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Chapter 6: Problem 47

The drag force on a supersonic aircraft of a particular shape depends on its size, its speed, the density of the air, and the sonic speed in the air. Determine a nondimensional functional relationship between the drag force and its influencing variables. Identify any named conventional dimensionless groups.

Short Answer

Expert verified
The nondimensional relationship is \( \frac{F_D}{\rho V^2 L^2} = f\left(\frac{V}{a}\right) \), involving the drag coefficient and Mach number as dimensionless groups.

Step by step solution

01

Identify Variables

To start, identify all the relevant variables involved in the problem: Drag force \( F_D \), size of the aircraft (we'll consider characteristic length \( L \)), speed of the aircraft \( V \), density of the air \( \rho \), and the sonic speed in air \( a \).
02

Define Dimensions of Each Variable

Define the dimensions for all the variables: - Drag force \( F_D \) has dimensions \([M L T^{-2}]\).- Characteristic length \( L \) has dimension \([L]\).- Speed \( V \) has dimension \([L T^{-1}]\).- Density \( \rho \) has dimension \([M L^{-3}]\).- Sonic speed \( a \) has dimension \([L T^{-1}]\).
03

Setup the Dimensional Matrix

Construct a dimensional matrix using the dimensions identified. The matrix will have rows for each dimension \( (M, L, T) \) and columns for each variable: \[\begin{array}{ccc}F_D & L & V & \rho & a \1 & 0 & 0 & 1 & 0 \1 & 1 & 1 & -3 & 1 \-2 & 0 & -1 & 0 & -1 \\end{array}\]
04

Apply Buckingham Pi Theorem

Using Buckingham Pi Theorem, determine the number of dimensionless groups, \( \pi \). Since there are 5 variables and 3 fundamental dimensions \((M, L, T)\), there will be \( 5-3=2 \) dimensionless groups. Select repeating variables \( \rho, V, L \).
05

Form the Dimensionless Groups

Form the dimensionless groups using the selected repeating variables:- \( \pi_1 = \frac{F_D}{\rho V^2 L^2} \), reminiscent of the drag coefficient.- \( \pi_2 = \frac{V}{a} \), which is the Mach number.
06

Express the Nondimensional Relationship

Express the nondimensional relationship in terms of \( \pi \) groups:\[\pi_1 = f(\pi_2)\]or equivalently, in terms of variables:\[\frac{F_D}{\rho V^2 L^2} = f\left(\frac{V}{a}\right)\]This indicates that the nondimensional drag force is a function of the Mach number.
07

Identify Conventional Dimensionless Groups

The identified dimensionless groups \( \frac{F_D}{\rho V^2 L^2} \) is the drag coefficient and \( \frac{V}{a} \) is the Mach number, both of which are well-known in fluid dynamics and aerodynamics.

