# Estimation of Aircraft-Dependent Bumpiness Severity in Turbulent Flight
## 2 Methodology
### Vertical Acceleration Response by UVLM
At each time step \(t_{k}\), the local velocity induced by unsteady motion at an arbitrary point can be divided by the far-field free flow velocity, the effects of aircraft angular motion, and instantaneous turbulence. As a result, the Neumann boundary condition at any collocation points is described as  
\[(\sum_{j=1}^{N}\mathbf{K}_{i}^{j}\Gamma_{j}(t_{k},))\cdot\mathbf{n}_{i}+\left[V_{W_{x}, t}(t_{k}),V_{W_{z},i}(t_{k})\right]^{\mathrm{T}}\cdot\mathbf{n}_{i}+[\,V_{x,i}(t_{k}),V_{ z,i}(t_{k})]^{\mathrm{T}}\cdot\mathbf{n}_{i}=0 \tag{9}\]  
An in-depth analysis of the boundary condition is shown in Figure 5, which shows the decomposition of boundary conditions along the wing chord. From Figure 5, the boundary condition at the collocation point is expressed as  
\[ \begin{array}{ll}V_{\infty}sin(\alpha+\alpha_{i}-\delta)&+(ry_{i}-qz_{i})sin (\alpha_{i}-\delta)+(qx_{i}-py_{i})cos(\alpha_{i}-\delta)\\ &=(W_{x}+u)sin(\alpha+\alpha_{i}-\delta)+(w+W_{z})cos(\alpha+\alpha_{i}-\delta )\end{array}  \tag{10}\]  
where \(V_{\infty}\) represents the far-field free flow, \(\delta\) stands for the direction of angle of attack due to control surface deflection. Besides, \(\alpha_{i}=tan^{-1}(-\frac{dz}{dx})\) shows the camber effects. \(\left[u,w\right]^{\mathrm{T}}\) represents the induced velocity decomposition along each axis, and \(\left[x,y,z\right]^{\mathrm{T}}\) stands for the coordinate of arbitrary point. A linear algebraic equation is derived by further expanding Equation (9). The circulation of each vortex ring is obtained by \(\Gamma_{1}(t_{k}),\Gamma_{2}(t_{k}),\dots,\Gamma_{N}(t_{k})\). In turbulent flight, in addition to the far-field free flow \(\mathbf{V}_{\infty}(t_{k})=\left[V_{\infty x}(t_{k}),V_{\infty y}(t_{k}),V_{ \infty z}(t_{k})\right]^{\mathrm{T}}\) and aircraft angular motion \([p(t_{k}),q(t_{k}),r(t_{k})]^{\mathrm{T}}\), the instantaneous turbulence \(\mathbf{W}(t_{k})=\left[W_{x}(t_{k}),0,W_{z}(t_{k})\right]^{\mathrm{T}}\) has major effects on local velocity. The local velocity of arbitrary point at any time step \(t_{k}\) is described by  
\[\left[ \begin{array}{c}V_{x}(t_{k})\\ V_{y}(t_{k})\\ V_{z}(t_{k})\end{array} \right]=\left[ \begin{array}{c}V_{\infty x}(t_{k})cos (\alpha_{k})cos\beta(t_{k})\\ -V_{\infty y}(t_{k})sin\beta(t_{k})\\ V_{\infty z}(t_{k})sin\alpha(t_{k})cos\beta(t_{k})\end{array} \right]+\left[ \begin{array}{c}W_{x}(t_{k})\\ 0\\ W_{z}(t_{k})\end{array} \right] \tag{11}\]  
The aerodynamic force is further computed by Kutta-Joukowski lift theorem based on the circulation of each vortex ring. Taking the right half wing as an example, any vortex and its neighboring vortices are superposed on the lattice panel to obtain total circulation distribution. As shown in Figure 6, the aerodynamic force at the collocation point is:  
Figure 5: Boundary condition decomposition in turbulence. \[ \begin{array}{l}\mathbf{F}_{i}(t_{k})=\rho\Big{(}\Gamma_{(n_{RW},n_{CW})}(t_{k})- \Gamma_{(n_{RW},n_{CW})}(t_{k})\Big{)}\big{(}\mathbf{V}_{b,i}(t_{k})+\big{[}V_{x,i}( t_{k}),V_{z,i}(t_{k})\big{]}^{T}+[V_{W_{x,i}}(t_{k}),V_{W_{y,i}}(t_{k})]^{T}\Big{)} \times\mathbf{r}_{AB}\\ +\rho\Big{(}\Gamma_{(n_{RW},n_{CW})}(t_{k})-\Gamma_{(n_{RW},n_{CW+1})}(t_{k}) \Big{)}\big{(}\mathbf{V}_{b,i}(t_{k})+\big{[}V_{x,i}(t_{k}),V_{z,i}(t_{k})\big{]}^{ T}+[V_{W_{x,i}}(t_{k}),V_{W_{z,i}}(t_{k})]^{T}\Big{)}\times\mathbf{r}_{BC}\\ +\rho\Big{(}\Gamma_{(n_{RW},n_{CW})}(t_{k})-\Gamma_{(n_{RW},n_{CW+1})}(t_{k}) \Big{)}\big{(}\mathbf{V}_{b,i}(t_{k})+\big{[}V_{x,i}(t_{k}),V_{z,i}(t_{k})\big{]}^{ T}+[V_{W_{x,i}}(t_{k}),V_{W_{z,i}}(t_{k})]^{T}\Big{)}\times\mathbf{r}_{CD}\end{array}  \tag{12}\]  
where \(\rho\) is air density, \(\mathbf{V}_{b,i}\) is the total induced velocity of airfoil-attached vortex at the collocation point. In order to differentiate the aerodynamic force induced by aircraft maneuvers and external turbulence, Equation (12) is solved twice, with or without turbulence effects, respectively, which are recorded as \(\mathbf{F}_{i}\) and \(\mathbf{F}_{i}^{0}\).