💨Fluid Dynamics Unit 8 – Computational fluid dynamics

Key Concepts and Fundamentals

• Computational Fluid Dynamics (CFD) involves using numerical methods to solve fluid flow problems
• CFD simulations provide detailed insights into fluid behavior, including velocity, pressure, and temperature distributions
• Navier-Stokes equations form the mathematical foundation of CFD, describing the conservation of mass, momentum, and energy in fluid flows
• Fluid properties such as density, viscosity, and thermal conductivity play crucial roles in CFD simulations
• CFD simulations require appropriate boundary conditions (inlet, outlet, walls) and initial conditions to define the problem domain
• Mesh generation involves discretizing the computational domain into smaller elements (cells) for numerical analysis
• Convergence criteria determine when a CFD simulation has reached a satisfactory solution, based on residuals or other metrics
• Verification and validation processes ensure the accuracy and reliability of CFD results by comparing with analytical solutions or experimental data

Governing Equations

• Conservation of mass (continuity equation) ensures that the net mass flow entering a control volume equals the net mass flow leaving it
  • For incompressible flows: $\nabla \cdot \vec{u} = 0$
  • For compressible flows: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$
• Conservation of momentum (Navier-Stokes equations) describes the balance of forces acting on a fluid element
  • Incompressible: $\rho \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g}$
  • Compressible: Additional terms for viscous stresses and heat transfer
• Conservation of energy (energy equation) accounts for the transfer and conversion of energy within the fluid
  • Includes terms for heat conduction, viscous dissipation, and external heat sources
• Equation of state relates fluid properties (density, pressure, temperature) and closes the system of equations
• Turbulence models (RANS, LES, DNS) introduce additional equations to capture the effects of turbulent fluctuations on the mean flow

Discretization Methods

• Finite Difference Method (FDM) approximates derivatives using Taylor series expansions and a structured grid
  • Suitable for simple geometries and straightforward implementation
• Finite Volume Method (FVM) divides the domain into control volumes and enforces conservation laws on each volume
  • Handles complex geometries and is widely used in commercial CFD software
  • Requires careful treatment of fluxes at cell faces
• Finite Element Method (FEM) uses a variational formulation and shape functions to approximate the solution
  • Provides high-order accuracy and flexibility in handling complex geometries
  • Computationally expensive compared to FDM and FVM
• Spectral methods represent the solution using a linear combination of basis functions (Fourier, Chebyshev)
  • Offers high accuracy for smooth solutions but limited to simple geometries
• Mesh types include structured (regular connectivity), unstructured (irregular connectivity), and hybrid (combination of structured and unstructured)

Numerical Schemes

• Explicit schemes calculate the solution at the next time step using only information from the current time step
  • Conditionally stable, requiring small time steps for stability
  • Examples: Forward Euler, Runge-Kutta methods
• Implicit schemes involve solving a system of equations that includes both current and future time step values
  • Unconditionally stable, allowing larger time steps
  • Examples: Backward Euler, Crank-Nicolson
• Upwind schemes consider the direction of information propagation when approximating convective terms
  • First-order upwind is stable but prone to numerical diffusion
  • Higher-order schemes (QUICK, MUSCL) reduce numerical diffusion but may introduce oscillations
• Pressure-velocity coupling algorithms (SIMPLE, PISO, COUPLED) ensure the satisfaction of continuity and momentum equations
  • Iterative process to update pressure and velocity fields until convergence
• Multigrid methods accelerate the convergence of iterative solvers by solving the problem on multiple grid levels
  • Smoothing of high-frequency errors on fine grids and correction of low-frequency errors on coarse grids

Boundary Conditions

• Inlet boundary conditions specify the flow properties entering the domain
  • Velocity inlet: Prescribed velocity profile (uniform, parabolic, user-defined)
  • Pressure inlet: Total pressure and flow direction specified
• Outlet boundary conditions define the flow behavior leaving the domain
  • Pressure outlet: Static pressure prescribed, velocity extrapolated from interior
  • Outflow: Zero-gradient condition for all variables except pressure
• Wall boundary conditions represent the interaction between the fluid and solid surfaces
  • No-slip condition: Zero velocity relative to the wall
  • Free-slip condition: Zero normal velocity, zero shear stress
  • Moving wall: Prescribed wall velocity
• Symmetry boundary conditions reduce computational cost by exploiting flow symmetry
  • Zero normal velocity and zero gradients for all variables
• Periodic boundary conditions connect two or more boundaries, enforcing identical flow conditions
  • Useful for simulating fully developed flows or repeated patterns

