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titleOverview

This circle boundary layer case involves a heat source being 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 ]. The , and the numerical heat flux at the inner boundary is checked with the analytical heat flux at the inner boundary to a specified tolerance. All boundaries boundary temperatures are set to a the analytical thermal boundary layer condition which is a function of r and θ. The applied heat source is also a function of r and θsolution, and the initial condition is set to 1.

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titleBoundary conditionsTemperature Distribution:

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

Picture of Source Boundary

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Vortex Flow
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  • Boundary Condition:
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  • Heat Source:
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    where
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  • Non-dimensional viscosity:
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    titleFlow Conditions:
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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
    • 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

    Temperature Distribution:

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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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