Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 4

Air (of kinematic viscosity \(\left.15 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\right)\) flows at \(10.5 \mathrm{~m} \cdot \mathrm{s}^{-1}\) past a smooth, rectangular, flat plate \(300 \mathrm{~mm} \times 3 \mathrm{~m}\) in size. Assuming that the turbulence level in the oncoming stream is low and that transition occurs at \(R e=5 \times 10^{5}\) calculate the ratio of the total drag force when the flow is parallel to the length of the plate to the value when the flow is parallel to the width.

Short Answer

Expert verified
The ratio of the total drag force for flow parallel to the length to that parallel to the width is 0.588.

Step by step solution

01

Calculate the Reynolds Number

The Reynolds number is given by: \[ Re = \frac{UL}{u} \] where U is the flow velocity, L is the characteristic length and \( u \) is the kinematic viscosity. We need to calculate it for both orientations of the plate.
02

Reynolds Number for Flow Parallel to Length

When the flow is parallel to the length (3m): \[ Re_{length} = \frac{10.5 \times 3}{15 \times 10^{-6}} = 2.1 \times 10^{6} \]
03

Reynolds Number for Flow Parallel to Width

When the flow is parallel to the width (0.3m): \[ Re_{width} = \frac{10.5 \times 0.3}{15 \times 10^{-6}} = 0.21 \times 10^{6} \]
04

Determine the drag force formulas

For a flat plate, the drag force in turbulent flow can be estimated using the formulas for skin friction coefficient: \[ C_f = 0.074 Re^{-\frac{1}{5}} \]
05

Calculate Drag Coefficient for Both Orientations

\[ C_{f, length} = 0.074 (2.1 \times 10^{6})^{-\frac{1}{5}} = 0.00258 \] \[ C_{f, width} = 0.074 (0.21 \times 10^{6})^{ - \frac{1}{5}} = 0.00439 \]
06

Calculate Drag Forces

The drag force is determined by:\[ D = 0.5 \rho U^{2}C_{f} A \]Assuming air density \(\rho = 1.2 \text{kg/m}^3\): For flow parallel to the length: \[ D_{length} = 0.5 \cdot 1.2 \cdot 10.5^{2} \cdot 0.00258 \cdot 0.3 \cdot 3 = 0.117 \text{N}\] For flow parallel to the width: \[ D_{width} = 0.5 \cdot 1.2 \cdot 10.5^{2} \cdot 0.00439 \cdot 3 \cdot 0.3 = 0.199 \text{N}\]
07

Calculate the Ratio of Drag Forces

Finally, the ratio of the drag forces is: \[ \text{Ratio} = \frac{D_{length}}{D_{width}} = \frac{0.117}{0.199} = 0.588 \]

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!

Key Concepts

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

Kinematic Viscosity
Kinematic viscosity is a measure of a fluid's resistance to flow under gravity. It is defined as the ratio of dynamic viscosity (\( \mu \)) to density (\( \rho \)). The formula is: \[ u = \frac{\mu}{\rho} \] Kinematic viscosity is expressed in units of \( \text{m}^2/\text{s} \). For air, it has a typical value in the range of \( 15 \text{mm}^2/\text{s} \). This property plays a crucial role in calculating the Reynolds number (\( Re \)), which helps determine the flow regime of the fluid motion.
Turbulent Flow
Turbulent flow is characterized by chaotic changes in pressure and flow velocity. Unlike laminar flow, where fluid flows in parallel layers, turbulent flow involves eddies and vortices. The transition from laminar to turbulent flow in a flat plate often occurs at a Reynolds number (\( Re \)) of about 5 x 10^5. The turbulent flow increases mixing and enhances heat and mass transfer but also increases friction and drag forces. Understanding turbulent flow is essential for designing systems with efficient flow characteristics.
Skin Friction Coefficient
The skin friction coefficient (\( C_f \)) quantifies the frictional resistance exerted by a fluid flowing over a surface. For turbulent flow over a flat plate, it can be calculated using the formula: \[ C_f = 0.074 Re^{-1/5} \] This coefficient is a function of the Reynolds number, indicating that as \( Re \) increases, the \( C_f \) decreases. Determining the skin friction coefficient is key to computing the drag force on objects subjected to fluid flow.
Drag Force
Drag force is the resistance caused by the interaction between a moving object and the fluid through which it moves. For a flat plate in turbulent flow, the drag force can be computed using the formula: \[ D = 0.5 \rho U^2 C_f A \] where: \( \rho \) is the density of the fluid, \( U \) is the flow velocity, \( C_f \) is the skin friction coefficient, and \( A \) is the surface area of the plate. The drag force is crucial for understanding how much energy is lost due to friction and for optimizing the shapes and surfaces of objects moving through fluids to minimize resistance.

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

A honeycomb type of flow straightener is formed from perpendicular flat metal strips to give \(25 \mathrm{~mm}\) square passages, \(150 \mathrm{~mm}\) long. Water of kinematic viscosity \(1.21 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\) approaches the straightener at \(1.8 \mathrm{~m} \cdot \mathrm{s}^{-1}\). Neglecting the thickness of the metal, the effects of the small pressure gradient and of threedimensional flow in the corners of the passages, calculate the displacement thickness of the boundary layer and the velocity of the main stream at the outlet end of the straightener. Applying Bernoulli's equation to the main stream deduce the pressure drop through the straightener.

What frequency of oscillation may be expected when air of kinematic viscosity \(15 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\) flows at \(22 \mathrm{~m} \cdot \mathrm{s}^{-1}\) past a \(3 \mathrm{~mm}\) diameter telephone wire which is perpendicular to the air stream?

When water (kinematic viscosity \(1.2 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\) ) flows steadily at the rate of \(18.5 \mathrm{~L} \cdot \mathrm{s}^{-1}\) through \(25 \mathrm{~m}\) of a \(100 \mathrm{~mm}\) diameter pipe, the head loss is \(1.89 \mathrm{~m}\). Estimate the relative roughness, the velocity on the axis of the pipe and the shear stress at the wall

A smooth flat plate \(2.4 \mathrm{~m}\) long and \(900 \mathrm{~mm}\) wide moves lengthways at \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\) through still atmospheric air of density \(1.21 \mathrm{~kg} \cdot \mathrm{m}^{-3}\) and kinematic viscosity \(14.9 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\). Assuming the boundary layer to be entirely laminar, calculate the boundary layer thickness (i.e. the position at which the velocity is \(0.99\) times the free-stream velocity) at the trailing edge of the plate, the shear stress half-way along and the power required to move the plate. What power would be required if the boundary layer were made turbulent at the leading edge?

Calculate the diameter of a parachute (in the form of a hemispherical shell) to be used for dropping a small object of mass \(90 \mathrm{~kg}\) so that it touches the earth at a velocity no greater than \(6 \mathrm{~m} \cdot \mathrm{s}^{-1}\). The drag coefficient for a hemispherical shell with its concave side upstream is approximately \(1.32\) for \(\operatorname{Re}>10^{3}\). (Air density \(=1.22 \mathrm{~kg} \cdot \mathrm{m}^{-3}\).)
See all solutions

Recommended explanations on Physics Textbooks

Physics of Motion

Read Explanation

Turning Points in Physics

Read Explanation

Force

Read Explanation

Fundamentals of Physics

Read Explanation

Scientific Method Physics

Read Explanation

Linear Momentum

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