# Calculation of Heat Transfer and Drag Coefficients for Aircraft Geometric Models
## 4 Results and Discussion  
Only the main points of the computational algorithm for calculating aerodynamic heating and the friction coefficient are given here. A more detailed description can be found in the references [3; 14]. In this work, a calculation technology based on an unstructured surface mesh was used to estimate heat fluxes and drag coefficients [22; 23; 24; 25]. The calculation of the integrals included in expressions (1)-(16) requires finding the metric coefficients (local mean radius of curvature) included in the integrand. In this case, when the aircraft surface is locally specified, the calculation of the local mean radius of curvature \(R_{ef}\) is based on the following symmetric form [19]:  
\[K=\frac{1}{2}\frac{\big{(}1+f_{u}^{2}\big{)}f_{vv}-2f_{u}f_{v}f_{uv}+\big{(}1 +f_{v}^{2}\big{)}f_{uu}}{\big{(}1+f_{u}^{2}+f_{v}^{2}\big{)}^{3/2}},\;R_{ef}= \frac{1}{K} \tag{16}\]  
The partial derivatives \(f_{u}f_{v}f_{uv}f_{uu},f_{vv}\) included in this relationship are:  
\[ \begin{array}{l}f\big{(}u,v\big{)}\approx f\big{(}u_{j},v_{j}\big{)}+f_{u} \big{(}u-u_{j}\big{)}+f_{v}\big{(}v-v_{j}\big{)}+\\ +f_{uu}\frac{\big{(}u-u_{j}\big{)}^{2}}{2}+f_{uv}\big{(}u-u_{j}\big{)}\big{(}v -v_{j}\big{)}+f_{vv}\frac{\big{(}v-v_{j}\big{)}^{2}}{2}.\end{array}  \tag{17}\]  
The calculation of an external inviscid flow (based on the Euler equations) near the surface of a streamlined body was performed using unstructured grids and the computational code from [17]. It is also noted that in the practical use of the developed methodology, it is necessary to be able to calculate the integrals of the form with the required degree of accuracy (above 2 nd): \(\int\limits_{0}^{S}\rho^{*}\mu^{*}|V|r^{2}\text{d}s\)--for the laminar flow; \(\int\limits_{0}^{S}\rho^{*}(\mu^{*})^{m}|V|r^{c_{3}}\text{d}s\), for the turbulent flow, where \(s\) is the distance along the streamline in the direction of flow on the surface of the streamlined body; \(r\) is the distance (for the axisymmetric case) from the axis \(\text{O}x\) to a point \(s(x,y,z)\) on the surface of the streamlined body \(F(x,y,z)\), or the metric coefficient (the local average curvature radius of the HSA surface \(R_{ef}\)[19]). For the purposes of 2D validation and the verification of the mathematical model of heat transfer in a high-velocity flow around blunt axisymmetric bodies, the calculated and experimental data given in [18] were used. According to the graphical dependences of work [3], the following error estimate for the 2D calculation of the convective heat flux \(q_{uv}\) can be made: 20-37%. Testing the numerical method for solving 2D Euler equations is based on the calculation and comparison of vector and scalar fields of gas-dynamic parameters, the estimation of the shock wave receding \(\Delta\), and the value of the radius of curvature of the shock wave \(R_{s}\) on the axis of symmetry for a sphere or cylinder [3; 26]. This testing showed that the gas-dynamic characteristics error is 5%; the error in the retraction \(\Delta\) and the curvature of the shock wave \(R_{s}\) is 9% and 8.5%, respectively. A comparison between the proposed mathematical model, experiments, and calculations [6; 27; 28; 29] was made for the geometric model HSA X-33 (geometric scaling factor is 0.0132), as well as for the model created by Lockheed Martin for the aircraft demonstrator X-33 single-stage-to-orbit (SSTO). The studies were carried out for different attack angles [26; 30; 31; 32]. In addition, a comparison was made for the HSA X-33 model [33]. The following parameters of the air flowing onto the vehicle were used (Table 1). In Table 2, the aerodynamic lift coefficients \(C_{y\alpha}\), drag forces \(C_{xx\alpha}\), as well as the aerodynamic quality \(K=C_{y\alpha}/C_{xx}\) are shown for the X-33. Amendments have been introduced in the far right of the column which allows the evaluation of the accuracy of the performed calculations of the aerodynamic coefficients. In Table 3 similar aerodynamic coefficients are presented for the X-43 vehicle.