Thermal wall function for high speed flows and flows with viscous heating

For flows with significant viscous heating, such as high speed flows, the heat flux depends on wall temperature and local conditions derived from thermal wall function.

The flow solver computes the wall heat flux as follows:

  • Tw is the wall temperature.
  • Tf is the fluid temperature at the near wall node.
  • h is the heat transfer coefficient.

For flows with significant viscous heating (such as high speed flows), the heat flux is not dependent on the difference between Tw and Tf but rather between Tw and an adiabatic wall temperature. This adiabatic wall temperature depends on the local flow conditions and can be derived from the thermal wall function. Denoting LS as low speed and HS as high speed, the general thermal wall function can be written as:

where

In the above equations:

  • T+LS is the standard thermal wall function for low speed flows.
  • T+HS is the correction to the standard thermal wall function which accounts for the contribution from the viscous heating on the temperature profile.
  • Pr is the laminar Prandtl number.
  • Prt is the turbulent Prandtl number.
  • yf is the normal distance from the wall to the near wall node.
  • u* is the shear velocity.
  • qw is the wall heat flux.
  • ρ is the fluid density.
  • μ is the fluid dynamic viscosity.
  • Uf is the fluid velocity at the near wall node.
  • UC is the fluid velocity at the intersection of the linear and logarithmic temperature profiles (for low speed flow).

Combining above equations gives:

which allows to express the wall heat flux as:

The heat flux qw is positive when heat flows from the solid. In these equations the terms:

  • represents the standard low speed flow wall heat flux.
  • is the contribution to the wall heat flux due to viscous heating.

Equation for qw can be rewritten as:

where:

In the literature for high speed flows, T*f is defined as the recovery temperature or the adiabatic wall temperature and is expressed as:

where r is the recovery factor.

If yf is located in the laminar sublayer (small y+f value), the equation for T*f is defined as:

which gives the standard recovery factor for the Couette flow equal to Pr.

For the thermal wall function for high speed flow, the flow solver uses the standard formulation for the heat transfer coefficient given by:

The effects of viscous heating are then accounted for by correcting the fluid temperature so that T*f replaces Tf when the heat flux is computed.