# External velocities may include gusts, turbulence, etc...

### Equations from LDVM (JFM 2014) are modified here to add u_ext(t) and w_ext(t) - These are uniform over the airfoil chord

The vorticity distribution on the airfoil is taken to be:

\begin{equation}
\gamma(\theta,t)=2U\left[A_{0}(t)\frac{1+\cos \theta}{\sin \theta} + \sum_{n=1}^{\infty}A_{n}(t) \sin(n\theta)\right]
\end{equation}

where $\theta$ is a variable of transformation related to
the chordwise coordinate $x$ as,

\begin{equation}
x=\frac{c}{2}(1-\cos\theta)
\end{equation}

in which $A_{0}(t)$, $A_{1}(t)$, ..., $A_{n}(t)$ are the
time-dependent Fourier coefficients, $c$ is the airfoil chord, and $U$
is the component of the airfoil's velocity in the negative $X$
direction.

\begin{equation}
A_{0}(t) = -\frac{1}{\pi}
\displaystyle \int_{0}^{\pi}\frac{W(x,t)}{U}d \theta
\end{equation}
\begin{equation}
A_{n}(t) =
\frac{2}{\pi} \displaystyle \int_{0}^{\pi}\frac{W(x,t)}{U} \cos n
\theta d \theta
\end{equation}


The downwash with no external flow velocity is, 

\begin{equation}
W(x,t)\equiv \frac{\partial\phi_{B}}{\partial z}=\frac{\partial
  \eta}{\partial x}(U \cos \alpha+\dot{h} \sin \alpha+\frac{\partial
  \phi_{lev}}{\partial x} +\frac{\partial \phi_{tev}}{\partial x}) -
   U \sin \alpha -\dot{\alpha}(x-ac)
  +\dot{h} \cos \alpha -\frac{\partial \phi_{tev}}{\partial z}-
  \frac{\partial \phi_{lev}}{\partial z}
\end{equation}

The terms multiplied with $\partial \eta/\partial x$ are all velocities tangential to (along) the chord. The other terms are velocities normal to and entering the chord.  

The new downwash will be: 

\begin{equation}
W(x,t)\equiv \frac{\partial\phi_{B}}{\partial z}=\frac{\partial
  \eta}{\partial x}(U \cos \alpha+\dot{h} \sin \alpha+\frac{\partial
        \phi_{lev}}{\partial x} +\frac{\partial \phi_{tev}}{\partial x} + u_{ext}\cos \alpha -w_{ext}\sin\alpha) -
   U \sin \alpha -\dot{\alpha}(x-ac)
  +\dot{h} \cos \alpha -\frac{\partial \phi_{tev}}{\partial z}-
    \frac{\partial \phi_{lev}}{\partial z} - u_{ext} \sin \alpha - w_{ext} \cos \alpha
\end{equation}

### Now modify the pressure equations as well

From the unsteady Bernoulli's equation, we have that:

\begin{equation}
    \Delta p(x) = p_l(x)-p_u(x) 
= \rho \left(\frac{1}{2}(V_{t_u}^2-V_{t_l}^2)+
\left(\frac{\partial \phi}{\partial t}\right)_u-
\left(\frac{\partial \phi}{\partial t}\right)_l\right)
\end{equation}


The velocity potential is comprised of components from bound
vorticity, leading-edge vortices and trailing-edge vortices,

\begin{equation}
\phi=\phi_b+\phi_{lev}+\phi_{tev}
\end{equation}


and the tangential velocities on upper and lower surface are expresse\
d as:

\begin{equation}
V_{t_u} = U\cos \alpha +\dot{h} \sin \alpha +
           \left(\frac{\partial \phi_{lev}}{\partial x}\right)_u +
           \left(\frac{\partial \phi_{tev}}{\partial x}\right)_u +
           \left(\frac{\partial \phi_{b}}{\partial x}\right)_u 
\end{equation}
\begin{equation}
V_{t_l} = U\cos \alpha +\dot{h} \sin \alpha +
           \left(\frac{\partial \phi_{lev}}{\partial x}\right)_l +
           \left(\frac{\partial \phi_{tev}}{\partial x}\right)_l +
           \left(\frac{\partial \phi_{b}}{\partial x}\right)_l
\end{equation}


These expressions are now modified to:

\begin{equation}
    V_{t_u} = U\cos \alpha +\dot{h} \sin \alpha + u_{ext}\cos \alpha - w_{ext}\sin \alpha +
           \left(\frac{\partial \phi_{lev}}{\partial x}\right)_u +
           \left(\frac{\partial \phi_{tev}}{\partial x}\right)_u +
           \left(\frac{\partial \phi_{b}}{\partial x}\right)_u 
\end{equation}

\begin{equation}
V_{t_l} = U\cos \alpha +\dot{h} \sin \alpha + u_{ext}\cos \alpha - w_{ext}\sin \alpha + 
           \left(\frac{\partial \phi_{lev}}{\partial x}\right)_l +
           \left(\frac{\partial \phi_{tev}}{\partial x}\right)_l +
           \left(\frac{\partial \phi_{b}}{\partial x}\right)_l
\end{equation}



