Problem 49 - Fundamentals of Physics

Question

Canal effect. Figure \(14-45\) shows an anchored barge that cxtcnds across a canal by distance \(d=30 \mathrm{~m}\) and into the water by distance \(b=12 \mathrm{~m} .\) The canal has a width \(D=55 \mathrm{~m},\) a water depth \(H=14 \mathrm{~m},\) and a uniform water-flow speed \(v_{i}=1.5 \mathrm{~m} / \mathrm{s} .\) Assume that the flow around the barge is uniform. As the water passes the bow, the water level undergoes a dramatic dip known as the canal effect. If the dip has depth \(h=0.80 \mathrm{~m},\) what is the water speed alongside the boat through the vertical cross sections at (a) point \(a\) and (b) point \(b ?\) The erosion due to the speed increase is a common concern to hydraulic cngincers.

Step-by-Step Solution

Answer

Speed at points 'a' and 'b' is approximately 4.18 m/s.

Step 1: Understanding the Problem

We need to calculate the water speed alongside the boat at two points, given the dimensional details of the canal and the barge, including the depth of the dip in water level. This involves using the concept of flow rate and continuity.

Step 2: Apply Bernoulli's Equation

At point 'a' or 'b', we can apply Bernoulli's equation to find the relation between water velocities and depths. The equation is given as \( P_1 + \frac{1}{2} \rho v_1^2 + \rho g z_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g z_2 \). Here we compare the conditions at the water before passes the barge and those alongside the bar.

Step 3: Determine Changes in Water Height

Identify that a decrease in water height at the bow, \( h = 0.80 \text{ m} \), results in a change in pressure and speed according to Bernoulli’s equation. The difference in height correlates to differences in flow speed.

Step 4: Apply Continuity Equation

Use the continuity equation \( A_1 v_1 = A_2 v_2 \) to relate the speeds and cross-sectional areas at the different points. Since the barge modifies the cross-section, account for it in your area calculation.

Step 5: Calculate Velocity at Point a

For point 'a', using Bernoulli's equation and continuity, solve for the new velocity \( v_a \) with the given dip: \( v_a = \sqrt{v_i^2 + 2gh} \), which simplifies to \( v_a = \sqrt{(1.5)^2 + 2*9.81*0.80} \), solving this gives \( v_a \approx 4.18 \text{ m/s} \).

Step 6: Calculate Velocity at Point b

For point 'b', since the flow remains uniform around the barge and no additional dip is involved, the velocity \( v_b \) remains similar: \( v_b \approx 4.18 \text{ m/s} \), following the same calculation as point 'a' since the flow was accounted for together across the barge.

Key Concepts

Canal Effect

The canal effect is a fascinating phenomenon that occurs in narrow waterways, particularly when there's a structure, like a barge, that partially obstructs water flow. This effect often results in a noticeable dip in the water level, as seen at the bow of the barge in our exercise.

What causes the water level to dip is the change in water speed as it flows around the obstruction. When water has to squeeze through a smaller area, it must speed up to maintain the flow rate. This increase in speed leads to a decrease in pressure, causing the water level to drop. This is due to Bernoulli's principle, which you'll find explained in further detail in this article.

Understanding the canal effect is important for engineers due to the potential erosion risks it poses to canal banks and structures. Addressing this issue involves careful planning and design to minimize the impact of increased water speeds resulting from such dips.

Water Speed Calculation

Calculating water speed in the presence of structures like the barge involves applying Bernoulli's Equation. This equation helps us understand how energy conservation works within a fluid in motion.

• Bernoulli’s Equation connects the velocity of the fluid, the pressure along the flow, and the height of the fluid at different points along its path.
• Using the equation, you can determine how much faster the water moves as it passes the bow, where the water level dips.

To calculate the speed at point 'a', where there's a dip, we use the formula: \( v_a = \sqrt{(v_i)^2 + 2gh} \) where:

• \( v_i \) is the initial velocity,
• \( g \) is the gravitational acceleration (\( 9.81 \ \mathrm{m/s^2} \)),
• \( h \) is the depth of the water level dip.

Plugging in the numbers gives us a faster velocity at point 'a' due to the reduced water height and the increased speed needed to maintain continuity of flow.

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics, crucial for solving problems involving fluid flow in pipes and open channels like canals. It expresses the conservation of mass in a fluid flow \( A_1 v_1 = A_2 v_2 \),where:

• \( A_1 \) and \( A_2 \) are the cross-sectional areas at two different points,
• \( v_1 \) and \( v_2 \) are the fluid velocities at those points.

In the exercise, as the water passes around the barge, it flows through a reduced cross-sectional area—due to the barge's presence. To maintain the same flow rate, the water must move faster, resulting in a velocity increase calculated using these principles.

Understanding the continuity equation is crucial for predicting fluid behavior in various engineering applications, ensuring that structures like canals and bridges are designed to handle changes in velocity and prevent possible erosion or structural compromises.