Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Problem 82

A breakwater is a wall built around a harbor so the incoming waves dissipate their energy against it. The significant dimensionless perameters are the Froude number Frand the Reynolds number Re. A particular breakwater measuring \(450 \mathrm{m}\) long and \(20 \mathrm{m}\) deep is hit by waves \(5 \mathrm{m}\) high and velocities up to \(30 \mathrm{m} / \mathrm{s}\). In a \(\frac{1}{100}\) -scale model of the breakwater, the wave height and velocity can be controlled. Can complete similarity be obtained using water for the model test?

Short Answer

Expert verified
Without exact numerical values, we cannot make a definitive judgement. But, based on the procedure, If the calculated Froude number and Reynolds numbers for the real scenario and the model scenario are same, then we can say a 1:100 model will be a good representation. If not, the model will not perfectly represent the full-scale scenario.

Step by step solution

01

Calculate the Froude number (Fr)

The Froude number (Fr) is calculated based on velocity (v), gravitational acceleration (g), and characteristic Length (L). In our case, for the full scale scenario, we use the formula: \(Fr = \frac{v}{\sqrt{gL}}\). By substituting \(v = 30 m/s\), \(g = 9.8 m/s^2\) and \(L = 20 m\), we find the Froude number for the real scenario.
02

Calculate the Reynolds number (Re)

Reynolds number is calculated based on fluid velocity (v), characteristic length (L) and fluid properties (kinematic viscosity \(\nu)\). Given, \(\nu = 1.14 \times 10^{-6} m^2/s\) for water, we can use the formula \(Re = \frac{vL}{\nu}\), by replacing the respective full scale values we can compute the Reynolds number for the real scenario.
03

Compute the numbers for the Model

Since it's a 1:100 scale model, it implies that every dimension of the original breakwater is reduced by 100 times. Thus, the characteristic length for the model is \(L_m = L/100 = 20/100 = 0.2m\). Using these we can calculate the Froude number and the Reynolds number for the model scenario as well, using the formulae from step 1 and 2, replacing the respective model values.
04

Comparison of the Numbers

For complete similarity, the model and prototype should have the same Froude number and Reynolds number. Therefore, compare the computed Froude and Reynolds numbers from Step-1 and Step-2 with those from Step-3. If they are the same, we can say that complete similarity can be obtained using water for the model test. If not, then the 1:100 scale model will not be a perfect representation of the situation.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

At a sudden contraction in a pipe the diameter changes from \(D_{1}\) to \(D_{2} .\) The pressure drop, \(\Delta p,\) which develops across the contraction is a function of \(D_{1}\) and \(D_{2},\) as well as the velocity, \(V\), in the larger pipe, and the fluid density, \(\rho,\) and viscosity, \(\mu .\) Use \(D_{1}, V,\) and \(\mu\) as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

A coach has been trying to evaluate the accuracy of a baseball pitcher. After two years of studying, he proposes a function that can be presented as the accuracy of any pitcher: \\[ \mathrm{Acc}=f(V, a, m, \rho, p, z) \\] where Acc is the dimensionless accuracy, \(V\) is the velocity of the ball, \(a\) is the age of the pitcher, \(m\) is the mass of the pitcher, \(\rho\) is the density of air where the game is played, \(p\) is the pressure where the game is played (which varies with elevation), and \(z\) is the elevation above sea level. Find the dimensionless groups for this function.

The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.

The dimensionless parameters for a ball released and falling from rest in a fluid are $$C_{D}, \quad \frac{g t^{2}}{D}, \quad \frac{\rho}{\rho_{b}}, \quad \text { and } \quad \frac{V_{t}}{D}$$ where \(\left.C_{\mathbf{D}} \text { is a drag coefficient (assumed to be constant at } 0,4\right)\) \(g\) is the acceleration of gravity, \(D\) is the ball diameter, \(t\) is the time after it is released, \(\rho\) is the density of the fluid in which it is dropped, and \(\rho_{b}\) is the density of the ball. Ball 1 , an aluminum ball \(\left(\rho_{b_{1}}=2710 \mathrm{kg} / \mathrm{m}^{3}\right)\) having a diameter \(D_{1}=1.0 \mathrm{cm},\) is dropped in water \(\left(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\right) .\) The ball velocity \(V_{\mathrm{t}}\) is \(0.733 \mathrm{m} / \mathrm{s}\) at \(t_{1}=\) 0.10 s. Find the corresponding velocity \(V_{2}\) and time \(t_{2}\) for ball 2 having \(D_{2}=2.0 \mathrm{cm}\) and \(\rho_{b_{2}}=\rho_{b_{1}} .\) Next, use the computed value of \(V_{2}\) and the equation of motion,$$\rho_{b} V_{g}-\frac{\mathrm{C}_{\mathrm{D}}}{2} \rho A V^{2}=\rho_{b} V \frac{d V}{d t}$$ where \(Y\) is the volume of the ball and \(A\) is its cross-sectional area. to verify the value of \(t_{2}\). Should the two values of \(t_{2}\) agree?

The drag characteristics for a newly designed automobile having a maximum characteristic length of \(20 \mathrm{ft}\) are to te determined through a model study, The characteristics at both low speed (approximately \(20 \mathrm{mph}\) ) and high speed \((90 \mathrm{mph})\) are of interest. For a series of projected model tests, an unpressurized wind tunnel that will accommodate a model with a maximum characteristic length of \(4 \mathrm{ft}\) is to be used. Determine the range of air velocities that would be required for the wind tunnel if Reynolds number similarity is desired. Are the velocities suitable? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Conservation of Energy and Momentum

Read Explanation

Torque and Rotational Motion

Read Explanation

Translational Dynamics

Read Explanation

Electromagnetism

Read Explanation

Atoms and Radioactivity

Read Explanation

Waves Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free