Now, 

\begin{equation}
\left(\frac{\partial \phi_b}{\partial x}\right)_u = \frac{\gamma(x)}{2} 
\end{equation}
\begin{equation}
\left(\frac{\partial \phi_b}{\partial x}\right)_l = -\frac{\gamma(x)}{2}
\end{equation}

\begin{equation}
V_{t_u}^2-V_{t_l}^2 = 2\left(U\cos \alpha +\dot{h}\sin \alpha + u_{ext}\cos \alpha - w_{ext}\sin \alpha
+\left(\frac{\partial \phi_{lev}}{\partial
x}\right)+\left(\frac{\partial \phi_{tev}}{\partial
x}\right)\right)\gamma(x)
\end{equation}


\begin{eqnarray}
\phi_u&=&\int_0^x\frac{\gamma(x)}{2}dx+
         \int_0^x\left(\frac{\partial \phi_{lev}}{\partial x}\right)dx+
         \int_0^x\left(\frac{\partial \phi_{tev}}{\partial x}\right)dx \\
\phi_l&=&-\int_0^x\frac{\gamma(x)}{2}dx +
         \int_0^x\left(\frac{\partial \phi_{lev}}{\partial x}\right)dx+
         \int_0^x\left(\frac{\partial \phi_{tev}}{\partial x}\right)dx
\end{eqnarray}

\begin{equation}
\left(\frac{\partial \phi}{\partial t}\right)_u-
\left(\frac{\partial \phi}{\partial t}\right)_l=
\frac{\partial}{\partial t}\int_0^x \gamma(x)dx
\end{equation}

\begin{equation}
  \Delta p(x)=\rho\left[ \left (U \cos \alpha+\dot{h}\sin \alpha + u_{ext}\cos \alpha - w_{ext}\sin \alpha + 
              \left(\frac{\partial \phi_{lev}}{\partial x}\right)+
              \left(\frac{\partial \phi_{tev}}{\partial x}\right)\right)\gamma(\
x) +
              \frac{\partial}{\partial t} \int_0^x \gamma(x)dx \right]
\end{equation}

The normal force on the airfoil is obtained by integrating
the pressure coefficient over the chord,

\begin{equation}
F_N = \rho \left [\int_{0}^{c} \left( U\cos \alpha +
       \dot{h} \sin \alpha + u_{ext}\cos \alpha - w_{ext}\sin \alpha + \left(\frac{\partial \phi_{lev}}{\partial x}\right) +
       \left (\frac{\partial \phi_{tev}}{\partial x}\right)\right)\gamma(x,t)dx + \int_{0}^{c} \frac{\partial}{\partial t} \int_{0}^{x}
\gamma(x_{0},t)dx_{0}dx \right]
\end{equation}



\begin{equation}
 F_N = \rho \pi cU \left[\left(U\cos \alpha +\dot{h}\sin \alpha + u_{ext}\cos\alpha - w_{ext}\sin\alpha \right)\left(A_{0}(t)+
\frac{1}{2}A_{1}(t)\right) + c \left(\frac{3}{4}\dot{A_{0}}(t)+
\frac{1}{4}\dot{A_{1}}(t)+\frac{1}{8}\dot{A_{2} }(t)\right)\right] \rho \int_{0}^{c}\left(\left(\frac{\partial \phi_{lev}}{\partial x}\right) +
\left(\frac{\partial \phi_{tev}}{\partial x}\right) \right)\gamma(x,t)dx
\end{equation}


\begin{equation}
F_S=\rho \pi c U^{2} A_{0}^{2}
\end{equation}


The moment on the airfoil,

\begin{equation}
M = \int_{0}^{c} \Delta p (x_{ref}-x)dx \\
= x_{ref} F_{N} - \rho \pi c^2 U \left[\left( U \cos \alpha +
\dot{h} \sin \alpha + u_{ext}\cos\alpha - w_{ext}\sin\alpha \right)\left(\frac{1}{4}A_{0}(t)+\frac{1}{4}A_{1}(t)-
\frac{1}{8}A_{2}(t) \right)
 + c \left(\frac{7}{16}\dot{A_{0}}(t)+\frac{11}{64}\dot{A_{1}}(t) +
\frac{1}{16} \dot{A_{2}}(t)-\frac{1}{64}\dot{A_{3}}(t)\right)\right]
-\rho \int_{0}^{c} \left(\left(\frac{\partial \phi_{lev}}{\partial x}\right)+
\left(\frac{\partial \phi_{lev}}{\partial x}\right)\right)\gamma(x,t)xdx
\end{equation}



```julia

```

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ipynb
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Notebooks/Accounting for external velocities in UNSflow.ipynb
Mahziron12/TEST1
dd1a9fa2605540fc7fa6625db886ea780e17bede
MIT
Notebooks/Accounting for external velocities in UNSflow.ipynb
Mahziron12/TEST1
dd1a9fa2605540fc7fa6625db886ea780e17bede
MIT
Notebooks/Accounting for external velocities in UNSflow.ipynb
Mahziron12/TEST1
dd1a9fa2605540fc7fa6625db886ea780e17bede
MIT
Qwen/Qwen-72B
1. YES
2. YES