###### tags: `三上`
# 空氣動力學
# CH 2
## 2.14 Stream Function
- $\overline{\psi}\ (x,y)$ is called the stream function

$\overline{\psi}\ (x,y)=c$
where c is an arbitrary constant
- equation for a streamline is given by setting the stream function equal to a constant
- The equivalence between $\overline{\psi}\ (x,y)=constant$ designating a streamline, and $\Delta\overline{\psi}$ equaling mass flow between streamlines, is nature
- If we know $\overline{\psi}$, then we can obtain the product $(\rho\overrightarrow V)$ by differentiating $\overline{\psi}$ in the direfction normal to $\overrightarrow V$

$\rho \overrightarrow V=\displaystyle\lim_{\Delta n\rightarrow 0}\dfrac{\Delta \overline{\psi}}{\Delta n}\equiv\dfrac{\partial\overline{\psi}}{\partial n}$

$\begin{cases}
\rho\overrightarrow u=\dfrac{\partial \overline{\psi}}{\partial y}\\
\rho \overrightarrow v=-\dfrac{\partial\overline{\psi}}{\partial x}\\
\end{cases}$

## 2.15 Velocity Potential
By comparing the equations for an irrotational flow
$\zeta=\nabla\times\overrightarrow V=0$
the curl of the gradient of a scalar function is identically zero
$\nabla\times(\nabla\phi)=0$
we obtain
$\overrightarrow V=\nabla\phi$
$\phi$ is the velocity potential

- Thus in Cartesian CS.

$u=\dfrac{\partial\phi}{\partial x}\quad v=\dfrac{\partial\phi}{\partial y}\quad w=\dfrac{\partial\phi}{\partial z}$

- In Cylindrical CS.

$V_r=\dfrac{\partial\phi}{\partial r}\quad V_{\theta}=\dfrac{1}{r}\dfrac{\partial\phi}{\partial\theta}\quad V_z=\dfrac{\partial\phi}{\partial z}$

- In Spherical CS.

$V_r=\dfrac{\partial\phi}{\partial r}\quad V_{\theta}=\dfrac{1}{r}\dfrac{\partial\phi}{\partial\theta}\quad V_{\Phi}=\dfrac{1}{rsin\theta}\dfrac{\partial\phi}{\partial\Phi}$

- Difference between $\phi\ \ and\ \ \overline{\psi}\ (or\ \psi)$
- the velocity potential is defined for irrotational flow only
- The stream function is defined for two-dimensional flows only
# CH 3
## 3.2 Bernoulli' equation
- ==Euler's equation==

$dp=-\rho\overrightarrow Vd\overrightarrow V$

* invisicid flow with no body forces
* it relates the change in velocity along a streamline $d\overrightarrow V$
* the change in pressure $dp$ along the same streamline

- ==Bernoulli's equtation==

For incompressible flow, $\rho=constant$
integrate between any two points 1 and 2 along a streamline, we obtain Bernoulli's equation

$p_1+\dfrac{1}{2}\rho V_1^2=p_2+\dfrac{1}{2}\rho V_2^2\quad or\quad p+\dfrac{1}{2}\rho V^2=constant$

* It hold along a streamline in rotational/irrotational flow
* For rotational flow, the constante will change from on streamline to the next
* For irrotatinal flow, then Bernoulli' s equation holds for between any two points in the flow

:::info
Bernoulli's equation is a statement of Newton's second law for an inviscid, incompressibleflow with no body force
:::
## 3.5 Pressure Coefficient
$Pressure\ \ coefficient\ \ \ C_p\equiv\dfrac{p-p_{\infty}}{q_{\infty}}\qquad where\ \ q_{\infty}=\dfrac{1}{2}\rho_{\infty}V^2_{\infty}$
- For ==incompreesible flow==, $C_p$ can be expreesed in terms of velocity only by Bernoulli's equation

$C_p=1-(\dfrac{V}{V_{\infty}})^2$

- At stagnation point (where V=0), $C_p$ is always equal to 1.0
- this is the highest allowable value of $C_p$ anywhere in the flow field
- $C_p$ can be negative (in the flow field where $V>V_{\infty}\ \ or \ \ p<p_{\infty}$)
- By rearranging,

$p=p_{\infty}+q_{\infty}C_p$

the value of $C_p$ is how much $p$ differs from $p_{\infty}$

- $C_p$ should depend only on the Mach number, Reynolds number, shape and orientation of the body, and location of the body

## 3.9 Uniform Flow
- The velocity potential for a uniform flow with velocity $V_{\infty}$ oriented in the positive x direction

$\phi =V_{\infty}x$

- Note it applies to any uniform flow, compressible or incompressible

- Stream Function

$\psi =V_{\infty}y$

- it is the stream function for an incompressible uniform flow oriented in the positive x direction
- the streamline is given by $\psi = constant$
and because $V_{\infty}$, then $y=constant$
therefore the streamlines are horizontal lines
and the equipotential lines are lines of contant x
- In polar CS.

