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This circle boundary layer case demonstrates the method of manufactured solutions in which an analytical heat source is derived from an analytical expression of the thermal boundary layer. This derived heat source is applied to a vortex flow around a circular annulus of radius r = [ 1 , 2 ], and the numerical heat flux at the inner boundary is checked with the analytical heat flux to a specified tolerance. All boundary temperatures are set to the analytical thermal boundary layer solution, and the initial condition is set to 1.

Note that the singularity at (θ=0) is disregarded.

• Equation Set: Scalar 2D
• Nonlinear Solver: Newton
• Basis Type: Hierarchical
• Initial/Max CFL: 10³⁰
• Target residual: 10^-11
• Preconditioning:
  • Type: ILU (Left)
  • Reordering: MDF
  • Outer GMRES iterations (# restarts): 20
  • Inner GMRES iterations (# Krylov vectors): 200

Tested Outputs

• L₂ error (compared to an exact solution function)
• Numerical heat flux on inner boundary
• Exact heat flux error on inner boundary
  
• L_∞ of the exact heat flux error

• Scalar Convective Diffusion Equation
  

• Vortex Flow
  
  where
  

• Boundary Condition:
  
  where
  

• Derived Heat Source:
  
  where
  

• Non-dimensional viscosity:
  

• Solution Order: 2
• Iterations to solve: 2
• All outputs tested to a tolerance of 10^-12

Job File:
Scalar2d_CircleBoundaryLayer

• Solution Order = 1
• Additional tested outputs: Heat flux adjoint residual

• Adaptation parameters:
  • Adaptation Iterations: 2
  • Adaptation Method: Fixed fraction
  • Output Adapted: Heat flux
  • Anisotropy Method: Hessian

• Iterations to solve on each mesh: 2
• All outputs tested to a tolerance of 10^-10

Job File:
Scalar2d_CircleBoundaryLayer_Adapt