Wall function for the thermal boundary layer

The flow solver models heat flux near adiabatic and convecting walls for laminar and turbulent flows using specific boundary conditions and wall functions.

For the energy equation, a solid wall can either be specified as:

  • Adiabatic, which means no heat transfer allowed across the wall. In this case, a heat flux of zero is imposed at the wall.
  • Convecting, where the wall temperature, the heat flux, or the heat load is specified at the wall.

For laminar flows, the heat flux between the wall and the fluid, qw, is related to the wall and fluid temperatures through:

  • k is the thermal conductivity of the fluid.
  • yf is the distance from the wall at which Tf is evaluated.

For turbulent flows, the temperature variation in the thermal boundary layer generally follows the general wall function. For more information, see Wall functions.

A dimensionless temperature, T+, is defined in terms of fluid properties, wall heat flux, and the near-wall velocity scale u*:

The following general equation describes the universal boundary layer profile for temperature:

  • Pr is the Prandtl number.

This definition of the thermal wall functions, proposed by B.A. Kader [9], is accurate right to the wall, and applies for a wide range of Prandtl numbers. Notice that there is no explicit reference to roughness and it is assumed that roughness effects are captured implicitly by the wall when y+ is defined through u*. Roughness τw and near wall k will increase so that u* will be increased suitably.

The equation for qw is re-arranged as follows:

Given a near wall fluid temperature, Tf, this equation can be used to compute qw which is substituted directly into the finite volume energy equation at the wall.

For consistency with the momentum treatment, the equation for qw is defined as follows:

The effect of the wall function is to amplify the laminar (i.e. molecular) thermal diffusion to the wall. The amplification term, (y+/T+)Pr, tends to 0 as y+ tends to 0. Because the dependent variable for the energy equation is not temperature but enthalpy, h. The equation for qw is modified so that an approximate dependency of qw on h is retained, while not changing the converged answer for qw. So, denoting the "new" solution with superscript " n" and the "old" solution by " o", Equation for qw is written as:

On convergence, the wall heat flux does not depend on enthalpy.