Handle Boundary Conditions

Handling boundary conditions correctly is crucial in computational fluid dynamics (CFD) to ensure that the numerical solution accurately represents the physical problem. Here’s a structured approach to implementing boundary conditions in the discretized equations using the Finite Volume Method (FVM):

1. Identify and Define Boundary Conditions

  • Inlet Boundary: Specifies the inflow conditions (e.g., velocity, temperature, concentration). The inlet conditions are typically prescribed at the nodes just outside the domain, allowing the boundary conditions to be easily incorporated into the equations at the internal nodes.
  • Outlet Boundary: Represents the outflow conditions, often with specified pressure or zero-gradient conditions for variables like velocity or temperature. The outlet boundary conditions ensure that the solution remains stable and realistic as the flow exits the domain.
  • Wall Boundary: Accounts for solid surfaces within the domain where no-slip conditions are applied (velocity = 0). Wall boundaries also handle heat transfer and other interactions between the fluid and the solid surface.
  • Prescribed Pressure: Used when pressure is known at a boundary. The pressure boundary condition is implemented by adjusting the pressure correction equation to enforce the specified value.
  • Symmetry Boundary: Implies that the flow and gradients normal to the boundary are zero. Symmetry conditions reduce computational effort by modeling only a portion of the domain.
  • Periodic (Cyclic) Boundary: Applied when the flow at one boundary repeats at another. Periodic boundaries are common in simulations of repeated patterns or cyclic flows, ensuring continuity between opposite faces.

2. Discretize the Domain with a Staggered Grid

  • In a staggered grid, different variables (e.g., velocity, pressure) are stored at different locations within the control volume, enhancing numerical stability and accuracy.
  • Surround the physical boundary with additional nodes that store boundary values. The physical boundaries coincide with the scalar control volume boundaries, allowing easy incorporation of boundary conditions.

3. Modify Discretized Equations for Boundary Nodes

  • Suppress Links to Boundary Sides: For nodes adjacent to the boundary, modify the discretized equations by setting the coefficient associated with the boundary-side node to zero. This effectively removes the influence of the nonexistent neighboring node outside the boundary.
  • Modify Source Terms: Introduce source terms Su and Sp in the discretized equation to represent boundary fluxes or set specific values at the boundary node.

    • For example, to fix the value of a variable ϕ at a node P, use the source term modification:

      $\ S_p = -10^{30} \ and \ \ S_u = 10^{30} ϕ_{fix}$

4. Implement Specific Boundary Condition Treatments

  • Inlet Conditions: Set the value of flow variables (e.g., velocity, temperature) at the inlet nodes. Modify the adjacent internal nodes to ensure the correct inflow conditions.
  • Outlet Conditions: Apply the specified pressure or zero-gradient condition at the outlet. Adjust the pressure-correction equation to enforce these conditions.
  • Wall Conditions: Implement no-slip conditions by setting the velocity at wall nodes to zero. For heat transfer, set the temperature or heat flux as required by the physical problem.
  • Symmetry and Periodic Boundaries: Ensure that the gradients normal to the boundary are zero for symmetry conditions. For periodic boundaries, link the variables on one side of the domain to the corresponding variables on the opposite side.

5. Verify Implementation

  • After incorporating the boundary conditions, validate the implementation by checking for physical consistency in the results (e.g., correct flow patterns, temperature distributions).
  • Conduct tests with known analytical solutions or benchmark cases to ensure that the boundary conditions are correctly applied and that the numerical solution converges properly.

6. Use Large Source Terms for Special Cases

  • For solid obstacles or specific fixed values within the domain, apply overwhelmingly large source terms in the discretized equation to force the solution to the desired value. This approach simplifies the handling of complex boundary conditions without needing special treatment of the obstacles.

By carefully implementing these steps, you ensure that the boundary conditions are accurately represented in the discretized equations, leading to a stable and physically realistic solution in CFD simulations.