# Numerical Study of Aircraft Wake Vortex Evolution under the Influence of Vertical Winds
## 2 Materials and Methods
### SST k-\(\omega\)Model Governing Equations  
The wake vortex flow field is a low-speed flow problem in the aircraft approach phase. Hence, the control equations use the Navier-Stokes (N-S) equations for incompressible fluids with constant viscosity, including the mass conservation equation and momentum equation [27]. The RANS turbulence model is more computationally efficient and stable for complex geometries than DNS and LES, making it suitable for steady flows and practical engineering applications with limited computational resources. The SST k-\(\omega\) turbulence model has the advantages of high accuracy, high stability in high-Reynolds-number flows, and high computational efficiency, and the model is especially capable of capturing the details of near-wall turbulence, which is very important for the safety and performance of airplanes, and is often used in the RANS method for the study of aircraft wake vortices. Several research articles [11; 28; 29; 30] have discussed the advantages of the model in detail and justified the choice of the model for the study.  
Mass conservation equation:  
\[\
abla\mathrm{U}=0, \tag{1}\]  
Momentum equation:  
\[\mathrm{U}\cdot\
abla\mathrm{U}=-\frac{\
abla\mathrm{P}}{\rho}+\upsilon\
abla^ {2}\mathrm{U}+\mathrm{g}-\
abla\cdot(\overline{u^{\prime}u^{\prime}}), \tag{2}\]  
where \(U\) is the velocity vector, \(\rho\) is the fluid density, \(P\) is the pressure, \(\upsilon\) is the kinematic viscosity (kinetic viscosity divided by density), \(\mathrm{g}\) is the volumetric force (e.g., gravity), \(\overline{u^{\prime}u^{\prime}}\) is the Reynolds stress tensor, which is the second-order moment of the velocity rise and fall.  
The SST k-\(\omega\) model is chosen for the turbulence model. The turbulent kinetic energy k and specific dissipation rate \(\omega\) can be obtained from the following equations.  
\[\mathrm{U}\cdot\
abla\mathrm{k}=\mathrm{P_{k}}-\beta^{*}\mathrm{k}\omega+ \
abla\cdot\left[\left(\upsilon+\sigma_{\mathrm{k}}\frac{\mathrm{k}}{\omega} \right)\
abla\mathrm{k}\right], \tag{3}\]  
\[\mathrm{U}\cdot\
abla\omega=\alpha\frac{\mathrm{P_{k}}}{\upsilon_{\mathrm{t}}} -\beta\omega^{2}+\
abla\left[\left(\upsilon+\sigma_{\mathrm{\omega}}\frac{ \mathrm{k}}{\omega}\right)\
abla\omega\right]+2(1-\mathrm{F_{1}})\frac{\sigma \omega_{2}}{\omega}\
abla\mathrm{k}\
abla\omega, \tag{4}\]  
\(\mathrm{P_{k}}\) is the turbulence production term; \(\beta^{*},\ \alpha,\ \beta\) are the model constants, determined through calibration with empirical or experimental data; \(\sigma_{\mathrm{k}},\sigma_{\omega}\) are the turbulent Prandtl numbers for the diffusion terms; \(\upsilon\) is the molecular viscosity; \(\upsilon_{\mathrm{t}}\) is the turbulent viscosity, representing the viscous effects of turbulence; \(\mathrm{F_{1}}\) is a blending function; \(\sigma_{\omega 2}\) is the turbulent Prandtl number for the cross-diffusion term.  
Combining the N-S governing equations with the SST k-\(\omega\) turbulence model ensures the closure of the numerical computation process.  
The simulation employs a coupled algorithm and uses a second-order upwind scheme to discretize the governing equations. Convergence is checked by observing residuals, with further iterations added after criteria are met to ensure solution stability.