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Chapter 6: Problem 94

A particular submarine is capable of cruising at \(20 \mathrm{~km} / \mathrm{h}\) at the surface of the ocean and far below the surface of the ocean. Tests of both of these conditions are to be performed using a 1: 40 scale model of the submarine. Determine the required model speed and the ratio of prototype drag to model drag for the following model conditions: (a) a model of cruising on the surface and (b) a model of cruising far below the surface. Assume that the prototype ocean water properties can be duplicated in the model tests.

Short Answer

Expert verified
Model speed is 3.16 km/h for surface, 800 km/h underwater (unrealistic); drag ratio: 1600 (surface), 40 (underwater).

Step by step solution

01

Understand the Conditions and Variables

The exercise requires us to find the model speed and the ratio of prototype to model drag for a submarine cruising at the surface and far below the surface. The prototype speed is given as 20 km/h, and a scale factor of 1:40 is applied to the model. The main physics concepts involved are Froude number similarity for surface cruising and Reynolds number similarity for underwater cruising.
02

Calculate Model Speed for Surface Cruising

For surface cruising, the Froude number similarity needs to be maintained, where the Froude number \( F_r \) is defined by \( F_r = \frac{V}{\sqrt{gL}} \), where \( V \) is the velocity, \( g \) is the acceleration due to gravity, and \( L \) is a characteristic length. For the model, the length \( L_m = \frac{L_p}{40} \). Equating prototype and model Froude numbers, \( \frac{V_p}{\sqrt{gL_p}} = \frac{V_m}{\sqrt{gL_m}} \). Solving for \( V_m \):\[ V_m = V_p \times \sqrt{\frac{L_m}{L_p}} = 20 \times \sqrt{\frac{1}{40}} = \frac{20}{\sqrt{40}} \approx 3.16 \text{ km/h} \].
03

Calculate Ratio of Prototype to Model Drag for Surface Cruising

Using Froude number similarity for surface conditions typically implies that drag ratios are related to the square of the linear scale:\[ \frac{D_p}{D_m} = \left( \frac{L_p}{L_m} \right)^2 = 40^2 = 1600 \].
04

Calculate Model Speed for Underwater Cruising

For cruising far below the surface, Reynolds number similarity is necessary, where the Reynolds number \( Re \) is defined as \( \frac{\rho VL}{\mu} \). Assuming dimensionless similarity (same fluid properties), equate prototype and model Reynolds numbers: \( \frac{V_pL_p}{u} = \frac{V_mL_m}{u} \). Thus, \( V_m = V_p \times \frac{L_p}{L_m} = 20 \times 40 = 800 \text{ km/h} \), which suggests that direct scaling of speed is unrealistic for practical tests and might be adjusted by changing fluid conditions.
05

Calculate Ratio of Prototype to Model Drag for Underwater Cruising

Reynolds number similarity implies that drag ratios can be directly related to linear scale:\[ \frac{D_p}{D_m} = \frac{L_p}{L_m} = 40 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scale Modelling in Fluid Dynamics
Understanding scale modelling in fluid dynamics is essential when testing prototypes such as submarines in a controlled setting. By creating a scaled-down version, engineers can simulate and analyze behavior under specific conditions, which is both cost-effective and practical. To achieve accurate results, it's crucial to maintain similarity in certain dimensionless numbers, depending on whether the object interacts primarily with the surface (like a ship) or if it operates predominantly underwater (like a submarine).
  • Froude Number Similarity: For models interacting with the water鈥檚 surface, such as a ship, the Froude number governs the dynamics to ensure wave-making resistance is similar between model and prototype.
  • Reynolds Number Similarity: For underwater models, the Reynolds number is used to match the flow characteristics, including turbulent and laminar flow transitions.
Scale models allow engineers to systematically adjust and test different scenarios in fluid environments, leading to innovations and improvements in design.
Reynolds Number
The Reynolds number (\( Re \)) is a key parameter in fluid mechanics that characterizes the flow of fluid around an object, such as a submarine. It is defined by the equation:\[Re = \frac{\rho VL}{\mu}\]where \( \rho \) is the fluid density, \( V \) is the fluid velocity, \( L \) is a characteristic length, and \( \mu \) is the dynamic viscosity of the fluid. This dimensionless number helps predict flow patterns in different fluid flow situations.
  • Laminar Flow: Characterized by smooth, constant fluid motion, typically occurring at lower Reynolds numbers.
  • Turbulent Flow: Characterized by chaotic and irregular fluid motion, occurring at higher Reynolds numbers and is crucial in submarine design due to the large sizes and velocities involved.
In underwater cruising for submarines, achieving Reynolds number similarity ensures that the model accurately replicates the flow characteristics of the prototype, which is crucial for predicting behavior like drag forces.
Drag Force in Fluid Mechanics
Drag force in fluid mechanics is the force exerted by a fluid against the motion of an object through it. It is a crucial component that must be understood to effectively design and operate vehicles such as submarines and aircraft. The drag force (\( D \)) depends on several factors:
  • Shape of the Object: Streamlined shapes reduce drag, enabling higher efficiency.
  • Surface Area: Larger areas facing the flow direction can increase drag.
  • Fluid Density and Velocity: Higher density and velocity amplify the drag force.
Drag force calculationsenable engineers to optimize designs and take necessary adjustments, linking closely with other concepts like the Reynolds number to create efficient and effective underwater vehicles. In scale models, ensuring that physical laws governing drag are maintained requires careful attention to the similarity of dimensionless numbers such as Reynolds and Froude numbers.

