Addressing Challenges in Complex Geometries

Challenges: Complex shapes and internal obstacles hinder accurate meshing and boundary condition applications

1. Advanced Mesh Generation Techniques

CFD methods for complex geometries are classified into two groups:

  • Structured curvilinear grid arrangements or body fitted grids
  • Unstructured grid arrangements

$\ Body \ Fitted \ Grids \ \ Block \ Structured \ Grids \ \ Unstructured \ Grid \ arrangements $ for detail about different grid arrangements see Additional Resources at the bottom section

  • Use unstructured meshes, adaptive mesh refinement, or hybrid meshing for better geometric representation.
  • Unstructured Meshes: Flexible meshes that adapt to complex shapes.
  • Adaptive Mesh Refinement (AMR): Refines mesh in critical areas to capture details.
  • Hybrid Meshing: Combines structured and unstructured meshes for efficiency.

2. Advanced Boundary Condition Implementation

  • Immersed Boundary Methods: Embed boundaries within the grid, simplifying complex geometries.
  • Ghost-Cell or Cut-Cell Methods: Modify equations near boundaries for accuracy.
  • Boundary-Conforming Grids: Align grids closely with boundaries to reduce errors.

3. Improved Numerical Methods

  • Higher-Order Schemes: More accurate methods that reduce errors near boundaries.
  • Implicit Time-Stepping: Stable for complex geometries, especially in fluid dynamics.
  • Nonlinear Solvers: Robust methods for accurately applying complex boundary conditions.

4. Optimized Computational Resources

  • Parallel Computing: Distributes tasks across processors to speed up computations.
  • GPU Acceleration: Uses graphics processors to enhance performance.
  • Memory-Efficient Algorithms: Reduces memory usage in large simulations.

5. Software and Algorithmic Innovations

  • Custom Boundary Condition Code: Tailors implementations for specific geometries.
  • Advanced Simulation Tools: Utilizes software like ANSYS and OpenFOAM for complex scenarios.
  • Validation and Verification: Regular checks against known solutions ensure accuracy.

7. Error Estimation and Adaptive Techniques

  • Error Estimation: Identifies high-error regions for targeted refinement.
  • Sensitivity Analysis: Assesses impact of changes in conditions on solutions.

Technical Terms and Resources

  • Unstructured Mesh: A type of mesh that does not follow a regular grid pattern, allowing for more complex shapes. Learn more

  • Adaptive Mesh Refinement (AMR): A technique to refine the mesh dynamically based on the solution’s needs. Learn more

  • Immersed Boundary Method: A numerical technique for simulating fluid flow with complex boundaries. Learn more

  • Ghost-Cell Method: A method used in computational fluid dynamics to handle boundaries. Learn more

  • Higher-Order Numerical Methods: Techniques that improve accuracy by using polynomial approximations of higher degree. Learn more

Additional Resources

Introduction to Grid Systems:

  1. Cartesian Grids:

    • Simplest to understand and implement.
    • Suitable for simple geometries but inefficient for complex geometries.
    • Issues include stepwise approximation errors, wasted computational resources, and difficulty in representing curved boundaries.

  2. Body-Fitted Grids:

    • Orthogonal Curvilinear Coordinates: Grid lines intersect at 90° angles, making them easier to implement for simpler curved geometries.
    • Non-Orthogonal Coordinates: More flexible but challenging to implement, especially for very complex shapes.
    • Useful for capturing geometric details more accurately than Cartesian grids.
    • Commonly used in engineering problems like flow over aerofoils, internal combustion engine chambers, and more.
    • Allows better representation of curved surfaces without the need for approximations.

    Advantages and Disadvantages:

  • Advantages of Body-Fitted Grids:

    • More efficient use of computational resources.
    • Accurate representation of complex geometries, improving flow predictions.

  • Disadvantages of Body-Fitted Grids:

    • Governing equations are more complex.
    • Time-consuming and difficult to generate for very complex geometries.
    • Potential for unnecessary grid resolution or skewed grids if the geometry is difficult to map.
  1. Block-Structured Grids:
    • Sub-Division of Flow Domain: The flow domain is divided into several blocks, each with its structured mesh.
    • Increased Flexibility: Allows using fine grids in critical areas while keeping coarser grids in less critical areas, improving computational efficiency.
    • Matching or Non-Matching Interfaces: Adjacent blocks may have either matching or non-matching grid interfaces.
  2. Unstructured Grids: Unstructured grids are a type of finite element method used to solve partial differential equations. The main advantage of unstructured grids is their flexibility in handling complex geometries and topologies, allowing for a more accurate representation of real-world problems. However, this flexibility comes at a computational cost, as the grid generation and meshing process can be time-consuming and complex. Additionally, the increased adaptability of unstructured grids can lead to inconsistent mesh quality, affecting the overall accuracy and reliability of the solution.

For simple geometries, Cartesian grids may be sufficient, but as geometries become more complex, body-fitted or block-structured grids are preferred despite the challenges they present. Unstructured grids, where each cell can be treated as a block, offer even greater flexibility, particularly for highly complex flows.