3.2.4.2. Conservation of Energy

For a material with constant density and specific heat $\specificheat$ , temperature $T$ , heat flux $\vector{q}$ and volumetric energy source $H_V$ , letting $b = \density\specificheat T$ , $\vector{f} = \vector{q}$ and $B_\Vol = H_V$ results in the conservation of energy.

(3.11) $\density\specificheat\pt{T} + \density\specificheat\v\bcdot\grad T = -\div\vector{q} + H_V.$

A common constitutive relationship for $\vector{q}$ is Fourier’s law, $\vector{q} = - \thermcond\grad T$ where $\thermcond$ is the thermal conductivity. However, we leave the heat flux as an option to be specified as part of the material properties (see Heat Conduction). Using equation (3.3), the G/FEM residual form is

(3.12) $\symRes_T^i = \int\limits_\Vol\left[\left(\density\specificheat\pt{T} + \density\specificheat\v\bcdot\grad T - H_V\right)\phi^i - \grad\phi^i\bcdot\vector{q}\right]\dV + \int\limits_\Surf q_n\phi^i\dS = 0$

where $q_n$ is the heat flux at the boundary. For example, the natural convection boundary condition gives $q_n = h(T-T_\infty)$ where $h$ is the heat transfer coefficient and $T_\infty$ is the bulk temperature away from the surface.

In Aria, each term in (3.12) is specified separately as identified in equation (3.13).

(3.13) $\symRes_T^i & = \underbrace{\int\limits_\Vol \density\specificheat\pt{T}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \density\specificheat\v\bcdot\grad T\phi^i\dV}_\mathrm{ADV} + \underbrace{\int\limits_\Vol \density\specificheat T \div \v \phi^i\dV}_\mathrm{DIV} \\ & - \underbrace{\int\limits_\Vol H_V \phi^i\dV}_{\mathrm{SRC}} - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{q}\dV}_\mathrm{DIFF} + \int\limits_\Surf q_n\phi^i\dS = 0$

For a purely incompressible form, we know that $\div\v\equiv 0$ and can remove terms. Equation (3.13) then simplifies to

(3.14) $\symRes_T^i & = \underbrace{\int\limits_\Vol \density\specificheat\pt{T}\phi^i\dV}_\mathrm{MASS} + \underbrace{\int\limits_\Vol \density\specificheat\v\bcdot\grad T\phi^i\dV}_\mathrm{ADV} - \underbrace{\int\limits_\Vol H_V \phi^i\dV}_{\mathrm{SRC}} \\ & - \underbrace{\int\limits_\Vol \grad\phi^i\bcdot\vector{q}\dV}_\mathrm{DIFF} + \int\limits_\Surf q_n\phi^i\dS = 0$

Note, however, that both equations still assume a constant specific heat $\specificheat$ .

3.2.4.2.1. SUPG

SUPG stabilization can be activated by added the SUPG term on the input file line. This capability is still under development.