2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Analytical solution.
Temperature varying linearly according to \(\mathrm{CD}\)
Isotherms parallel to faces \(\mathrm{CHKF}\) and \(\mathrm{DIJE}\).
In the frame: \((\frac{\mathrm{CD}}{\parallel \mathrm{CD}\parallel },\frac{\mathrm{CH}}{\parallel \mathrm{CH}\parallel },\frac{\mathrm{CF}}{\parallel \mathrm{CF}\parallel },)\), we have:
\(\left[\begin{array}{}{\varphi }_{x}\\ {\varphi }_{y}\\ {\varphi }_{z}\end{array}\right]=\left[\begin{array}{}-({\lambda }_{X}{\mathrm{cos}}^{2}\alpha +{\lambda }_{Y}{\mathrm{sin}}^{2}\alpha )\frac{\partial T}{\partial x}\\ -({\lambda }_{X}-{\lambda }_{Y})\mathrm{cos}\alpha \mathrm{sin}\alpha \frac{\partial T}{\partial x}\\ 0\end{array}\right]\)
with:
\({\varphi }_{x}=1200{\varphi }_{y}=400\alpha =({X}_{\mathrm{1,}}\mathrm{CD})\) \(T(x)=\frac{-{\varphi }_{x}}{{\lambda }_{{X}_{1}}{\mathrm{cos}}^{2}\alpha +{\lambda }_{Y}{\mathrm{sin}}^{2}\alpha }x+T(A)\)
That is: \(T(x)=-\mathrm{1600.x}+20.\)
If \(\beta =(\mathrm{CD},{X}_{0})\): \(\begin{array}{cc}\varphi \mathrm{.}{X}_{0}=\mathrm{cos}\beta \mathrm{.}{\varphi }_{X}-\mathrm{sin}\beta {\varphi }_{y}& \text{soit}720\\ \varphi \mathrm{.}{Y}_{0}=\mathrm{sin}\beta \mathrm{.}{\varphi }_{X}+\mathrm{cos}\beta {\varphi }_{L}& \text{soit}720\end{array}\)
2.2. Benchmark results#
Temperature at points \(A,B,G\).
Flow following directions \({X}_{0}\) and \({Y}_{0}\).
\(T(A)=100T(B)=20T(G)=60\Phi \mathrm{.}{X}_{0}=720\Phi \mathrm{.}{Y}_{0}=1040\)
2.3. Bibliographical references#
RICHARD: Technical note HM-18/94/0011, « Development of thermal anisotropy in Aster software ».