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titleOverview

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.

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titleTemperature Distribution:

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

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titleSolver Parameters:
  • 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
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  • L∞ of the exact heat flux error
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titleFlow Conditions:
  • Scalar Convective Diffusion Equation
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  • Vortex Flow
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    where
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  • Boundary Condition:
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    where
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  • Derived Heat Source:
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    where
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  • Non-dimensional viscosity:
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Case Specific Details

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titleCase 1 (Non-Adaptive):
  • Solution Order: 2
  • Iterations to solve: 2
  • All outputs tested to a tolerance of 10-12

Job File:
Scalar2d_CircleBoundaryLayer

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titleCase 1 Mesh

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titleCase 2 (Adaptive):
  • 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

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titleCase 2 Adapted Meshes:

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graphs:

  • geom
  • grids
  • solution

what is tested?

  • L2 error of ExactSolutionFunction
  • Heat Flux on Boundary Group 1
  • Exact Heat Flux Error on Boundary Group 1
  • Linf of the Exact Heat Flux Error
  • Heat Flux Adjoint Residual (Mesh Adaptive Case)

equations

  • Scalar 2D
  • Basis Type = Hierarchical

flow conditions

  • SourceFunction = ScalarCircleBoundaryLayerSource
  • Viscosity = 1e-2
  • VelocityFunction = Vortex Flow

numerical parameters

  • CFL = 1e30
  • target residual = 1e-11
  • Nonlinear Solver = Newton
  •  Preconditioner = ILU
  • Preconditioner Side = Left
  • ReorderingType = MDF
  • # GMRES outer iterations = 20
  • # GMRES inner iterations = 200

adaptation

  •  

results or convergence

  • Tolerances (in 2 iterations)
    • Residual Norm = 6.425e-14
    • L2Error converges to 1.003e-07
    • Heat Flux converges to 3.794e-01
    • Heat Flux Error converges to 3.268e-06
    • Linf of Heat Flux Error converges to 1.059e-04
    • Compared to truth files with tolerance of 

tol: 1.0 1.0 1.0 0.1 e-12 e-12 e-12 e-12
Adapt tol : 1.0 e-10 e-10 e-10

Case Details

Scalar2d_CircleBoundaryLayer

Scalar2d_CircleBoundaryLayer_Adapt

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