💻Applications of Scientific Computing Unit 8 – Intro to Computational Fluid Dynamics

Key Concepts and Foundations

• Fluid dynamics studies the behavior and motion of fluids (liquids and gases) under various conditions
• Involves understanding the fundamental properties of fluids such as density, viscosity, pressure, and temperature
• Explores the interaction between fluids and solid boundaries, including concepts like no-slip condition and boundary layer
• Analyzes the forces acting on fluids, including pressure gradients, viscous forces, and external forces (gravity, electromagnetic fields)
• Investigates different flow regimes, such as laminar flow (smooth, orderly) and turbulent flow (chaotic, irregular)
  • Laminar flow occurs at low Reynolds numbers, while turbulent flow occurs at high Reynolds numbers
• Considers the conservation laws of mass, momentum, and energy, which form the basis for the governing equations of fluid dynamics
• Studies the concept of compressibility, distinguishing between incompressible fluids (constant density) and compressible fluids (density varies with pressure)
• Examines the role of dimensionless numbers, such as Reynolds number and Mach number, in characterizing flow behavior and similarity

Governing Equations of Fluid Dynamics

• Conservation of mass (continuity equation) ensures that mass is neither created nor destroyed within a fluid system
  • Mathematically represented as $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$, where $\rho$ is density and $\mathbf{u}$ is velocity vector
• Conservation of momentum (Navier-Stokes equations) describes the balance of forces acting on a fluid element
  • Includes pressure gradient, viscous forces, and external forces (gravity, electromagnetic fields)
  • For incompressible flow, the equation is $\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g}$, where $p$ is pressure, $\mu$ is dynamic viscosity, and $\mathbf{g}$ is gravitational acceleration
• Conservation of energy (energy equation) accounts for the transfer and conversion of energy within a fluid system
  • Considers heat transfer, work done by pressure forces, and viscous dissipation
  • Temperature is a key variable in the energy equation, influencing fluid properties and flow behavior
• Equation of state relates fluid properties (pressure, density, temperature) and closes the system of governing equations
  • Ideal gas law is commonly used for compressible fluids, while incompressible fluids have a constant density
• Simplifications and assumptions, such as incompressibility, steady-state, or inviscid flow, can be applied to the governing equations depending on the problem at hand

Discretization Methods

• Discretization converts the continuous governing equations into a discrete form suitable for numerical solution
• Finite Difference Method (FDM) approximates derivatives using Taylor series expansions
  • Divides the domain into a structured grid and uses finite differences to approximate spatial and temporal derivatives
  • Offers simplicity and efficiency for regular geometries but may struggle with complex boundaries
• Finite Volume Method (FVM) divides the domain into control volumes and enforces conservation laws locally
  • Integrates the governing equations over each control volume and uses flux balances to update the solution
  • Handles complex geometries well and is widely used in commercial CFD software
• Finite Element Method (FEM) discretizes the domain into elements and uses variational principles to derive the weak form of the equations
  • Approximates the solution using basis functions defined on each element
  • Provides flexibility in handling irregular geometries and is commonly used in structural mechanics and multiphysics problems
• Spectral methods represent the solution using a linear combination of basis functions (Fourier series, Chebyshev polynomials) and solve the equations in the spectral space
  • Offers high accuracy for smooth solutions but may struggle with discontinuities or complex geometries
• Mesh generation is a crucial step in discretization, involving the creation of a computational grid that represents the physical domain
  • Structured meshes have a regular connectivity and are efficient but may struggle with complex geometries
  • Unstructured meshes allow for flexible mesh adaptation and can handle complex geometries but require more computational overhead

Numerical Schemes and Algorithms

• Explicit schemes calculate the solution at the next time step using only information from the current time step
  • Conditionally stable and require small time steps to maintain stability
  • Examples include Forward Euler, Runge-Kutta methods, and explicit finite difference schemes
• Implicit schemes involve solving a system of equations that includes both current and future time step information
  • Unconditionally stable and allow for larger time steps but require more computational effort per time step
  • Examples include Backward Euler, Crank-Nicolson, and implicit finite difference schemes
• Pressure-velocity coupling algorithms ensure the satisfaction of the continuity equation and the correct pressure field
  • SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) is a widely used algorithm that iteratively solves for pressure and velocity corrections
  • Other variants include SIMPLER, SIMPLEC, and PISO (Pressure-Implicit with Splitting of Operators)
• Advection schemes handle the discretization of the convective terms in the governing equations
  • Upwind schemes (first-order, second-order) consider the direction of flow and use information from upstream nodes
  • Central differencing schemes use a symmetric stencil and can provide higher accuracy but may introduce oscillations
  • High-resolution schemes, such as QUICK (Quadratic Upstream Interpolation for Convective Kinematics) and TVD (Total Variation Diminishing) schemes, aim to balance accuracy and stability
• Temporal discretization schemes approximate the time derivatives in the governing equations
  • Explicit schemes (Forward Euler, Runge-Kutta) are straightforward to implement but have stability limitations
  • Implicit schemes (Backward Euler, Crank-Nicolson) are more stable but require the solution of a system of equations at each time step

