## 2 Vertical Acceleration and Turbulence Severity Estimation
### Vertical Acceleration Computation in Turbulent Flight
#### 2.3.2 Unsteady Aerodynamics in Turbulence  
At any collocation point, the boundary condition of zero normal flow through the airfoil's surface is satisfied, which can be described by Equation (5):  
\[\
abla\Phi\cdot\mathbf{n}=0 \tag{5}\]  
where \(\Phi\) is the potential function of the flow field. Velocity components at any collocation point are described as \(\
abla\Phi=\frac{\partial\Phi}{\partial x}i+\frac{\partial\Phi}{\partial y}j +\frac{\partial\Phi}{\partial z}\mathbf{k}\). At any time \(t_{k}\), as shown in Figure 3, the aircraft moves along the flight path, vortices on the trailing edge fall off at local velocity, generating a series of wake vortices. Local velocity at the trailing edge is defined by  
\[ \begin{array}{rl}\mathbf{V}_{m}(t_{k-1})=&\left(\sum\limits_{j}^{N}\mathbf{K}_{m}^{ j}\Gamma_{j}(t_{k-1})\right)+\left[V_{x,m}(t_{k-1}),V_{y,m}(t_{k-1}),V_{z,m}(t_{k-1}) \right]^{T}\\ &+[V_{W_{x},m}(t_{k-1}),V_{W_{y},m}(t_{k-1}),V_{W_{z},m}(t_{k-1})]^{T}\end{array}  \tag{6}\]  
where \(\sum\limits_{j}^{N}\mathbf{K}_{m}^{j}\Gamma_{j}(t_{k-1})\) is the induced velocity of airfoil-attached vortices at trailing edge points \(m\) at time \(t_{k-1}\). \(\left[V_{W_{x},m}(t_{k-1}),V_{W_{y},m}(t_{k-1}),V_{W_{z},m}(t_{k-1})\right]^{T}\) is the induced velocity of wake vortices. The trailing edge continues to move forward with the aircraft and the corner points determined before each time step move forward along the airflow. By determining a new couple of corner points corresponding to the trailing edge points, a new wake vortex ring is constructed. The corner points determined at \(t_{k-1}\) move forward by \(\mathbf{V}_{m}(t_{k-1})\Delta t\). This is repeated at each time step to establish a series of wake vortices. The latest wake vortex ring has the same strength as the trailing edge vortex at \(t_{k}\). In other word,it is not affected by aerodynamic force. The wake vortices move at local velocity, which is achieved by modifying the positions of the corner point \(n\) at each time step:  
\[ \begin{split}(\Delta x,\Delta y,\Delta z)_{n}^{T}=(\sum\limits_{j=1 }^{N}\mathbf{K}_{n}^{j}\Gamma_{j}(t_{k-1}))\Delta t+\\ [V_{W_{x},n}(t_{k-1}),V_{W_{y},n}(t_{k-1}),V_{W_{z},n}(t_{k-1}) ]^{T}\Delta t+[W_{x}(t_{k-1}),W_{y}(t_{k-1}),W_{z}(t_{k-1})]^{T}\Delta t\end{split}  \tag{7}\]  
As a result, the induced velocity of wake vortices at collocation points at \(t_{k}\) is obtained. The boundary condition of zero normal flow is adjusted by  
\[(\sum\limits_{j=1}^{N}\mathbf{K}_{n}^{j}\Gamma_{j}(t_{k}))\cdot\mathbf{n}_{i}+[V_{ x,i}(t_{k}),V_{y,i}(t_{k}),V_{z,i}(t_{k})]^{T}\cdot\mathbf{n}_{i}+[V_{W_{x},i}(t_{k}),V_ {W_{y},i}(t_{k}),V_{W_{z},i}(t_{k})]^{T}\cdot\mathbf{n}_{i}=0 \tag{8}\]  
By further expanding the above equation, the following linear algebraic equation is obtained:  
\[ \begin{split}\left[\begin{array}{cccc}\mathbf{K}_{1}^{1}\mathbf{n}_ {1}&K_{1}^{2}\mathbf{n}_{1}&...&\mathbf{K}_{1}^{N}\mathbf{n}_{1}\\ \mathbf{K}_{2}^{1}\mathbf{n}_{2}&\mathbf{K}_{2}^{2}\mathbf{n}_{2}&...&\mathbf{K}_{2}^ {N}\mathbf{n}_{2}\\...&...&...&...\\ \mathbf{K}_{1}^{N}\mathbf{n}_{N}&K_{N}^{2}\mathbf{n}_{N}&...&\mathbf{K}_{N}^{N}\mathbf{n} _{N}\end{array}\right]+\\ \left[\begin{array}{cccc}\mathbf{V}_{1}(t_{k})&...&...&0\\ 0&\mathbf{V}_{2}(t_{k})&...&0\\...&...&...&...\\ 0&...&...&\mathbf{V}_{N}(t_{k})\end{array}\right]\left[\begin{array}{c}n_{1} \\ n_{2}\\...\\...\\ 0&...\\ \end{array}\right]+\left[\begin{array}{cccc}\mathbf{V}_{W1}(t_{k})&...&...&0\\ 0&\mathbf{V}_{W2}(t_{k})&...&0\\...&...&...&...\\ 0&...&...&\mathbf{V}_{WN}(t_{k})\end{array}\right]+\left[\begin{array}{c} \mathbf{n}_{1}\\ n_{2}\\...\\ \mathbf{n}_{N}\end{array}\right]=\mathbf{0}\end{split}  \tag{9}\]  
By solving Equation (9), the circulation of each vortex ring \(\Gamma_{1}(t_{k}),\Gamma_{2}(t_{k}),\ldots,\Gamma_{N}(t_{k})\) is acquired. It should be noted that there is no vortex ring at \(t=0\). At the first time step \(t_{1}\), the circulation of each vortex ring is also obtained without wake effects. Here is an analysis about the boundary condition on each vortex ring.