✈️Aerodynamics Unit 8 – Computational fluid dynamics (CFD)

Key Concepts and Fundamentals

• CFD involves the numerical solution of fluid flow equations using computers to simulate and analyze fluid dynamics
• Fundamental principles of fluid mechanics, such as conservation of mass, momentum, and energy, form the basis of CFD
• Navier-Stokes equations describe the motion of viscous, incompressible fluids and are commonly used in CFD simulations
• Computational domain represents the physical space where the fluid flow is simulated and is discretized into smaller elements (mesh or grid)
• Boundary conditions specify the fluid behavior at the edges of the computational domain (inflow, outflow, walls)
• Numerical methods, such as finite difference, finite volume, and finite element methods, are employed to discretize and solve the governing equations
• Convergence refers to the reduction of numerical errors as the solution progresses towards a stable and accurate result
• Turbulence modeling is essential for capturing complex flow phenomena, such as eddies and vortices, in high Reynolds number flows

Governing Equations

• Conservation of mass equation (continuity equation) ensures that the mass of fluid entering a control volume equals the mass leaving it
• Conservation of momentum equation (Navier-Stokes equations) describes the balance of forces acting on a fluid element, including pressure, viscous, and body forces
• Conservation of energy equation accounts for the energy transfer and exchange within the fluid, including heat transfer and work done by the fluid
• Equation of state relates fluid properties, such as pressure, density, and temperature, and closes the system of equations
• Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous, incompressible fluids
  • Incompressible flow assumes constant fluid density, simplifying the equations
  • Viscous forces are represented by the viscous stress tensor in the equations
• Euler equations are a simplified form of the Navier-Stokes equations, neglecting viscous effects and suitable for inviscid flows
• Additional equations, such as the transport equation for turbulence quantities (k-ε, k-ω models), may be solved alongside the governing equations

Discretization Methods

• Discretization involves converting the continuous governing equations into a discrete form suitable for numerical solution
• Finite difference method (FDM) approximates derivatives using Taylor series expansions and is straightforward to implement
  • FDM is commonly used for structured grids and simple geometries
• Finite volume method (FVM) divides the computational domain into control volumes and enforces conservation laws on each volume
  • FVM is versatile and can handle unstructured grids and complex geometries
  • Flux balance is maintained across control volume faces, ensuring conservation
• Finite element method (FEM) discretizes the domain into elements and approximates the solution using basis functions
  • FEM is well-suited for structural analysis and can handle irregular geometries
  • Weighted residual formulation is used to minimize the error in the approximation
• Spectral methods represent the solution as a sum of basis functions (Fourier series, Chebyshev polynomials) and are accurate for smooth solutions
• Temporal discretization schemes, such as explicit (Euler, Runge-Kutta) and implicit (backward differentiation) methods, are used for time-dependent problems

Boundary Conditions and Initial Conditions

• Boundary conditions specify the fluid behavior at the boundaries of the computational domain and are crucial for obtaining a unique solution
• Inflow boundary conditions prescribe the velocity, pressure, or other properties at the inlet of the domain
  • Velocity inlet specifies the velocity profile (uniform, parabolic) at the inlet
  • Pressure inlet specifies the total pressure and other scalar properties at the inlet
• Outflow boundary conditions define the fluid behavior at the outlet of the domain
  • Pressure outlet sets a static pressure at the outlet and extrapolates other properties from the interior
  • Outflow boundary condition imposes zero gradient for all properties at the outlet
• Wall boundary conditions represent the interaction between the fluid and solid surfaces
  • No-slip condition enforces zero velocity relative to the wall (viscous flows)
  • Free-slip condition allows tangential velocity but sets normal velocity to zero (inviscid flows)
• Symmetry boundary conditions reduce computational cost by modeling symmetric flow patterns
• Periodic boundary conditions connect the flow properties between corresponding faces of the domain
• Initial conditions specify the flow field at the start of the simulation and are required for time-dependent problems
  • Uniform initial conditions assign constant values to all flow properties
  • Spatially varying initial conditions can be used to trigger specific flow phenomena (vortex shedding)

Numerical Solvers and Algorithms

• Numerical solvers compute the solution to the discretized governing equations by iteratively updating the flow variables
• Pressure-based solvers solve for pressure and velocity sequentially, using pressure correction methods (SIMPLE, PISO)
  • Staggered grid arrangement is often used to avoid checkerboard pressure oscillations
• Density-based solvers solve for all flow variables simultaneously and are suitable for compressible flows
  • Explicit density-based solvers (Lax-Wendroff, MacCormack) are limited by the CFL condition for stability
  • Implicit density-based solvers (Beam-Warming, Roe) allow larger time steps but require the solution of a large linear system
• Multigrid methods accelerate convergence by solving the equations on a hierarchy of grids, capturing both high and low-frequency errors
• Preconditioning techniques (Jacobi, Gauss-Seidel, ILU) improve the conditioning of the linear system and speed up convergence
• Krylov subspace methods (CG, GMRES) are efficient iterative solvers for large, sparse linear systems arising from implicit methods
• Adaptive time-stepping adjusts the time step size based on the local stability and accuracy requirements, improving efficiency

