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titleCommon Case Information:Overview

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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  • Flow parameters:
    • Vortex Flow
    • Heat source on inner radius boundary
    • Non-dimensional Viscosity: 10^-2^
    • Solver parameters
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    titleSolver Parameters
    :
    • Equation Set: Scalar 2D
    • Nonlinear Solver: Newton
    • Basis Type: Hierarchical
    • Initial/Max CFL: 1030
    • target Target residual: 10^10-11^11
    • Preconditioning:
      • Type: ILU (Left)
      • Reordering: MDF
      • Outer GMRES iterations (# restarts): 20
      • Inner GMRES iterations (# Krylov vectors): 200

    Tested Outputs

    • L2 L2 error (compared to an exact solution function)
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    • Heat Numerical heat flux on inner boundary
    • Exact heat flux error on inner boundary
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    • L∞ Linf of the exact heat flux error
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    conditions
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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:
    • Picture of Source Boundary
    • Solution Order = 2 (non-adaptive)
      • Heat Flux Adjoint Residual (Mesh Adaptive Case)

    adaptation

    • Solution Order =1 
    • Adaptation Iterations = 2
    • Adaptation Method = Fixed Fraction
    • Output Adapted = Heat Flux
    • Anisotropy Method = Hessian

    results or convergence

    • Both cases reach tolerances in 2 iterations
    • Compared to truth files with tolerance of 
      • tol: e-12
      • Adapt tol : e-10

    Case Details

    Scalar2d_CircleBoundaryLayer

    Scalar2d_CircleBoundaryLayer_Adapt

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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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    ...