where $x=rcos\theta\quad y=rsin\theta$
we obtain

$\phi =V_{\infty}rcos\theta\\
\psi =V_{\infty}rsin\theta$

- Circulation

the circulation around any close curve in a uniform flow is 0

$\Gamma=-\oint_C V_{\infty} \cdot ds=-V_{\infty}\cdot\oint_C ds=V_{\infty}\cdot 0=0$

## 3.10 Source Flow
### Assumptions
- Consider a two-dimensional, incompressible flow where all the streamlines are straight emanating from a central point O
- let the velocity along each of the streamlines vary inversly with distance from point O


### Known
- $V_{\theta}=0$
- sourve flow is a physically possible incompressible flow, that is $\nabla\cdot V=0$, at every point except the origin
- source flow is irrotational at every point
- $V_r=\dfrac{c}{r}\qquad V_{\theta}=0\quad$where c is constant

### Analysis
- The elemental mass flow across the surface element $dS$ is
$\rho\ V\cdot dS=\rho\ V_r\ (r\ d\theta)\ (l)$

for $V_r$ is the same value at any $\theta$ location for the fixed radius r, the total mass flow across the surface of the cylinder is

$\dot m=\int_0^{2\pi}\rho\ V_r\ (r\ d\theta)\ l=\rho\ r\ l\ V_r\ \int_0^{2\pi}d\theta=2\pi\ rl\rho V_r$

the rate of volume flow $\dot v=\dfrac{\dot m}{\rho}=2\pi\ rlV_r$

- Denote volume flow rate per unit length as

$\Lambda=\dfrac{\dot v}{l}=2\pi\ rV_r\quad or\quad V_r=\dfrac{\Lambda}{2\pi\ r}$

and $\Lambda$ defines the ==source strength==
- positive value represents a **source**
- negative value represents a **sink**

- The ==velocity potential== for 2-D source flow

by integrating the velocity

$\phi=\dfrac{\Lambda}{2\pi}ln\ r$

- The ==stream function== for 2-D source flow

$\psi=\dfrac{\Lambda}{2\pi}\ \theta$

- The stream lines

setting stream function equal to constant

$\psi=\dfrac{\Lambda}{2\pi}\ \theta=constant$

we get $\theta =constant\quad$
in polar CS. is the equation of a straight line from the origin

## 3.12 Doublet Flow
- the strength of the doublet is denoeted by $\kappa$ is defined as

$\kappa\equiv l\Lambda$

as the distance $l$ approach 0 and the product $l\ \Lambda$ remains constant
let r and b denote the distances to point P from the source and sink
$a=l\ sin\theta\quad b=r-l\ cos\theta$
$d\theta=\dfrac{a}{b}\rightarrow\dfrac{l\ sin\theta}{r-l\
cos\theta}$

- the stream function for a doublet

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

- the velocity potential for a doublet

$\phi=\dfrac{\kappa}{2\pi}\dfrac{cos\theta}{r}$

- streamlines of a doublet flow

$\psi=-\dfrac{\kappa}{2\pi}\dfrac{sin\theta}{r}=constant=c$

or $\quad r=-\dfrac{\kappa}{2\pi c}\ sin\theta$

in polar CS. $r=d\ sin\theta\quad$, where d is the diameter
the streamline for a doublet are a family fo circles with diameter $\dfrac{\kappa}{2\pi c}$

- By convention, we designate the direction of the doublet by an arrow drawn from the sink to the source


## 3.13 Nonlifting Flow over a Circular Cylinder
The combination of a uniform flow and a doublet produces the flow over a circular cylinder


let $R^2\equiv\dfrac{\kappa}{2\pi V_{\infty}}$
and we get the stream function

$\psi =(V_{\infty}\ r\ sin\theta)(1-\dfrac{R^2}{r^2})$

this stream function is also the stream function for the flow over a circular cylinder of radius R

- Obtain the velocity by differentiating stream function

$V_r=\dfrac{1}{r}\dfrac{\partial \psi}{\partial \theta}=(1-\dfrac{R^2}{r^2})V_{\infty}\ cos\theta$

$V_{\theta}=-\dfrac{\partial\psi}{\partial r}=-(1+\dfrac{R^2}{r^2})V_{\infty}\ sin\theta$

there are two stagnation points, $(r,\theta)=(R,0)\ \ and \ \ (R,\pi)$
the streamline $\psi=0$ goes through the stagnation points is the ==dividing streamline==

- As the streamline pattern shown
the flow field is symmetrical about both the horizontal and vertiacl axes through the center of the cylinder, hence the pressure distribution is also symmetrical about both axes

the surface pressure coefficient over a cylinder is

$C_p=1-4\ sin^2\theta$