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Chapter 7: Problem 53

(See The Wide World of Fluids article "Modeling Parachutes in a Water Tunnel," Section \(7.8 .1 .\) ) Flow characteristics for a \(30-f t\) diameter prototype parachute are to be determined by tests of a 1-fit-diameter model parachute in a water tunnel. Some data collected with the model parachute indicate a drag of 17 lb when the water velocity is \(4 \mathrm{f}\) Us. Lse the model data to predict the drag on the prototype parachute falling through air at \(10 \mathrm{ft} / \mathrm{s}\). Assume the drag to be a function of the velocity, \(V\), the fluid density, \(\rho\), and the parachute diameter, \(D\).

Short Answer

Expert verified
The above steps derived the relationship of the drag force, the fluid density, the velocity, and the diameter of the parachute. Using the given model data, we are able to calculate the constant of proportionality. With this, we can predict the drag on the prototype by substituting the known values into the equation.

Step by step solution

01

Identify the proportionalities

The drag force is a function of the velocity (\(V\)), fluid density (\(\rho\)), and parachute diameter (\(D\)). Therefore, we can write the force (\(F\)) proportionality as: \[ F \propto V^n \cdot \rho^m \cdot D^p \]where the powers \(n\), \(m\), and \(p\) need to be determined.
02

Use dimensional analysis

We use dimensional analysis to work out the powers. The basic dimensions for these quantities are - Force (\(F\)): (mass length/time虏) or (density脳length鲁 length/time虏) = \[\rho^1 \cdot V^0 \cdot D^4\]- Velocity (\(V\)): length/time = \[\rho^0 \cdot V^1 \cdot D^0\]- Density (\(\rho\)): mass/length鲁 = \[\rho^1 \cdot V^0 \cdot D^{-3}\]- Diameter (\(D\)): length = \[\rho^0 \cdot V^0 \cdot D^1\]Replacing these in our proportionality gives \[\rho^1 \cdot V^0 \cdot D^4 \propto (\rho^0 \cdot V^1 \cdot D^0)^n \cdot (\rho^1 \cdot V^0 \cdot D^{-3})^m \cdot (\rho^0 \cdot V^0 \cdot D^1)^p\], equating coefficients we get \(n=2\), \(m=-2\), \(p=1\). Our proportionality then becomes \[ F \propto \rho V^2D\].
03

Use model data to find constants

Substitute the proportionality relationship into an equality by introducing a constant of proportionality (\(C\)):\[ F = C \rho V^2D \]We can use the given model data (that is, the force of 17 lb, the water density, the water velocity of 4 ft/s, and the model diameter of 1 ft) to calculate the constant of proportionality.
04

Predict drag on prototype

We can then use the determined constant of proportionality, the air density, the velocity of the prototype parachute (10 ft/s), and the diameter of the prototype (30 ft) to calculate the resultant force (drag) on the prototype.

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Key Concepts

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

Fluid Mechanics
Fluid Mechanics is an essential discipline when studying how objects move through fluids like air or water. This field examines the forces and flows that occur in fluids, and it applies to a myriad of engineering challenges, including aerospace and marine applications.
  • **Fluid flow** can be categorized into types such as laminar or turbulent. Laminar flow is smooth and orderly, whereas turbulent flow is chaotic and characterized by eddies and vortices.
  • **Density** and **viscosity** are crucial properties that affect how fluids move. Density measures how compact a fluid's mass is, while viscosity describes its resistance to deformation.
Understanding these properties helps predict how a fluid will interact with an object, such as a parachute, and how forces are transmitted through the fluid to the object.
Drag Force
The concept of drag force is central in both fluid mechanics and parachute design. Drag force is the resistance an object experiences as it moves through a fluid. It directly impacts how quickly an object can descend.
  • The drag force depends on the **velocity** of the object through the fluid, **density** of the fluid, and the **surface area** perpendicular to the flow, often associated with the object's shape or diameter鈥攍ike a parachute.
  • Mathematically, drag force is expressed as: \[ F = \frac{1}{2} \cdot C_d \cdot \rho \cdot V^2 \cdot A \\] where \(C_d\) is the drag coefficient, \(\rho\) is the fluid density, \(V\) is the velocity, and \(A\) is the reference area.
This relationship illustrates that increasing velocity or fluid density increases drag force, a concept utilized in Dimensional Analysis to estimate forces in engineering scenarios.
Parachute Modeling
In parachute modeling, engineers aim to predict how a parachute will behave under different conditions by testing models in controlled environments. Parachutes need to be designed to provide enough drag to slow down safely the descent of the payload. This involves scaling model behaviors to predict full-size outcomes.
  • In submerged model tests, parachutes are assessed in water tunnels, providing a low-cost way to observe flow patterns and forces like drag.
  • Scalable physical properties, like **model size** and **fluid properties** (water for models and air for prototypes), play crucial roles in accurate predictions. Engineers employ principles of **similarity and scaling** according to Reynolds numbers to ensure that model tests accurately reflect real-world conditions.
Applying these techniques helps in minimizing risk and optimizing performance for various uses, such as in aerospace and recreational parachuting.
Prototype Testing
Prototype testing is a vital step in the design process that involves using models to predict the performance of full-scale designs. Through this phase, engineers translate model data into reliable predictions for real-world conditions.
  • While testing, parameters such as speed, density, and scale are adjusted to reflect those expected for the prototype, ensuring the results from models inform real scenarios.
  • Dimensional Analysis helps bridge the gap between model and prototype testing, providing a way to derive relationships that remain valid through scale changes.
This strategic phase helps in fine-tuning designs, reducing costs, and enhancing safety by identifying potential problems before full-scale or production stages are reached.

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