# 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  
In references [3; 14], a version of the effective length method is proposed, which allows the finding of the different coefficients. The area of applicability of the effective length method is of the flows at relatively large Reynolds numbers (Re > \(10^{4}\div 10^{5}\)). This approximate method is applicable only in flow regions with low-pressure gradients along the streamline and in the absence of separation zones. The effective length is determined on the HSA surface (here the symbol denotes the length of the curve along the streamline) as:  
\[\ell_{eff}=\frac{\int\limits_{0}^{x}\left[\Pi^{2}K^{2}K_{1}^{2}u_{0}\rho_{0}U _{0}C_{p,0}^{2}(T_{e}-T_{w})^{2}Pr_{0}^{-4/3}\right]\text{d}s}{\left[\Pi^{2}K ^{2}K_{1}^{2}u_{0}\rho_{0}U_{0}C_{p,0}^{2}(T_{e}-T_{w})^{2}Pr_{0}^{-4/3} \right]} \tag{1}\]  
Under the integral are the variables varying from the beginning of the formation of the boundary layer (the critical point is the stagnation point of the flow incident on the HSA) to the considered section. The integrand values are variables that vary from the beginning of the formation of the boundary layer (the critical point-braking point flux incident on the HSA) before the section under consideration, where \(\Pi=R_{ef}(s)\) is the metric coefficient and average radius of curvature (for a non-axisymmetric body), respectively, determined and serving to take into account the increase or decrease in the thickness of the boundary layer due to runoff or spreading of streamlines; \(\mu_{0},\rho_{0},U_{0},M_{0},Pr_{0}\)--dynamic viscosity coefficient, density, velocity, local Mach, and Prandtl number, respectively, taken at the outer edge of the boundary layer, at the point coordinate of the current line; \(T_{w},T_{e}\) are the body temperature and wall temperature, respectively:  
\[T_{e}=T_{0}\bigg{(}1+r\frac{\gamma-1}{2}M_{0}^{2}\bigg{)},\quad r=\sqrt{Pr_{0}} \tag{2}\]  
where \(r\) is the recovery factor. \(K_{1}\) and \(K\) (effect of compressibility) are introduced as:  
\[K_{1}=\bigg{[}1+0.16\bigg{(}1+\frac{T_{w}}{T_{0}^{*}}\bigg{)}\bigg{(}\frac{2m} {m+1}\bigg{)}^{1/3}\bigg{]}^{1/2} \tag{3}\]  
\[K=\bigg{(}\frac{\rho_{0}\mu_{0}}{\rho_{w}\mu_{w}}\bigg{)}^{1/3} \tag{4}\]  
The dimensionless velocity gradient can be expressed as:  
\[m=\frac{x}{V_{0}}\frac{\partial V_{0}}{\partial x} \tag{5}\]  
The Reynolds and Stanton numbers are determined as:  
\[St_{eff}=0.332(m+1)^{1/2}\text{Re}_{eff}^{1/2}\text{Pr}_{0}^{-2/3}K\cdot K_{1} \tag{6}\]  
\[\text{Re}_{eff}=\frac{\rho_{w}V_{0}\ell_{eff}}{\mu_{w}} \tag{7}\]  
Convective heat can be found as follows:  
\[q_{w,L}=\big{(}C_{p}\big{)}_{cp}^{*}\rho_{0}V_{0}(T_{e}-T_{w})St_{eff} \tag{8}\]  
The effective length for the turbulent regime can be obtained as follows:  
\[\ell_{eff}=\frac{\int\limits_{0}^{x}\bigg{[}\bigg{[}\Pi^{5/4}\Big{(}1+r\frac{ \gamma-1}{2}M_{0}^{2}\Big{)}^{0.1375}\rho_{0}U_{0}\text{Pr}_{0}^{-0.7125}\mu_ {0}^{0.25}C_{p,0}^{1.25}\bigg{]}\big{(}T_{w}/T_{e}\big{)}^{0.5}(T_{e}-T_{w})^{ 1.25}\bigg{]}\text{d}s}{\bigg{[}\bigg{[}\Pi^{5/4}\Big{(}1+r\frac{\gamma-1}{2}M _{0}^{2}\Big{)}^{0.1375}\rho_{0}U_{0}\text{Pr}_{0}^{-0.7125}\mu_{0}^{0.25}C_{p,0}^{1.25}\bigg{]}\big{(}T_{w}/T_{e}\big{)}^{0.5}(T_{e}-T_{w})^{1.25}\bigg{]}} \tag{9}\]  
where \(r=\sqrt[3]{\text{Pr}_{0}}\), and the heat flux is determined as follows:  
\[q_{w,L}=\big{(}C_{p}\big{)}_{cp}^{*}\rho_{0}V_{0}(T_{e}-T_{w})St_{eff},\text { Re}_{eff}=\frac{\rho_{w}V_{0}\ell_{eff}}{\mu_{w}} \tag{10}\]  
In Equations (1) and (9), the parameter \(\Pi\), in the case of plane flow, \(\Pi=R_{ef}=1\), and for the cylindrical case in Equation (1), \(\Pi^{2}=R_{ef}^{2}\), and in Equation (9), \(\Pi^{5/4}=R_{ef}^{5/4}\). The Stanton number is written as follows:  
\[T_{e}=T_{0}\bigg{(}1+r\frac{\gamma-1}{2}M_{0}^{2}\bigg{)},\quad r=\sqrt{Pr_{0}} \tag{2}\]0  
\(\text{Re}_{eff}=10^{5}\div 10^{6}\) or our flow conditions [15].