Heat Transfer Condition over Non-Conforming Interfaces
Summary
The Heat PDE is used to model heat conduction problems in openCFS. The implementation already contains some useful features like non-conforming grids, infinite matting layers, or convective terms for modeling material motion, i.e. the advection-diffusion equation
\rho c_{\rm p} \frac{\partial T}{\partial t} +
\rho c_{\rm p} \bm v \cdot \nabla T - \nabla \cdot \lambda(T) \nabla T
= \dot{q}_{\rm d}
where T
denotes the temperature, \bm v
the convention velocity, \dot{q}_{\rm d}
a heat source density and \rho
, c_p
and \lambda
are material parameters in standard notation.
For certain modeling problems, a heat transfer boundary condition between two media via non-conforming interfaces would allow us to conveniently take varying reference temperatures of flowing media into account. Consider the situation in the sketch below:
The inductor transmits heat to a cooling medium via a heat transfer condition
\bm q \cdot \bm n = \alpha(T_f-T_k)
which will raise the temperature of the cooling medium T_k
which will rise over the depth direction (as the medium flows) of the inductor.
The temperature distribution in the cooling medium is only interesting in the depth direction and will be taken into account correctly via the convective term in the heat PDE.
By using the above heat transfer condition instead of the continuity of the unknown condition, one can modify the derivation of the classical Mortar method for non-conforming interfaces, arriving at the required formulation that allows for a temperature jump over the interface. Alternatively, a Nitsche-type formulation can be obtained by directly inserting the expression for the heat flux into the boundary term.
Tasks
-
derive the Mortar or Nitsche formulation -
implement the required new bi-linear forms -
validate by defining suitable test cases, e.g. use an artificial thin layer of equivalent "heat transfer" material -
Compare both "formulations" (optional)
Possible Solutions
- use the existing non-conforming interface definition in the XML input and add a "heat transfer coefficient" that triggers the new implementation
- validate by using an artificial thin layer of equivalent "heat transfer" material as a reference solution
Resources and Hints
- Slides for Multiphysics II, Lecture video on Non-conforming Interfaces
- The Mortar interface is defined in
SinglePDE::DefineMortarCoupling
, the Nitsche coupling inDefineNitscheCoupling
right below - HeatMortar testscase: https://gitlab.com/openCFS/Testsuite/-/tree/master/TESTSUIT/Singlefield/Heat/HeatMortar
- Testcase with convective term in HeatPDE: https://gitlab.com/openCFS/Testsuite/-/tree/master/TESTSUIT/Singlefield/Heat/ConvDiff2D