The energy conservation equation described in Energy Conservation Equation is applied in the near wall region, with the following assumptions applied:

With these assumptions, Equation 2–5 is reduced to a one-dimensional energy conservation equation in the near wall region, written as

(3–9)

with

(3–10)

Where T is temperature, J is heat flux, y is the normal distance to the wall, is the flow velocity in the normal-to-wall direction, ρ is the gas density, is the specific heat of the gas mixture, p is pressure, and is volumetric heat release rate, respectively. k and are laminar and turbulent thermal conductivity, and are associated with laminar and turbulent viscosity by Prandtl numbers . According to Rakopoulos et al. [75], the left-hand-side of Equation 3–9, that is, the term , is negligible, and is assumed to be zero.

Following the approach taken by Han and Reitz [31], integration of Equation 3–9 over the distance y from the wall (y=0) with consideration of the variations of gas density, turbulent viscosity, and turbulent Prandtl number in the turbulent boundary layer. As a result, a temperature wall function is formulated as

(3–11)

where , , , and are defined as

(3–12)

and

(3–13)

Here, u* is the friction velocity defined in terms of the turbulence kinetic energy, k , as , y is the normal distance to the wall, is the heat flux through the wall (positive if heat goes to the wall, negative if heat comes from the wall), is the averaged volumetric chemical heat release rate, is the averaged pressure fluctuation rate, and is the laminar kinematic viscosity. The corresponding formulation for the wall heat flux is given as

(3–14)

where T and are the gas temperature and wall temperature, respectively. Following the implementation of Han and Reitz [31], we neglect the source term , such that Equation 3–14 becomes

(3–15)

As seen in Equation 3–15, the pressure fluctuation rate near the wall could affect heat transfer. When the pressure is rising during the engine compression stroke and combustion, it promotes heat transfer to the wall. In contrast, if the pressure is decreasing, it reduces heat transfer to the wall. However, the pressure fluctuation’s effect is considered secondary when compared to the effect of a temperature difference between the gas and the wall. If the pressure fluctuation’s effect is also neglected, Equation 3–15 becomes

(3–16)

Equation 3–16 states that due to the density variation, the wall heat flux is proportional to the logarithm of the ratio of the gas temperature to the wall temperature. Note that this equation is slightly different from Han and Reitz [31], which is written as . Even though we have largely followed the derivation approach suggested in Han and Reitz [31], we believe that there was a mathematical error in their derivation that resulted in this difference. Therefore, our own equation (Equation 3–16) is used in Ansys Forte.

Other energy-equation boundary conditions on rigid walls are introduced by directly specifying either the wall temperature or the wall heat flux . For adiabatic walls, we set equal to zero. For fixed-temperature walls that are also either free slip or no slip, the wall temperature is prescribed and is determined implicitly from Equation 2–5.

The Amsden model was first implemented in the KIVA-3V code [5]. It makes the same assumptions as those in the Han-Reitz model. In the Amsden model, for a given wall temperature , if the wall uses the turbulent law-of-the-wall condition, heat flux is calculated as

(3–17)

where

(3–18)

and

(3–19)

In these equations, ϱ, , are the density, laminar kinematic viscosity, and specific heat at constant pressure of the fluid, respectively, is the laminar Prandtl number, is the turbulent Prandtl number, κ is von Karman's constant with value 0.43 in the implementation, and are the temperature and turbulent kinetic energy evaluated at the wall boundary cell center, is the distance from the cell center to the wall, is a constant in the turbulence model, and is also a model constant related to the k-ε turbulence model, with value 5.5.

When a turbulent law-of-the-wall slip condition or no-slip condition is used, frictional heating serves as a source to the internal energy and it has the form

(3–20)

where is the heating per unit area of wall, and v is the tangential gas velocity relative to the wall.