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section
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titleOverview
The

This circle boundary layer case

involves a heat source being applied to the inner boundary of

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 ]

. This regression test checks

, and the numerical heat flux at the inner boundary

to

is checked with the analytical heat flux

with

to a specified tolerance.

The

All boundary

is

temperatures are set to

a

the analytical thermal boundary layer

condition which is a function of r and theta.
The applied heat source on the inner boundary is a function of r and theta.
The

solution, and the initial condition is set to 1.

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titleGeneral Case InformationTemperature Distribution:

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

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  • Flow parameters:
    • Vortex Flow - Vx = V∞sin(theta) Vy = V{}∞cos(theta)
    • Heat source on inner radius boundary
    • Non-dimensional viscosity: 10-2
Solver parameters
Panel
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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titleBoundary conditions:Flow Conditions:
  • Scalar Convective Diffusion Equation
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  • Vortex Flow
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    where
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  • Boundary Condition:
    Image Added
    where
    Image Added
  • Derived Heat Source:
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    where
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  • Non-dimensional viscosity:
    Image AddedPicture of Source Boundary

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 and Heat Contour:

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Picture of Mesh and Heat Contour

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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 and Heat Contours:
Picture of Adapted Meshes and Heat Contours

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Image Added