Turbulence Modeling

• Turbulence is characterized by chaotic, unsteady motion with a wide range of spatial and temporal scales
• Direct Numerical Simulation (DNS) resolves all turbulent scales but is computationally expensive and limited to low Reynolds numbers
• Reynolds-Averaged Navier-Stokes (RANS) equations model the effect of turbulence on the mean flow by introducing Reynolds stresses
  • Boussinesq approximation relates Reynolds stresses to mean velocity gradients through an eddy viscosity
• Spalart-Allmaras model is a one-equation model that solves for a turbulent viscosity-like variable and is suitable for attached boundary layers
• k-ε models (standard, RNG, realizable) are two-equation models that solve for turbulent kinetic energy (k) and dissipation rate (ε)
  • k-ε models are robust and widely used but have limitations in predicting flow separation and adverse pressure gradients
• k-ω models (standard, SST) are two-equation models that solve for turbulent kinetic energy (k) and specific dissipation rate (ω)
  • k-ω models are more accurate for wall-bounded flows and can handle flow separation better than k-ε models
• Reynolds Stress Models (RSM) directly solve transport equations for the Reynolds stresses, capturing anisotropic turbulence effects
• Large Eddy Simulation (LES) resolves large-scale turbulent motions and models the effect of small-scale motions using a subgrid-scale model
  • LES is more accurate than RANS but computationally more expensive
• Detached Eddy Simulation (DES) combines RANS near the walls and LES in the free-shear regions, balancing accuracy and computational cost

Mesh Generation and Refinement

• Mesh generation involves discretizing the computational domain into smaller elements (cells) to form a grid or mesh
• Structured meshes have a regular connectivity pattern and are suitable for simple geometries
  • Cartesian meshes are composed of rectangular elements and are easy to generate
  • Curvilinear meshes follow the contours of the geometry and can handle mildly complex shapes
• Unstructured meshes have an irregular connectivity pattern and are suitable for complex geometries
  • Triangular (2D) and tetrahedral (3D) elements are commonly used in unstructured meshes
  • Prismatic and hexahedral elements can be used in boundary layer regions for better resolution
• Hybrid meshes combine structured and unstructured elements to leverage the advantages of both approaches
• Mesh quality measures, such as skewness, aspect ratio, and orthogonality, assess the suitability of the mesh for CFD simulations
• Adaptive mesh refinement (AMR) automatically refines the mesh in regions with high gradients or errors, improving accuracy and efficiency
  • h-refinement subdivides elements into smaller ones to capture local flow features
  • p-refinement increases the polynomial order of the approximation within elements
• Overset (Chimera) meshes use overlapping grids to handle moving or deforming geometries, such as in fluid-structure interaction problems

Validation and Verification

• Validation assesses the accuracy of the CFD model by comparing simulation results with experimental data or analytical solutions
  • Validation ensures that the model captures the essential physics of the problem
  • Validation experiments should be carefully designed to isolate specific flow phenomena
• Verification assesses the numerical accuracy and convergence of the CFD solution
  • Grid convergence study refines the mesh systematically to ensure that the solution is grid-independent
  • Temporal convergence study reduces the time step size to ensure that the solution is time-step-independent
  • Code verification tests the implementation of the numerical methods against known solutions (method of manufactured solutions)
• Uncertainty quantification (UQ) assesses the impact of input uncertainties (geometry, boundary conditions, material properties) on the simulation results
  • Sensitivity analysis identifies the most influential input parameters on the output quantities of interest
  • Stochastic methods (Monte Carlo, polynomial chaos) propagate input uncertainties through the CFD model to obtain output probability distributions
• Best practices for validation and verification include using established benchmark cases, documenting the simulation setup, and reporting the numerical settings and convergence criteria

Applications in Aerodynamics

• External aerodynamics involves the study of flow around vehicles, such as aircraft, cars, and rockets
  • Lift and drag prediction is crucial for aircraft design and performance analysis
  • Flow separation and wake dynamics affect the stability and control of the vehicle
• Internal aerodynamics deals with the flow through propulsion systems, such as jet engines and turbomachinery
  • Compressor and turbine blade design requires accurate prediction of flow angles and losses
  • Combustion modeling involves the simulation of reacting flows and pollutant formation
• Hypersonic aerodynamics studies the flow around vehicles at high Mach numbers (typically >5)
  • Chemical reactions, such as dissociation and ionization, occur in the flow and affect the aerodynamic properties
  • Radiation heat transfer becomes significant and must be accounted for in the energy equation
• Aeroelasticity is the study of the interaction between aerodynamic forces and structural deformation
  • Flutter analysis predicts the onset of self-excited oscillations that can lead to structural failure
  • Fluid-structure interaction (FSI) simulations couple the CFD and structural dynamics solvers to capture the coupled response
• Aeroacoustics deals with the generation and propagation of noise from aerodynamic sources
  • Jet noise prediction requires accurate simulation of turbulent mixing and shear layer instabilities
  • Airframe noise sources, such as landing gear and high-lift devices, can be modeled using CFD to guide noise reduction strategies
• Multidisciplinary design optimization (MDO) integrates CFD with other disciplines, such as structures and propulsion, to optimize the overall system performance
  • Adjoint methods efficiently compute the sensitivity of objective functions (drag, lift) to design variables (shape, angle of attack)
  • Surrogate models (response surfaces, Kriging) can be used to reduce the computational cost of optimization by approximating the CFD response