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Most popular questions from this chapter

A model differential equation for chemical reaction dynamics in a plug-flow reactor is as follows: $$u \frac{\partial C}{\partial x}=D \frac{\partial^{2} C}{\partial x^{2}}-k C-\frac{\partial C}{\partial t}$$ where \(u\) is the velocity, \(D\) is the diffusion coefficient, \(k\) is the reaction-rate constant, \(x\) is the distance along the reactor, and \(C\) is the (dimensionless) concentration for a given chemical in the reactor. Determine the appropriate dimensions of \(D\) and \(k\). Using a characteristic length scale \(L,\) average velocity \(V,\) and background concentration \(C_{0}\) as parameters, rewrite the given equation in normalized form and comment on any dimensionless groups that appear.

The temperature distribution, \(T(x, y)\), within a two-dimensional flow field of an incompressible and nonviscous fluid is governed by the following (energy) equation: $$\rho c_{p}\left[u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}\right]=k\left(\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}\right)$$ where \(\rho\) and \(c_{p}\) are the density and specific heat of the fluid, respectively, \(u\) and \(v\) are the velocity components in the \(x\) - and \(y\) -directions, respectively, and \(k\) is the thermal conductivity of the fluid. The relevant scales are the length scale, \(L ;\) the velocity scale, \(V\); and the temperature scale, \(T_{0}\). Express Equation 6.31 in normalized form, using the Prandtl number, Pr, defined as \(\operatorname{Pr}=\mu c_{p} / k\), and the Reynolds number, Re, defined as \(\mathrm{Re}=\rho V L / \mu,\) where \(\mu\) is the dynamic viscosity of the fluid.

Under laminar flow conditions, the average velocity, \(V,\) in a pipe of diameter \(D\) depends on the pressure gradient in the direction of flow, \(\partial p / \partial x\), and the dynamic viscosity, \(\mu,\) of the fluid. Use dimensional analysis to determine the functional relationship between \(V\) and the influencing variables.

A prototype ship \(35 \mathrm{~m}\) long is designed to cruise at \(11 \mathrm{~m} / \mathrm{s}\). Its drag is to be simulated by a 1 -m-long model pulled in a tow tank. For Froude scaling, determine the tow speed, the ratio of prototype to model drag, and the ratio of prototype to model power.

Consider the energy equation for flow of a fluid in a closed conduit, which can be expressed in the form $$\frac{p_{1}}{\gamma}+\frac{V_{1}^{2}}{2 g}+z_{1}=\frac{p_{2}}{\gamma}+\frac{V_{2}^{2}}{2 g}+z_{2}+h_{\mathrm{f}}$$ where \(p_{1}, V_{1}\), and \(z_{1}\) are the pressure, average velocity, and centerline elevation at an upstream section of the conduit, \(p_{2}, V_{2},\) and \(z_{2}\) are the corresponding variables at a downstream section, \(h_{f}\) is the head loss between the two sections, \(\gamma\) is the specific weigh of the fluid, and \(g\) is the gravity constant. State the SI unit of energy and state the SI unit(s) of the terms (not the individual variables) in Equation 6.27. Explain why each of the terms in Equation 6.27 represents energy and express Equation 6.27 as a nondimensional equation.
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