# Estimation of Aircraft-Dependent Bumpiness Severity in Turbulent Flight
## 2 Methodology
### Vertical Acceleration Response by UVLM  
In this paper, aircraft vertical acceleration response excited by turbulence is computed via non-planar Unsteady Vortex Lattice Method (UVLM). The Laplace's equation governs the principle of irrotational, inviscid, and incompressible air flow [15]. Based on linear Laplace's equation, the flow around complex geometries can be solved by superposing elementary solutions. To improve the computing accuracy, the vortex rings, each composed of four vortex filaments, are placed on the mean camber surface of the lifting body. Compared with the horseshoe vortices assigned on the flat surface of the lifting body, the accuracy can be improved by assigning vortex rings. In high-altitude turbulent flight, the longitudinal and vertical turbulence components make the longitudinal aerodynamic force change rapidly, further leading to the fluctuation of vertical acceleration. In order to obtain the longitudinal aerodynamic performance with better computing efficiency, target aircraft is simplified to a lifting body of wing-horizontal tail assembly. As shown in Figure 1, vortex rings are assigned on the whole reference area of the assembly and distributed on the mean camber surface. The wing-tail assembly is further divided by dozens of vortex rings. Figure 2 shows the detailed grid arrangement and geometric parameters on the right-hand side. Semi-circle division method is used to design the neighboring lattices at the wingtip, wing root, and leading edge because the circulation distribution changes rapidly there [34]. In addition, the boundaries of each vortex ring are close to the geometric edge of control surfaces since elevators and spoilers can be deflected by pilot's manual operation or Automatic Flight Control System (AFCS). Figure 1: Vortex rings on the wing-tail assembly. Figure 2: Grid design on the wing-tail assembly. The arrangement of vortex ring is illustrated in Figure 3. According to the vortex lattice method, the leading edge of the vortex ring is placed at the 1/4 chord of the panel, while the Neumann boundary condition of no penetration is enforced by locating the collocation point at the 3/4 chord. In this way, the lift curve slope of the lifting body conforms to the thin airfoil theory [35]. As shown in Figure 3, the induced velocity of the vortex ring is described as the vector sum of the induced velocity superposed by four vortex filaments,  
\[\mathbf{V}=\Gamma(\mathbf{k}_{AB}+\mathbf{k}_{BC}+\mathbf{k}_{CD}+\mathbf{k}_{DA})=\Gamma\mathbf{K} \tag{7}\]  
where \(\mathbf{k}\) is the coefficient vector defined according to the Biot-Savart theorem [15]:  
\[\mathbf{V}=\frac{\Gamma}{4\pi}\frac{\mathbf{r}_{1}\times\mathbf{r}_{2}}{|\mathbf{r}_{1}\times \mathbf{r}_{2}|^{2}}\cdot\mathbf{r}_{0}\cdot\left(\frac{\mathbf{r}_{1}}{\mathbf{r}_{2}}-\frac{ \mathbf{r}_{2}}{\mathbf{r}_{1}}\right)=\Gamma\cdot\mathbf{k} \tag{8}\]  
where \(\Gamma\) is the strength of the vortex filament, and \(\mathbf{r}_{0}\) is the unit vector of the vortex filament. \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are the position vectors of both ends to the arbitrary point in space. A time-marching computing process is used to perform the unsteady simulation. At each time step \(t_{k}\), the newly generated vortex rings at the trailing edge of the lifting body are released into the wake. As shown in Figure 4, the wing-tail assembly releases force-free wakes with no aerodynamic loads. The circulation \(\Gamma\) of the newly generated wake panel stays unchanged for the remainder of the simulation according to [PERSON]'s theorem [15]. Figure 4: Unsteady motion of wake vortices. Figure 3: Vortex ring and induced velocity. Since the scale of the high-altitude turbulence is large enough compared to the aircraft, it is feasible to analyze the boundary condition and further compute the aerodynamic force by UVLM according to the change of turbulence at each time step.