Turbulence Modeling

• Reynolds-Averaged Navier-Stokes (RANS) models decompose the flow into mean and fluctuating components
  • Turbulence effects represented by additional terms (Reynolds stresses) in the governing equations
  • Eddy viscosity models (Spalart-Allmaras, $k-\epsilon$, $k-\omega$) relate Reynolds stresses to mean flow gradients
  • Reynolds Stress Models (RSM) solve transport equations for each component of the Reynolds stress tensor
• Large Eddy Simulation (LES) directly resolves large-scale turbulent structures and models small-scale structures
  • Filtering operation separates resolved and subgrid scales
  • Subgrid-scale models (Smagorinsky, dynamic) represent the effects of unresolved scales on the resolved flow
• Detached Eddy Simulation (DES) combines RANS near walls and LES in the free stream, reducing computational cost compared to pure LES
• Direct Numerical Simulation (DNS) resolves all scales of turbulence without any modeling
  • Extremely computationally expensive, limited to low Reynolds numbers and simple geometries
• Turbulence model selection depends on the flow complexity, required accuracy, and available computational resources

Simulation Setup and Preprocessing

• Geometry creation involves defining the computational domain and any relevant geometric features
  • CAD software or built-in tools in CFD packages
  • Simplifications and assumptions (2D vs. 3D, symmetry) to reduce complexity
• Mesh generation discretizes the domain into smaller elements
  • Mesh quality factors: Skewness, aspect ratio, orthogonality
  • Refinement in regions of high gradients or complex flow features
  • Boundary layer meshing for accurate near-wall treatment
• Material properties and fluid models are specified based on the problem requirements
  • Density, viscosity, thermal conductivity, specific heat
  • Incompressible vs. compressible, Newtonian vs. non-Newtonian
• Initial and boundary conditions are set to define the starting point and external influences on the simulation
  • Initial velocity, pressure, and temperature fields
  • Inlet, outlet, wall, and other relevant boundary conditions
• Solver settings include the choice of numerical schemes, convergence criteria, and solution controls
  • Spatial and temporal discretization schemes
  • Under-relaxation factors for stability
  • Monitoring of residuals, forces, or other quantities of interest

Post-processing and Result Analysis

• Visualization of flow fields helps in understanding the fluid behavior and identifying key features
  • Contour plots, vector plots, streamlines, isosurfaces
  • Colormaps and legends for quantitative information
• Extraction of quantitative data allows for detailed analysis and comparison with experimental or analytical results
  • Line plots, surface integrals, volume averages
  • Drag, lift, pressure drop, heat transfer coefficients
• Verification involves checking the numerical accuracy and consistency of the solution
  • Grid convergence studies, time step sensitivity analysis
  • Comparison with analytical solutions or benchmark cases
• Validation assesses the agreement between CFD results and experimental data
  • Quantitative comparison of flow quantities (velocity, pressure, temperature)
  • Qualitative comparison of flow patterns and trends
• Sensitivity analysis investigates the impact of input parameters on the simulation results
  • Boundary conditions, material properties, turbulence models
  • Design optimization and uncertainty quantification

Applications and Case Studies

• Aerodynamics: Analysis of flow around vehicles, aircraft, and buildings
  • Drag reduction, lift enhancement, wake characterization
  • Examples: Formula 1 car design, wind turbine optimization
• Turbomachinery: Design and performance evaluation of pumps, compressors, and turbines
  • Flow through rotating and stationary components
  • Examples: Jet engine compressor, hydroelectric turbine
• Heat transfer and thermal management: Investigation of cooling systems and heat exchangers
  • Conjugate heat transfer, natural and forced convection
  • Examples: Electronics cooling, HVAC systems
• Environmental flows: Simulation of atmospheric and oceanic phenomena
  • Weather prediction, pollutant dispersion, sediment transport
  • Examples: Urban air quality modeling, coastal erosion studies
• Biomedical applications: Analysis of blood flow and medical device design
  • Cardiovascular systems, respiratory flows, drug delivery
  • Examples: Aneurysm hemodynamics, inhaler design optimization
• Chemical and process engineering: Modeling of reactors, mixers, and separators
  • Multiphase flows, chemical reactions, mass transfer
  • Examples: Fluidized bed reactor, distillation column
• Renewable energy: Investigation of wind and tidal energy systems
  • Flow through wind and tidal turbines, wake interactions
  • Examples: Offshore wind farm layout, tidal turbine array optimization