114 年 國立臺灣大學應用力學研究所乙組《流體力學（A）》

1. Water enters a rotating lawn sprinkler through its base at the steady rate of 1000 ml/s as sketched in the following figure. The exit area of each of the two nozzles is 30 mm². Determine the average speed of the water (m/s) leaving the nozzle, relative to the nozzle, if

(a) (15%) the rotary sprinkler head is stationary,
(b) (5%) the sprinkler head rotates at 600 rpm, and
(c) (5%) the sprinkler head accelerates from 0 to 600 rpm.

🖼️【此處有附圖，請對照原卷】The figure shows a rotating lawn sprinkler. Water enters from the base (Section 1) with a flow rate $Q = 1000$ ml/s. The sprinkler head has two nozzles, each with an exit area $A_2 = 30$ mm². Section (3) is the control volume. $W_2$ is the velocity of the water leaving the nozzle relative to the nozzle. The sprinkler head rotates.

2. The two-dimensional flow of a non-viscous, incompressible fluid in the vicinity of the 90° corner as shown in the following figure is described by the stream function $\psi = 2r^2 \sin 2\theta$, where $\psi$ has units of m²/s when $r$ is in meters. Assume the fluid density is $\rho = 10^3$ kg/m³ and the x-y plane is horizontal, that is, there is no difference in elevation between points (1) and (2).

(a) (15%) Determine the velocity potential function $\phi$.
(b) (10%) If the pressure at point (1) on the wall is 30 kPa, what is the pressure at point (2)?

🖼️【此處有附圖，請對照原卷】The figure shows a 90° corner (in the first quadrant, with the corner at the origin). Streamlines are shown in polar coordinates $(r, \theta)$. Point (1) is on the wall at $(r=1, \theta=0)$. Point (2) is at $(r=0.5, \theta=\pi/4)$. The distance scale is indicated: 1 m on the x-axis, 0.5 m on the y-axis. The streamlines are curved lines originating from the corner.

3. Consider an incompressible Newtonian fluid with dynamic viscosity $\mu$ and density $\rho$, which is flowing steadily through a horizontal long circular straight pipe with diameter $D$ at an average speed $V$. We can express the pressure drop per unit axial length, $\Delta p/\Delta x$, as $\Delta p/\Delta x = f (D, \rho, \mu, V)$.

(a) (10%) Show that $\Delta p/\Delta x$ can be expressed in dimensionless form as $\frac{\Delta p}{\rho V^2 / D} = f(\text{Re})$, where $\text{Re} = \frac{\rho V D}{\mu}$ is the Reynolds number, through the use of dimensional analysis and Buckingham pi theorem.
(b) (15%) Considered the flow as fully-developed and axisymmetric. The velocity field can be expressed as $u = u(r) e_r$ in a cylindrical coordinates system, $(r, \theta, x)$, fixed with respect to the pipe, with the coordinate center at the central point of the inlet plane, and x-axis along the pipe axis. According to the momentum balance, we can write:
$\mu \left( \frac{d^2 u}{dr^2} + \frac{1}{r} \frac{du}{dr} \right) = \frac{dp}{dx}$
Solve the above equation for $u(r)$ subject to $u(D/2)=0$, and derive that $\frac{\Delta p}{\rho V^2 / D} = \frac{32}{\text{Re}}$.

4. The following figure shows the steady incompressible boundary layer flow over a flat plate.
   (a) (15%) Derive the momentum integral equation using the conservation of linear momentum for the flow through the control volume indicated. The result should be
   $\frac{d}{dx} \int_0^\delta \rho u (U-u) dy + \tau_w = \rho U^2 \frac{d\delta}{dx}$
   where $\delta$ is the momentum thickness and $\tau_w$ is the shear stress at wall, respectively.
   (b) (10%) Determine the boundary layer thickness and $\tau_w$, assuming a linear velocity profile across the boundary layer, i.e., $u = U \frac{y}{\delta}$ for $0 \le y \le \delta$ and $u = U$ for $y > \delta$, with $\delta = \delta(x)$ the boundary layer thickness.

🖼️【此處有附圖，請對照原卷】The figure shows a flat plate aligned with the flow. A control volume is indicated, extending from $x$ to $x+\Delta x$ and from $y=0$ (plate surface) to $y=\delta(x)$ (boundary layer edge). The outer flow velocity is $U$. Inside the boundary layer, the velocity $u(x,y)$ increases from 0 at $y=0$ to $U$ at $y=\delta(x)$. Streamlines are shown within the boundary layer. The shear stress at the wall is $\tau_w$. The momentum thickness is labeled as $\theta$.