Coordinate and Basis Vector Systems

Influence of Coordinate and Basis Vector Systems in CFD Formulation

In Computational Fluid Dynamics (CFD), the formulation of conservation equations—such as those for mass, momentum, and energy—depends significantly on the choice of coordinate and basis vector systems. These systems define how we describe the geometry of the problem, represent vector and tensor quantities, and discretize the governing equations. The choice of coordinate system and basis vectors can affect both the mathematical complexity of the equations and the computational efficiency of the solution.

1. Coordinate Systems

Coordinate systems provide a framework for describing the position of points in space and for expressing the conservation equations. The choice of a coordinate system depends on the geometry of the problem and the flow characteristics.

Cartesian Coordinate System (x, y, z)

  • Description: The Cartesian coordinate system is the most straightforward and commonly used system. It is defined by three mutually perpendicular axes: x, y, and z.
  • Application:
    • Suitable for problems with simple, rectangular geometries.
    • Example: Flow over a flat plate, flow in a rectangular duct.

Cylindrical Coordinate System (r, θ, z)

  • Description: The cylindrical coordinate system is defined by the radial distance r, azimuthal angle theta, and axial distance z.
  • Application:
    • Ideal for problems with rotational symmetry or cylindrical geometries.
    • Example: Flow in a pipe, swirling flows, flow around a cylinder.

Spherical Coordinate System (r, θ, φ)

  • Description: The spherical coordinate system uses the radial distance r, polar angle theta, and azimuthal angle phi.
  • Application:
    • Useful for problems involving spherical geometries or radial symmetry.
    • Example: Flow around a sphere, astrophysical flows, or problems with radial symmetry.

Curvilinear and Orthogonal Coordinate Systems

  • Description: Curvilinear coordinate systems are generalized coordinates that can follow the shape of the geometry. Orthogonal systems have coordinate lines that intersect at right angles, while non-orthogonal systems do not.
  • Application:
    • Suitable for complex geometries where Cartesian or cylindrical systems are inadequate.
    • Example: Flow in complex-shaped domains, such as around airfoils or in turbomachinery.
    • Formulation: The conservation equations in curvilinear coordinates are expressed using generalized derivatives, which can be challenging to discretize. The equations account for metric coefficients that describe how the coordinates deform from the Cartesian system.

2. Basis Vectors and Tensor Representation

In CFD, vectors (like velocity) and tensors (like stress) can be expressed in different bases, which influence the form of the equations.

Covariant and Contravariant Components

  • Covariant Components: These are components of a vector that align with the basis vectors of the coordinate system. They change with the scale of the coordinate system.

  • Contravariant Components: These are components that represent the projection of the vector onto the coordinate axes, independent of the scale of the coordinate system.

Fixed vs. Variable Basis

  • Fixed Basis: Basis vectors do not change with position. Common in Cartesian coordinates.

    • Example: Cartesian velocity components u, v, w remain constant in direction.

  • Variable Basis: Basis vectors change with position. Common in curvilinear coordinates.

3. Influence on Conservation Equations

The choice of coordinate system and basis vectors affects how the conservation equations are written and solved:

  • Mathematical Complexity: In non-Cartesian systems, the equations often include additional terms representing the geometry’s curvature or non-uniformity, making the equations more complex.

  • Discretization: The coordinate system influences how the domain is discretized. For example, in cylindrical or spherical systems, grid spacing may vary with radius or angle, requiring careful treatment in numerical methods like FDM, FVM, or FEM.

  • Boundary Conditions: Coordinate systems can simplify the application of boundary conditions. For instance, using cylindrical coordinates simplifies the application of boundary conditions for flow in a pipe.

Examples of Coordinate System Usage in Specific Flow Problems

  • Pipe Flow (Cylindrical Coordinates): For analyzing flow within a circular pipe, cylindrical coordinates are ideal because they align with the geometry of the problem. This alignment simplifies the momentum equation, particularly the radial and axial components.

  • Flow Around a Sphere (Spherical Coordinates): In problems involving spherical objects (e.g., flow around a ball), spherical coordinates allow for a natural expression of the flow field, simplifying the analysis of radial and angular variations.

  • Airfoil Flow (Curvilinear Coordinates): For flow around an airfoil, curvilinear coordinates can conform to the shape of the airfoil, leading to more accurate representation of the boundary layer and wake regions.

Conclusion

The choice of coordinate system and basis vectors is crucial in CFD as it directly impacts the complexity of the conservation equations and the accuracy and efficiency of numerical solutions. Understanding the flow characteristics and geometry is key to selecting the appropriate system for a given problem, ensuring that the mathematical formulation is both accurate and computationally feasible.