## Mathematical Model
### Boundary Conditions for Turbulence Models  
Two-equation turbulent models can be solved if appropriate boundary conditions are applied. When a near-wall approach with no wall functions is used, the boundary conditions can be computed by a near wall series expansion, as reported in [22]. In Table 2 the expansion is computed for the mean and fluctuating velocity, and for the mean and fluctuating temperature, by considering a small, plane surface area with \(x\)-\(z\) plane coordinates and \(y\) the wall distance. We remark that the velocity normal to the surface is assumed to vanish like its fluctuating components. By using similar expansions we calculate the near wall behaviors of \(k\), \(\varepsilon\), \(k_{\theta}\) and \(\varepsilon_{\theta}\), which are labeled by the lower-script \(w\) (\"wall\"). Then we combine these expansions and derive those for \(\Omega\) and \(\Omega_{\theta}\). Dynamical turbulence variables can be expanded around a wall point as  
\[k_{w} \simeq\frac{1}{2}(a_{1}^{2}+c_{1}^{2})y^{2}=\frac{1}{2}\xi y^{2}\,, \tag{24}\] \[\varepsilon_{w} \simeq\
u(a_{1}^{2}+c_{1}^{2})=\
u\xi\,,\] (25) \[\Omega_{w} \simeq\ln\left(\frac{2\
u}{C_{\mu}y^{2}}\right)\,. \tag{26}\]  
From Equation (24) one can see that the \(k_{w}\) tends to zero as \(y\) vanishes. The constant \(\xi\) is not known but the \(k_{w}\) boundary conditions can be written in a closure form by taking the derivative of \(k\). In fact we can write  
\[\left.\frac{\partial k}{\partial y}\right|_{w}=\xi y=\frac{2k_{w}}{y}\,. \tag{27}\]  
The form in Equation (27) is a well known result that allows us to determine Dirichlet boundary conditions for fluctuations in terms of their derivatives. We see that the dissipation of turbulent kinetic energy, \(\varepsilon\), in the near wall region does not depend on the wall distance. The near wall behavior is \(\varepsilon_{w}=\
u\xi\), where \(\xi\) is a constant value which depends on the fluctuating velocity components. This value cannot be determined _a priori_ and it should be calculated as \(\xi=2k_{w}/y^{2}\). On wall surfaces we thus impose a Dirichlet b.c. \(\varepsilon_{w}=\
u\xi\) with \(\xi\) varying iteration by iteration. This could be a reason for the KE model showing poor numerical stability. \(\Omega_{w}\) depends only on the known kinematic viscosity \(\
u\), wall distance \(y\) and model constant \(C_{\mu}\) so that the KLW model is not affected by this instability issue. If we take the \(\Omega\) derivative in the \(y\) direction we obtain  
\[\left.\frac{\partial\Omega}{\partial y}\right|_{w}=-\frac{2}{y}\,, \tag{28}\]  
so we can also impose a Neumann b.c. on wall surfaces. The boundary conditions for the dynamical turbulent variables of the KLW model are summarized in Table 3. \begin{table}
\begin{tabular}{l l} \hline
**Mean Components** & **Fluctuating Components** \\ \hline \(u=A_{1}y+A_{2}y^{2}+A_{3}y^{3}\) & \(u^{\prime}=a_{1}y+a_{2}y^{2}+a_{3}y^{3}\) \\ \(v=\phantom{-}+B_{2}y^{2}+B_{3}y^{3}\) & \(v^{\prime}=\phantom{-}+b_{2}y^{2}+b_{3}y^{3}\) \\ \(w=C_{1}y+C_{2}y^{2}+C_{3}y^{3}\) & \(u^{\prime}=c_{1}y+c_{2}y^{2}+c_{3}y^{3}\) \\ \(T=D_{0}+D_{1}y+D_{2}y^{2}\) & \(T^{\prime}=d_{0}+d_{1}y+d_{2}y^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Expansion for the components of the velocity and temperature for plane coordinates \(x\)-\(z\) around wall points and \(y\) distance from the wall. The issue of boundary conditions on fluctuating variables is still an open question [8; 9; 16; 17; 27]. As it concerns the energy equation, in this paper we apply only boundary conditions with uniform heat flux on the wall surfaces. Boundary conditions of this kind have to be satisfied for the total variation which is the sum of an instantaneous and an averaged temperature. For this reason the derivative of the fluctuating temperature tends to vanish as we tend to the wall. In this paper we use the mixed boundary condition (MX) in which the temperature fluctuation and its derivative on the wall--so the terms \(d_{0}\) and \(d_{1}\) in the near wall expansion in Table 2--are set to zero.