Question

Models of torpedoes are sometimes tested in a horizontal pipe of flowing water, much as a wind tunnel is used to test model airplanes. Consider a circular pipe of internal diameter \(25.5 \mathrm{~cm}\) and a torpedo model, aligned along the axis of the pipe, with a diameter of \(4.80 \mathrm{~cm}\). The torpedo is to be tested with water flowing past it at \(2.76 \mathrm{~m} / \mathrm{s} .(a)\) With what speed must the water flow in the unconstricted part of the pipe? (b) Find the pressure difference between the constricted and unconstricted parts of the pipe.

Short answer

The speed of the water in the unconstricted part of the pipe must be approximately 0.213 m/s. The pressure difference between the constricted and unconstricted parts of the pipe is approximately 2395.0 Pa.

Step-by-step solution

Calculate the cross-sectional areas

Calculate the cross-sectional areas of the constricted and unconstricted parts of the pipe using the formula for the area of a circle \(A=\pi r^2\). The radius of the constricted part is the radius of the pipe minus the radius of the torpedo model.

Apply the principle of continuity

According to the principle of continuity, the quantity that remains constant is the product of the cross-sectional area and the flow speed. From this, we get \(A_1v_1 = A_2v_2\), where \(A_1\), and \(v_1\) are the area and velocity in the constricted part and \(A_2\), \(v_2\) are the area and velocity in the unconstricted part. Use this equation to find \(v_2\), the speed of the water in the unconstricted part.

Applying Bernoulli's equation

To find the pressure difference, apply Bernoulli’s equation. The owing such equation: \(P_1 + \frac{1}{2}\rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gh_2\). Since the pipe is horizontal, the potential energy term \(gh\) at both points is the same and cancels out. Moreover, we are interested in the pressure difference, we can use the equation \(\Delta P = \frac{1}{2}\rho(v_2^2 - v_1^2)\)to find the difference in pressure between the constricted and unconstricted part.

Key concepts

Bernoulli's Equation

Bernoulli's equation is a statement of the conservation of energy principle for flowing fluids. It relates the speed of the fluid, its density, and its pressure at various points along the flow path. The equation can be represented as: \[\[\begin{align*}\rightleftharpoons P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}\end{align*}\]\]where:

• (\( P \)) is the pressure within the fluid,
• (\( \rho \)) is the density of the fluid,
• (\( v \)) is the velocity of the fluid,
• (\( g \)) is the acceleration due to gravity, and
• (\( h \)) is the height above a reference point.

Bernoulli's equation is powerful because it shows us how these different physical quantities are interconnected. A change in one (like velocity or elevation) can directly affect the others (like pressure), which is exactly what happens in the torpedo model testing scenario. The equation simplifies even further for horizontal flow because the term involving height (\( h \)) remains constant and therefore cancels out, leading us to only consider the relationship between pressure and velocity.

Principle of Continuity

The principle of continuity, also known as the continuity equation, is a principle derived from the conservation of mass. It tells us that for an incompressible fluid flowing through a pipe, the mass flow rate must remain constant from one cross-section to another. Mathematically, this can be expressed as:\[\[\begin{align*}A_1v_1 = A_2v_2\end{align*}\]\]where:

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

Applying this principle to the torpedo model testing situation, it means the flow speed must increase when passing the torpedo due to the reduced cross-sectional area at that point. This ensures the same amount of water moves past the torpedo as flows through the rest of the pipe, aligning with the conservation of mass.

Torpedo Model Testing

Torpedo model testing in a water tunnel is akin to aerodynamic testing in a wind tunnel, providing valuable data on how the torpedo will perform. By observing the behavior of water flowing around the torpedo model, researchers can measure various parameters, such as lift, drag, and pressure distribution. A scaled-down model is used due to practicality and cost efficiency, and the readings taken can be mathematically adjusted to predict real-world performance. The importance of accurately measuring the water’s velocity and pressure at different points of the water tunnel, using principles like Bernoulli’s equation and the principle of continuity, is imperative for precise results and can improve the design of the torpedo for optimal operation.

Cross-Sectional Area Calculation

Calculating cross-sectional area is essential in fluid dynamics to understand how fluids flow under various conditions. For a circular pipe, the area is found using the formula:\[\[\begin{align*}A = \pi r^2\end{align*}\]\]where (\( r \)) is the radius of the circular cross-section. In the exercise involving the torpedo model, calculating the constricted area through which the water flows is crucial for applying the principle of continuity. Here, the effective radius is the radius of the pipe minus the radius of the torpedo model. We must understand that as the cross-sectional area of the pipe decreases due to the presence of the torpedo, the speed of the fluid must increase to keep the flow rate constant. Likewise, when the area opens back up, the velocity changes again. These calculations directly impact the findings for velocity and pressure changes dictated by Bernoulli's equation and the principle of continuity.