# Calculation of Heat Transfer and Drag Coefficients for Aircraft Geometric Models
## 3 Mathematical Model for Calculating Convective Heat Fluxes for HSA of Complex Spatial Shape
For the laminar case, the evaluation procedure is carried out as follows:  
\[T_{e}=T_{0}\bigg{(}1+r\frac{\gamma-1}{2}M_{0}^{2}\bigg{)},\quad r=\sqrt{Pr_{0}} \tag{2}\]7  
After finding the local coefficient of friction at each point on the HSA surface \(C_{f}=\frac{\left|\vec{V}\right|_{\text{tr}}}{\rho_{\text{tr}}V_{\text{tr}}^{ 2}/2}\), the total coefficients of friction \(C_{fx}\), \(C_{fy}\) have the form:  
\[ \begin{split} C_{fx}=\frac{1}{s_{\text{mid}}}\iint\limits_{S} \frac{\left(\vec{v}_{ex}\right)}{\left|\vec{v}\right|}\left(\frac{\left|\vec{ v}\right|_{\text{tr}}}{\rho_{\text{tr}}V_{\text{tr}}^{2}/2}\right)\text{d}S= \frac{1}{s_{\text{mid}}}\iint\limits_{S}\frac{\left(\vec{v}_{ex}\right)}{ \left|\vec{v}\right|}C_{f}\text{d}S,\\ C_{fy}=\frac{1}{s_{\text{mid}}}\iint\limits_{S}\frac{\left(\vec{ v}_{ex}\right)}{\left|\vec{v}\right|}\left(\frac{\left|\vec{v}\right|_{\text{tr}}}{ \rho_{\text{tr}}V_{\text{tr}}^{2}/2}\right)\text{d}S=\frac{1}{s_{\text{mid}}} \iint\limits_{S}\frac{\left(\vec{v}_{ex}\right)}{\left|\vec{v}\right|}C_{f} \text{d}S\end{split}  \tag{14}\]  
The \(C_{f0}\) of in the turbulent case can be calculated using a number of empirical dependences, for example, \(C_{f0}=0.059\)\(\text{Re}_{eff}^{-0.2}\), \(\text{Re}_{eff}=\rho_{0}V_{0}\epsilon_{eff}/\mu_{0}\). This dependence gives the most acceptable results in the considered range of Mach numbers, which is confirmed by numerous calculations. It is most widespread when used in the range of Reynolds number from \(10^{6}\) to \(10^{8}\). For the range of Reynolds numbers from \(10^{5}\) to \(10^{10}\), various simple formulas can be used to determine the local coefficient of friction, \(C_{f0}\), depending on the local Reynolds number:  
\[ \begin{split} 10^{5}\ \leq\ \text{Re}_{eff}\leq\ 10^{6};\ C_{f0}=0.042 \text{Re}_{eff}^{-0.18},\ 10^{6}\ \leq\ \text{Re}_{eff}\leq\ 10^{7};\ C_{f0}=0.032 \text{Re}_{eff}^{-0.16}\\ 10^{7}\ \leq\ \text{Re}_{eff}\leq\ 10^{8};\ C_{f0}=0,023\text{Re}_{ eff}^{-0.14}\\ 10^{8}\ \leq\ \text{Re}_{eff}\leq\ 10^{9};\ C_{f0}=0.016\text{Re}_{ eff}^{-0.12},\ 10^{9}\ \leq\ \text{Re}_{eff}\leq 10^{10};\ C_{f0}=0.011\text{Re}_{ eff}^{-0.1}\end{split}  \tag{15}\]  
Using this approach in some cases allows you to get a more accurate result. In addition to estimating the coefficient of friction drag on the body, within the framework of the approximate semi-empirical approach presented above, it is possible to determine at each specific point the characteristic parameters of the boundary layer (BL), such as BL thickness \(\delta\) and integral thicknesses of BL--the displacement thickness \(\delta\) * and the thickness of the momentum loss \(\delta\) **. The displacement thickness \(\delta\) * is a measure of the decrease in flow through the section due to the velocity decrease. The thickness of the momentum loss \(\delta\) ** characterizes the decrease in the momentum in the section due to the deceleration of the flow. A scheme for computing the characteristic boundary layer thickness at the point on the body in the case of a turbulent boundary layer is as follows: for each point of the body, the effective length is calculated; the calculated \(\ell_{eff}\) determines the effective Reynolds number; further, for the incompressible case, the thickness \(\delta\), displacement thickness \(\delta\) *, and momentum loss thickness \(\delta\) ** are determined as follows: \(\delta_{H}=0,37\cdot\mathscr{L}_{eff}\cdot\text{Re}_{eff}^{-0.2}\), \(\delta_{H}^{*}=0,125\ \delta_{H}\), \(\delta_{H}^{**}=0,097\ \delta_{H}\); after their determination, the characteristic thicknesses (compressible layer) are calculated using the following relations:  
\[ \begin{split}\delta_{\text{C,nc}}=\delta_{H}\Big{(}1+0.72r\frac{ \gamma-1}{2}M_{0}^{2}\Big{)}^{0.34},\ \delta_{\text{C,nc}}^{*}=\delta_{H}^{*}\Big{(}1+0.72r\frac{\gamma-1}{2}M_{0}^{2} \Big{)}^{0.34}\\ \delta_{\text{C,nc}}^{**}=\delta_{H}^{**}\Big{(}1+0.72r\frac{ \gamma-1}{2}M_{0}^{2}\Big{)}^{-0.66},\ r=\sqrt[3]{\text{Pr}_{0}}\end{split}