Boundary Conditions and Initial Conditions

• Boundary conditions specify the fluid behavior at the boundaries of the computational domain
• Inlet boundary conditions prescribe the flow properties (velocity, pressure, temperature) at the entrance of the domain
  • Velocity inlet specifies the velocity profile, which can be uniform, parabolic, or user-defined
  • Pressure inlet specifies the total pressure and other scalar properties at the inlet
• Outlet boundary conditions define the flow behavior at the exit of the domain
  • Pressure outlet sets a static pressure at the outlet and extrapolates other flow quantities from the interior
  • Outflow boundary condition assumes a fully developed flow at the outlet, with zero normal gradients for all variables except pressure
• Wall boundary conditions represent the interaction between the fluid and solid surfaces
  • No-slip condition assumes that the fluid velocity matches the wall velocity (zero for stationary walls)
  • Wall functions are used to model the near-wall flow behavior and reduce the computational cost of resolving the boundary layer
• Symmetry boundary conditions exploit the geometric symmetry of the problem to reduce the computational domain size
  • Normal gradients of all variables are set to zero at the symmetry plane
• Periodic boundary conditions are used for flows with repeating patterns, such as turbomachinery or heat exchangers
  • Flow properties at the periodic boundaries are matched, allowing for a reduced computational domain
• Initial conditions specify the flow field at the start of the simulation
  • Should be physically realistic and consistent with the boundary conditions
  • Can be uniform, analytically prescribed, or interpolated from previous simulations or experimental data

Stability and Convergence

• Stability refers to the ability of a numerical scheme to produce bounded solutions without excessive growth of errors
• Courant-Friedrichs-Lewy (CFL) condition relates the time step size to the spatial discretization and the local velocity
  • For explicit schemes, the CFL number should be less than a critical value (usually 1) to ensure stability
  • Implicit schemes are generally more stable and allow for larger CFL numbers
• Von Neumann stability analysis is a technique to assess the stability of linear numerical schemes
  • Analyzes the growth or decay of Fourier modes in the numerical solution
  • Provides insights into the stability limits and the required time step size for explicit schemes
• Convergence refers to the property of a numerical solution approaching the exact solution as the mesh is refined or the time step is reduced
• Mesh convergence studies involve systematically refining the mesh and comparing the solutions to assess the spatial convergence
  • Richardson extrapolation can be used to estimate the grid-independent solution and the order of convergence
• Temporal convergence is evaluated by reducing the time step size and comparing the solutions at different time levels
• Residual monitoring tracks the magnitude of the residuals (errors) in the governing equations during the iterative solution process
  • Convergence is typically assumed when the residuals fall below a specified tolerance (e.g., 10^-6)
• Conservation errors quantify the deviation from the conservation laws (mass, momentum, energy) and should be minimized for a physically consistent solution

CFD Software and Tools

• Commercial CFD packages offer comprehensive tools for pre-processing, solving, and post-processing
  • Examples include ANSYS Fluent, ANSYS CFX, STAR-CCM+, and Siemens Simcenter STAR-CCM+
  • Provide user-friendly interfaces, extensive physical models, and advanced numerical algorithms
  • Often include built-in mesh generation, visualization, and data analysis capabilities
• Open-source CFD software provides freely available tools for CFD simulations
  • OpenFOAM is a popular open-source CFD toolbox based on C++ libraries
  • SU2 is an open-source suite for multiphysics simulation and optimization
  • These tools offer flexibility and customization options but may require more user expertise
• Mesh generation software is used to create high-quality computational grids
  • Examples include ANSYS Meshing, Pointwise, and Gmsh
  • Provide tools for geometry handling, mesh generation, and mesh quality assessment
• Visualization and post-processing tools help analyze and interpret CFD results
  • ParaView is an open-source, multi-platform data analysis and visualization application
  • Tecplot is a commercial software for visualizing and analyzing CFD and other engineering data
  • These tools offer features like contour plots, vector fields, streamlines, and animations
• High-performance computing (HPC) resources are often required for large-scale CFD simulations
  • Parallel computing techniques, such as domain decomposition and message passing (MPI), are used to distribute the workload across multiple processors
  • GPU acceleration can significantly speed up certain CFD operations, such as linear algebra and explicit schemes

Real-World Applications and Case Studies

• Aerodynamics and aerospace engineering
  • Aircraft design and optimization (wing shape, fuselage, control surfaces)
  • Turbomachinery (jet engines, turbines, compressors) performance analysis
  • Hypersonic flows and reentry vehicles
• Automotive and ground transportation
  • Vehicle aerodynamics (drag reduction, lift forces, stability)
  • Internal combustion engines (fuel injection, mixing, combustion)
  • Tire and road interaction, including aquaplaning and hydroplaning analysis
• Environmental and atmospheric flows
  • Weather prediction and climate modeling
  • Pollutant dispersion and air quality assessment
  • Wind engineering and pedestrian comfort around buildings
• Biomedical and biological flows
  • Cardiovascular system (blood flow in arteries, heart valves)
  • Respiratory system (airflow in lungs, drug delivery)
  • Microfluidics and lab-on-a-chip devices
• Chemical and process engineering
  • Mixing and separation processes in reactors and tanks
  • Multiphase flows (bubbles, droplets, particles) in pipelines and equipment
  • Heat and mass transfer in heat exchangers and packed beds
• Renewable energy and sustainability
  • Wind turbine design and optimization
  • Tidal and wave energy converters
  • Solar thermal collectors and concentrators
• Case studies showcase the application of CFD to real-world problems
  • Airflow around a Formula 1 car to optimize aerodynamic performance
  • Blood flow in a stented artery to assess the risk of thrombosis
  • Pollutant dispersion from a power plant to evaluate environmental impact
  • Optimization of a heat exchanger design to improve energy efficiency