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Key Concepts

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

Understanding Drag Force
Drag force is a resistance force that acts opposite to the relative motion of any object moving with respect to a surrounding fluid. It is crucial in many areas of engineering and science, particularly in aerodynamics. Imagine an aircraft soaring through the air; it must constantly overcome this drag force to maintain its speed. The drag force, denoted as \( F_D \), depends on various factors, such as the size of the object, its speed, and the characteristics of the fluid it's moving through. Generally, drag force factors in:
  • The shape and size of the object, contributing to how fluid (like air) flows around it
  • The object's velocity and the fluid's speed, density, and viscosity
Drag force can be quite significant, and understanding it helps improve the efficiency of transporting vehicles and optimizing their design for better performance.
Applying the Buckingham Pi Theorem
The Buckingham Pi Theorem is a pivotal concept in dimensional analysis, allowing the simplification of physical equations by deriving dimensionless parameters. It states that if you have an equation involving \( n \) variables, and these variables use \( k \) fundamental dimensions (like mass, length, time), the equation can be reduced to a set of \( n - k \) independent dimensionless products. In practical use, this theorem assists in reducing complex problems into simpler, dimensionless forms. For example:
  • Identify the physical variables and their dimensions.
  • Formulate a dimensional matrix to organize the relationships.
  • Determine how the dimensionless groups (or "pi groups") can be constructed.
This theorem is essential for creating relationships like that between drag force and the involved physical variables, assisting engineers in designing experiments and interpreting results efficiently.
Significance of Mach Number
In aerodynamics, the Mach number is a dimensionless value representing the ratio of an object's speed to the speed of sound in the surrounding medium. Named after Ernst Mach, it is central to the study of airflow around objects when they travel at high speeds.Calculating Mach number is straightforward: it's the vehicle's velocity \( V \) divided by the local speed of sound \( a \) : \[\text{Mach number} = \frac{V}{a}\]Different ranges of Mach numbers define the flow regime:
  • Subsonic: Mach < 1
  • Transonic: Mach 鈮� 1
  • Supersonic: Mach > 1
  • Hypersonic: Mach > 5
Understanding Mach number is crucial as it affects aerodynamic properties like pressure distribution and drag. Particularly at supersonic speeds, phenomena such as shock waves significantly impact designs for aircraft and missiles.
Drag Coefficient and Its Role
The drag coefficient, symbolized as \( C_d \), is another essential dimensionless number that aids in understanding the drag force. It represents the resistance an object faces in a fluid environment, factoring in its shape, size, and speed.The drag coefficient is calculated using:\[C_d = \frac{F_D}{\frac{1}{2} \rho V^2 A} \]Here, \( F_D \) is the drag force, \( \rho \) is the fluid density, \( V \) is the velocity, and \( A \) is the reference area.A low drag coefficient means an object will face little resistance moving through a fluid, making it an essential consideration in the design of vehicles, boats, and even sportswear. By optimizing the drag coefficient, engineers and designers can improve fuel efficiency and overall performance of transport and athletic gear.

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Most popular questions from this chapter

The momentum equation that describes the motion of a nonviscous incompressible fluid in a three-dimensional \((x y z)\) flow field can be expressed in the form $$\rho\left[\frac{\partial \mathbf{V}}{\partial t}+u \frac{\partial \mathbf{V}}{\partial x}+v \frac{\partial \mathbf{V}}{\partial y}+w \frac{\partial \mathbf{V}}{\partial z}\right]=-\nabla p-\rho g \mathbf{k}$$ where \(\rho\) is the density of the fluid, \(\mathbf{V}\) is the velocity vector with components \(u, v,\) and \(w, p\) is the pressure, and \(g\) is the gravity constant. Consider the case in which the only relevant scales are the length scale, \(L,\) and the velocity scale, \(V\). Note that time can be normalized by \(L / V\) and pressure can be normalized by \(\rho V^{2}\). Express Equation 6.29 in normalized form, using the Froude number, Fr, defined as \(\mathrm{Fr}=V / \sqrt{g L},\) in the final expression. What happens to the effect of gravity on the flow as the Froude number becomes large?

When an object of a given shape falls very slowly in a fluid, the steady-state velocity of the object is observed to depend only on the size of the object and the density and viscosity of the fluid. Use dimensional analysis to determine the functional relationship between the steady-state velocity and the influencing variables.

A 1: 60 scale model of a ship is used in a water tank to simulate a ship speed of 10 \(\mathrm{m} / \mathrm{s}\). What should be the model speed? If a towing force of \(10 \mathrm{~N}\) is measured in the model, what force is expected on the prototype? Neglect viscous effects.

Kerosene at \(20^{\circ} \mathrm{C}\) flows through a section of 170 -mm- diameter pipeline with a velocity of \(2.5 \mathrm{~m} / \mathrm{s}\). The pressure loss in the section of pipeline is to be studied using a scale model that has a diameter of \(30 \mathrm{~mm}\), and water at \(20^{\circ} \mathrm{C}\) is to be used in the model study. What water velocity should be used in the model? If a pressure loss of \(15 \mathrm{kPa}\) is measured in the model, what is the corresponding pressure drop in the actual pipeline section?

The length, \(L_{w},\) of the wake behind a particular model aircraft depends on the speed of the aircraft, the size of the aircraft, \(L,\) and the density and viscosity of the ambient air. Develop a nondimensional functional relationship that relates the size of the wake to its influencing variables.
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