[aspect-devel] Question regarding heat flux output

[Wolfgang Bangerth]

> Thanks for the quick reply! From heat_flux_stastics.cc (lines 96-103)
> there is
>
> local_normal_flux
> +=
> -thermal_conductivity *
> (temperature_gradients[q] *
> fe_face_values.normal_vector(q)) *
> fe_face_values.JxW(q);
>
> Is this the equation you are referencing below? If it is, I'm not sure
> what normal_vector(q) and JxW refer to. Is normal_vector(q) a
> directional multiplier?

This formula is the numerical approximation to the integral

   \int_{\Gamma_i}   - k  nabla T . n  dS

that Timo referenced. Here,  n=normal_vector(q)  at quadrature point q,
has no physical units, and  JxW(q)=area element dS has units meters in
2d and meters^2 in 3d.  Gamma_i is the part of the boundary of the
domain with boundary indicator i.

k, the thermal conductivity, is documented as having units W/m/K, so in
3d I indeed get units  W  for the entire expression. (Timo: The
derivative has units K/m, not K/s.) That is exactly the (integrated)
heat flux through the surface we were looking for.

John: Do you agree? This may of course not be what you need from a
simulation -- if so, let us know what it is that you want and we can see
how to implement it.

Cheers
  W.

PS: The units of the conductivity are
   heat flux per unit area perpendicular to the flux direction
   - per -
   unit gradient in temperature
In 3d, that is  (W/m^2) / (K/m)  =  W/(K m) as documented. In 2d, on the
other hand, it is  (W/m) / (K/m)  =  W/K. In other words, in 2d, the
units of the above formula would be
   W/K  *  K/m  *  1  *  m
which is again just Watts. I've fixed the description of the units in
the documentation to this end.

------------------------------------------------------------------------
Wolfgang Bangerth               email:            bangerth at math.tamu.edu
                                 www: http://www.math.tamu.edu/~bangerth/