#fluids
# lecture 1

## symbol changes
up until now, $x$ is the streamwise direction, $y$ is the vertical direction and $z$ is the spanwise direction

from here on, $x$ is the streamwise direction, $z$ is the vertical direction and $y$ is the spanwise direction
$x$ velocity is $u$, $z$ velocity is $w$

### new euler equations
$\frac{\partial u}{\partial x} + \frac{\partial w}{\partial x}=0$
$u \frac{\partial u}{\partial x} + w \frac{\partial u}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial x}$
$u \frac{\partial w}{\partial x} + w \frac{\partial w}{\partial z} = -\frac{1}{\rho} \frac{\partial p}{\partial y}$

### new curl
vorticity is not positive in the clockwise direction and negative in the counterclockwise direction
$\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x}$

### stream functions
$u = \frac{\partial \psi}{\partial z}, w = -\frac{\partial \psi}{\partial x}$
**way more on slide write them up**

## laplace equations

you can add 2 stream functions together to get another stream function
![[Pasted image 20221102102133.png]]
### foundational solutions to laplace equations
these are what is used to design airfoils and wings
they can be added together as above to find geometry

**coefficient of pressure**
coefficient of pressure is given by $C_p = 1-\frac{V_r^2 + V_\theta^2}{U_\infty^2}$ or $C_p = 1-\frac{u^2 + w^2}{U_\infty^2}$
if there is no freestream velocity(only a source/sink/vortex), there is a reference pressure and velocity. If these are 0, you can only get pressure rather than pressure coefficient

for all of them other than uniform flow, 
$\theta = \tan^{-1} \frac{x-x_0}{z-z_0}$
$r = \sqrt{(x-x_0)^2 + (z-z_0)^2}$
this is because all of the flow is relative to the point

#### uniform flow
$u=U_\infty = \frac{\partial\psi}{\partial z}$
$w=0=\frac{\partial\psi}{\partial x}$

integrate these to get the stream function $\psi(z) = U_\infty z$
the different streamlines all have different values

the velocity potential is then $\phi(x) = U_\infty x$
this makes sense because stream functions an velocity potentials are perpendicular

##### uniform flow at an angle
$u = U_\infty \cos \alpha$
$w = U_\infty \sin \alpha$
using the same approach as above:
$\psi(x, z) = U_\infty (z\cos \alpha - x \sin \alpha)$
$\phi(x, z) = U_\infty (x\cos \alpha + z \sin \alpha)$

in polar coordinates:
$\psi(r, \theta) = U_\infty (r\sin\theta \cos \alpha - r \cos \theta \sin \alpha)$


#### line source/sink
these are 2D flows where all the streamlines are straight and eminate from a single point.
For a source, all the streamlines point out. for a sink, they all point in

$V_r = f(r)$
$V_\theta = 0$
the speed only depends on the distance from the point

in 3D, $\dot q = \int^{2\pi}{0} = V_r l r d\theta$ so $2 \pi r lV_r$
in 2D, mass flow rate per length is $\Lambda \equiv \frac{\dot q}{l} = 2 \pi r V_r$, so $V_r = \frac{\Lambda}{2 \pi r}$
$\Lambda$ is knows as the strength of the flow. the point is a source when $\Lambda > 0$

**for velocity potential:**
$\phi = \frac{\Lambda}{2\pi} \ln r$

**for stream function**
$\psi = \frac{\Lambda}{2 \pi} \theta$

#### doublet
these have a source and sink of equal strength
the stream functions can be added together to give $\psi = \frac{\Lambda}{2 \pi}(\theta_1 - \theta_2)$
![[Pasted image 20221103160449.png|200]]
$\theta_1 = \arctan\frac{z}{x+b}$ , $\theta_2 = \arctan\frac{z}{x-b}$ 

when they are moved together with $\Lambda \times l$ constant, the strength is now called $\kappa$

$\psi = - \frac{\kappa}{2\pi} \frac{\sin \theta}{r}$

#### vortex
although potential flow has to be irrotational, vorticies can be used
##### circulation
$\Gamma = \int_c \bf V \cdot d\bf s$
integral of tangential velocity
it is also the integral of curl in the area
it is 0 if the flow is irrotational
to keep the flow irrotational, circulation is taken from all over the flow to at just one point.

$V_\theta = \frac{\text{constant}}{r}$
from circulation $\Gamma = -2\pi c$
$V_\theta = -\frac{\Gamma}{2\pi r}$

$\phi = -\frac{\Gamma}{2\pi}\theta$
$\psi = \frac{\Gamma}{2\pi} \ln r$