💨Mathematical Fluid Dynamics Unit 9 – Computational Fluid Dynamics (CFD)

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
• Fundamental principles of fluid mechanics, such as conservation of mass, momentum, and energy, form the basis of CFD
• CFD enables the analysis of complex geometries and flow conditions that are difficult to study experimentally
• Meshes discretize the computational domain into smaller elements (cells) for numerical analysis
  • Structured meshes have regular connectivity and are suitable for simple geometries
  • Unstructured meshes offer flexibility for complex geometries but require more computational resources
• Accuracy of CFD results depends on factors such as mesh resolution, numerical schemes, and turbulence modeling
• Verification and validation processes ensure the reliability of CFD simulations by comparing them with analytical solutions and experimental data

Governing Equations

• Navier-Stokes equations describe the motion of viscous fluids and form the foundation of CFD
  • Continuity equation ensures mass conservation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0$
  • Momentum equation represents the balance of forces acting on a fluid element: $\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \rho \mathbf{g}$
  • Energy equation accounts for heat transfer and temperature distribution in the fluid: $\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u} \cdot \nabla T\right) = k \nabla^2 T + \Phi$
• Simplifications of the Navier-Stokes equations lead to specialized forms for different flow regimes
  • Euler equations neglect viscous effects and are suitable for inviscid flows
  • Potential flow equations assume irrotational flow and simplify the governing equations
• Additional equations may be required for multiphase flows, chemical reactions, or turbulence modeling
• Boundary conditions and initial conditions complement the governing equations to define the specific problem

Discretization Methods

• Discretization converts the continuous governing equations into a system of algebraic equations
• Finite Difference Method (FDM) approximates derivatives using Taylor series expansions
  • Suitable for structured grids and simple geometries
  • Requires special treatment for irregular boundaries
• Finite Volume Method (FVM) divides the domain into control volumes and enforces conservation laws
  • Flexible for both structured and unstructured grids
  • Widely used in commercial CFD software due to its robustness and conservation properties
• Finite Element Method (FEM) approximates the solution using a weighted residual approach
  • Handles complex geometries and unstructured grids effectively
  • Commonly used in structural analysis and can be extended to fluid flow problems
• Spectral methods represent the solution using a series of basis functions (Fourier or Chebyshev)
  • Offer high accuracy for smooth solutions but may struggle with discontinuities
  • Suitable for problems with periodic boundary conditions

Numerical Schemes

• Numerical schemes discretize the governing equations and determine the accuracy and stability of the solution
• 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 (CFL condition)
  • Examples include Forward Euler, Lax-Wendroff, and MacCormack schemes
• Implicit schemes involve solving a system of equations that includes both current and future time steps
  • Unconditionally stable, allowing larger time steps
  • Examples include Backward Euler, Crank-Nicolson, and Implicit Runge-Kutta schemes
• Higher-order schemes (2nd order or above) provide better accuracy but may introduce oscillations near discontinuities
  • Upwind schemes (1st order) are more stable but introduce numerical diffusion
  • Flux limiters and slope limiters can be used to combine the benefits of higher-order and upwind schemes
• Time integration schemes advance the solution in time
  • Explicit schemes (Runge-Kutta) are simple to implement but have stability limitations
  • Implicit schemes (Backward Differentiation Formula) are more stable but require solving a system of equations at each time step

Boundary Conditions

• Boundary conditions specify the fluid behavior at the boundaries of the computational domain
• Inlet boundary conditions prescribe the flow properties (velocity, pressure, temperature) entering the domain
  • Velocity inlet specifies the velocity profile and direction
  • Pressure inlet defines the total pressure and flow direction
• Outlet boundary conditions describe the flow properties leaving the domain
  • Pressure outlet sets a static pressure at the outlet
  • Outflow boundary condition assumes fully developed flow at the outlet
• Wall boundary conditions represent the interaction between the fluid and solid surfaces
  • No-slip condition assumes zero velocity relative to the wall
  • Free-slip condition allows tangential velocity but no normal velocity
  • Wall functions model the near-wall flow behavior to avoid resolving the viscous sublayer
• Symmetry boundary conditions reduce computational cost by exploiting the geometric symmetry of the problem
• Periodic boundary conditions connect the flow properties between corresponding faces of the domain

Turbulence Modeling

• Turbulence is characterized by chaotic and unsteady fluid 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 effects of turbulence on the mean flow
  • Turbulence models (k-ε, k-ω, Spalart-Allmaras) provide closure for the RANS equations by relating turbulent stresses to mean flow properties
  • Eddy viscosity concept assumes that turbulent stresses are proportional to the mean strain rate
• Large Eddy Simulation (LES) resolves large-scale turbulent structures and models the smaller scales
  • Subgrid-scale models (Smagorinsky, dynamic models) represent the effects of unresolved scales on the resolved flow
• Detached Eddy Simulation (DES) combines RANS near the walls and LES in the free stream to balance accuracy and computational cost
• Turbulence modeling introduces additional equations and empirical constants that need to be carefully selected based on the flow problem

Stability and Convergence

• Stability refers to the ability of a numerical scheme to produce bounded solutions without excessive growth of errors
• CFL (Courant-Friedrichs-Lewy) condition relates the time step size to the spatial discretization and local velocity for explicit schemes
  • CFL number should be less than 1 for stability: $CFL = \frac{u \Delta t}{\Delta x} \leq 1$
• Von Neumann stability analysis determines the stability of a numerical scheme by examining the growth of Fourier modes
• Implicit schemes are unconditionally stable but may suffer from slow convergence or numerical diffusion
• Convergence refers to the property of a numerical solution approaching the exact solution as the mesh is refined or the time step is reduced
• Residuals measure the imbalance in the discretized equations and provide a criterion for assessing convergence
  • Iterative methods (Jacobi, Gauss-Seidel, Multigrid) are used to solve the system of equations until the residuals fall below a specified tolerance
• Grid independence study verifies that the solution is not sensitive to further mesh refinement
• Temporal convergence study ensures that the solution is independent of the time step size

Practical Applications and Case Studies

• Aerodynamics: CFD is extensively used in the aerospace industry for aircraft and vehicle design
  • Examples include wing design, drag reduction, and optimization of aerodynamic shapes
• Turbomachinery: CFD aids in the design and analysis of turbines, compressors, and pumps
  • Flow through complex blade geometries and rotating components can be simulated to improve efficiency and performance
• Automotive engineering: CFD is applied to optimize the design of cars, trucks, and racing vehicles
  • Simulations help in reducing drag, improving engine cooling, and enhancing vehicle stability
• Environmental flows: CFD models atmospheric and oceanic flows to study weather patterns, pollutant dispersion, and climate change
  • Examples include wind farm optimization, urban air quality modeling, and coastal erosion studies
• Biomedical engineering: CFD is used to analyze blood flow in the cardiovascular system and airflow in the respiratory tract
  • Applications include the design of medical devices (stents, heart valves) and drug delivery systems
• Chemical and process engineering: CFD simulates mixing, reaction, and separation processes in industrial equipment
  • Examples include the design of reactors, heat exchangers, and multiphase flow systems
• Case studies demonstrate the successful application of CFD in real-world problems
  • Validation against experimental data builds confidence in the CFD methodology
  • Best practices and lessons learned from case studies guide future CFD analyses