves) but requires fine meshes to resolve interface details

Eulerian-Eulerian approach

• Treats all phases as interpenetrating continua, with each phase having its own set of conservation equations
• Interaction terms are introduced to account for interfacial forces, mass, and heat transfer between phases
• Eulerian-Eulerian models are suitable for dispersed flows (bubbly, particulate) and can handle high volume fractions but require closure relations for interfacial terms

Lagrangian particle tracking

• Hybrid approach that combines an Eulerian description of the continuous phase with a Lagrangian treatment of the dispersed phase
• Particles or droplets are tracked individually, with their motion governed by Newton's laws and influenced by the continuous phase flow
• Lagrangian particle tracking is suitable for dilute dispersed flows and can provide detailed information on particle trajectories and residence times

Verification and validation

• Processes for assessing the accuracy and reliability of the numerical model and its implementation
• Verification ensures that the model equations are solved correctly, while validation compares the model predictions with real-world data
• Rigorous verification and validation are essential for establishing confidence in the numerical results and guiding model improvements

Grid convergence studies

• Systematic refinement of the computational mesh to assess the spatial discretization error and the order of accuracy
• Solutions are obtained on a series of successively finer grids, and the error is estimated using Richardson extrapolation or grid convergence indices
• Grid convergence studies help determine the appropriate mesh resolution for a desired level of accuracy and quantify the uncertainty due to discretization

Analytical test cases

• Comparison of numerical results with known analytical solutions for simplified problems or limiting cases
• Analytical test cases, such as the lid-driven cavity flow or the Poiseuille flow, provide a rigorous check on the correctness of the numerical implementation
• Successful reproduction of analytical solutions builds confidence in the code's ability to handle more complex problems

Experimental data comparison

• Validation of the numerical model against experimental measurements or benchmarks for realistic multiphase flow scenarios
• Experimental data may include velocity fields (PIV), phase distributions (X-ray, ECT), or global quantities (pressure drop, heat transfer coefficients)
• Comparison with experimental data assesses the model's predictive capability and identifies areas for improvement, such as refined physical models or enhanced numerical methods