13.3 Computational fluid dynamics (CFD) basics

Computational fluid dynamics (CFD) is a powerful tool for solving complex fluid flow problems. It uses numerical methods to analyze and predict fluid behavior, heat transfer, and mass transport in various systems.

CFD simulations involve creating a mesh, applying governing equations, and solving them numerically. This process allows engineers to study intricate flow phenomena, optimize designs, and make informed decisions without costly physical experiments.

Computational Fluid Dynamics Principles

Fundamentals and Applications

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• Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems involving fluid flows, heat transfer, and mass transfer
• CFD is based on the fundamental governing equations of fluid dynamics, including the continuity equation, momentum equation, and energy equation, which are derived from the conservation laws of physics
• The primary applications of CFD include aerodynamics (aircraft design), hydrodynamics (ship hull optimization), heat exchanger design, combustion modeling, and environmental engineering (pollutant dispersion), among others
• CFD enables the study of complex fluid flow phenomena, such as turbulence, boundary layer effects, and multiphase flows (gas-liquid, solid-liquid), which are difficult to analyze using experimental or analytical methods

Workflow and Components

• The basic workflow of a CFD analysis involves pre-processing (geometry and mesh generation), solving (numerical computation), and post-processing (visualization and analysis of results)
• Pre-processing involves creating a computational domain, generating a mesh of control volumes or elements, and specifying boundary conditions and initial conditions
• The solving stage involves the numerical solution of the governing equations using appropriate numerical methods, such as finite volume, finite difference, or finite element methods
• Post-processing involves visualizing and analyzing the computed flow fields, such as velocity, pressure, temperature, and species concentrations, using various graphical and quantitative techniques

Governing Equations and Methods in CFD

Navier-Stokes Equations

• The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances, taking into account the conservation of mass, momentum, and energy
• The continuity equation ensures the conservation of mass, stating that the rate of change of fluid density in a control volume is equal to the net rate of mass flow into the control volume: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v}) = 0$
• The momentum equation is derived from Newton's second law and relates the forces acting on a fluid element to its acceleration, considering pressure gradients, viscous stresses, and body forces: $\rho \frac{D \vec{v}}{D t} = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$
• The energy equation accounts for the conservation of energy, incorporating the effects of heat conduction, convection, and dissipation due to viscous stresses: $\rho c_p \frac{D T}{D t} = \nabla \cdot (k \nabla T) + \Phi$

Numerical Methods

• The finite volume method (FVM) is a common numerical technique used in CFD, which divides the computational domain into a set of control volumes and solves the governing equations by conserving fluxes across the boundaries of these volumes
• The finite difference method (FDM) approximates the derivatives in the governing equations using Taylor series expansions and solves the resulting algebraic equations at discrete grid points
• The finite element method (FEM) divides the computational domain into a set of elements, approximates the solution variables using basis functions, and solves the resulting system of equations by minimizing a residual or error function
• Spectral methods represent the solution variables using a linear combination of basis functions (Fourier series, Chebyshev polynomials) and solve the governing equations in the spectral space, offering high accuracy for smooth solutions but limited applicability to complex geometries

CFD Simulation Setup and Solution

Domain and Mesh Generation

• Setting up a CFD simulation involves defining the computational domain, generating a suitable mesh, specifying boundary conditions, and selecting appropriate physical models and numerical schemes
• The computational domain represents the physical space in which the fluid flow, heat transfer, or mass transfer problem is solved, and it is discretized into a mesh of control volumes or elements
• Mesh generation involves creating a grid of nodes and elements that conforms to the geometry of the computational domain and provides sufficient resolution to capture the relevant flow features and gradients
• Structured meshes (hexahedral elements) offer higher accuracy and efficiency but are limited to simple geometries, while unstructured meshes (tetrahedral elements) provide more flexibility for complex geometries but may require more computational resources

Boundary Conditions and Physical Models

• Boundary conditions specify the flow variables (velocity, pressure, temperature) at the boundaries of the computational domain, such as inlets, outlets, walls, and symmetry planes
• Inlet boundary conditions prescribe the velocity, pressure, or mass flow rate entering the domain, while outlet boundary conditions specify the pressure or outflow conditions leaving the domain
• Wall boundary conditions impose no-slip (zero velocity) or slip conditions, as well as thermal (temperature, heat flux) or mass transfer (concentration, flux) conditions at solid surfaces
• Physical models are used to represent the underlying physics of the problem, such as turbulence (RANS, LES, DNS), heat transfer (conduction, convection, radiation), chemical reactions (combustion, multiphase flows), and are selected based on the complexity and accuracy requirements of the simulation

Numerical Schemes and Solvers

• Numerical schemes, such as upwind, central, or high-resolution schemes, are chosen to discretize the governing equations and ensure numerical stability, accuracy, and convergence
• Upwind schemes (first-order, second-order) are stable but may introduce numerical diffusion, while central schemes (second-order) are more accurate but may suffer from oscillations or instabilities
• High-resolution schemes (QUICK, MUSCL) combine the advantages of upwind and central schemes by using flux limiters or slope limiters to preserve monotonicity and reduce numerical diffusion
• Solvers for the resulting system of algebraic equations can be classified as direct (Gaussian elimination, LU decomposition) or iterative (Gauss-Seidel, Jacobi, Krylov subspace methods), with the choice depending on the problem size, sparsity, and computational resources available

Interpretation and Evaluation of CFD Results

Visualization and Analysis

• Post-processing of CFD results involves visualizing and analyzing the computed flow fields, such as velocity, pressure, temperature, and species concentrations, using various graphical and quantitative techniques
• Velocity vector plots, streamlines, and contours can be used to visualize the flow patterns, recirculation zones, and boundary layer effects in the computational domain
• Pressure and temperature distributions can be analyzed to identify high and low-pressure regions, thermal gradients, and heat transfer characteristics in the flow field
• Quantitative analysis of CFD results may involve extracting data along lines, planes, or volumes of interest, computing integral quantities (drag, lift, heat flux), and comparing the results with experimental data or analytical solutions

Verification and Validation

• Critically evaluating CFD results requires an understanding of the assumptions and limitations of the physical models, numerical methods, and boundary conditions used in the simulation
• Verification is the process of assessing the accuracy of the numerical solution by comparing it with analytical solutions, manufactured solutions, or highly refined simulations
• Mesh independence studies should be conducted to ensure that the solution is not sensitive to the mesh resolution and that the discretization errors are within acceptable limits
• Validation of CFD results against experimental data or benchmarks is essential to assess the accuracy and reliability of the simulations and to identify potential sources of error or uncertainty
• Sensitivity analyses can be performed to investigate the influence of input parameters, such as material properties, boundary conditions, and model constants, on the simulation results and to quantify the associated uncertainties