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Chapter 5: Problem 67

A student needs to measure the drag on a prototype of characteristic dimension \(d_{p}\) moving at velocity \(U_{p}\) in air at standard atmospheric conditions. He constructs a model of characteristic dimension \(d_{m},\) such that the ratio \(d_{p} / d_{m}\) is some factor \(f .\) He then measures the drag on the model at dynamically similar conditions (also with air at standard atmospheric conditions). The student claims that the drag force on the prototype will be identical to that measured on the model. Is this claim correct? Explain.

Short Answer

Expert verified
Yes, the claim is correct under conditions of dynamic similarity.

Step by step solution

01

Understand Dynamic Similarity

For dynamic similarity, the Reynolds number of both the prototype and model should be the same. The Reynolds number is given by \( Re = \frac{\rho U L}{\mu} \), where \( \rho \) is the fluid density, \( U \) is the velocity, \( L \) is the characteristic dimension, and \( \mu \) is the dynamic viscosity.
02

Apply Reynolds Number for Prototype and Model

Calculate the Reynolds numbers for the prototype and the model. For the prototype, \( Re_p = \frac{\rho U_p d_p}{\mu} \), and for the model, \( Re_m = \frac{\rho U_m d_m}{\mu} \). For dynamic similarity, \( Re_p = Re_m \).
03

Analyze the Velocity Relationship

Since the densities and viscosities are equal (same fluid), the Reynolds number similarity condition becomes \( U_p d_p = U_m d_m \). Thus, \( U_p = U_m \frac{d_m}{d_p} \), or \( U_p = U_m \frac{1}{f} \) because \( \frac{d_p}{d_m} = f \).
04

Examine the Claim on Drag Force

In dynamically similar situations, the drag ratio is proportional to the square of the velocity ratio squared multiplied by the area ratio. The drag force on a body is given by \( F_D = \frac{1}{2} \rho C_D A U^2 \), where \( C_D \) is the drag coefficient, and \( A \) is the area.
05

Apply the Drag Force Equation

For dynamic similarity, \( F_{D,p} = F_{D,m} \left( \frac{U_p}{U_m} \right)^2 \left( \frac{A_p}{A_m} \right) \). However, since \( U_p = U_m \frac{1}{f} \), then \( F_{D,p} = F_{D,m} \left( \frac{1}{f} \right)^2 \left( f^2 \right) \), which indicates the forces are equal.
06

Final Conclusion

The student's claim that the prototype's drag force is identical to the model's drag force is correct under the conditions of dynamic similarity, as the factors influencing the drag are balanced between scale and velocity adjustments.

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Key Concepts

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

Reynolds number
In fluid mechanics, the Reynolds number is a dimensionless quantity that helps us understand the flow characteristics of a fluid around an object. It is crucial when we model and analyze fluid flow conditions. The formula for the Reynolds number is given by \[ Re = \frac{\rho U L}{\mu} \]. This equation involves:
  • \( \rho \): the fluid density
  • \( U \): the flow velocity relative to the object
  • \( L \): a characteristic length or size of the object
  • \( \mu \): the dynamic viscosity of the fluid, which represents the internal friction
The Reynolds number helps to predict flow patterns in different fluid flow situations. With identical Reynolds numbers, two different geometries or flow conditions will experience similar flow characteristics. Hence, when creating scale models in experiments, matching the Reynolds number ensures the model faithfully represents the prototype's flow behavior.
drag force calculation
Calculating drag force is essential when determining how much resistance an object encounters when moving through a fluid. The drag force equation is expressed as \[ F_D = \frac{1}{2} \rho C_D A U^2 \]. It consists of:
  • \( F_D \): the drag force
  • \( \rho \): the fluid density
  • \( C_D \): the drag coefficient, which depends on the shape of the object and its flow conditions
  • \( A \): the reference area, typically the frontal area of the object
  • \( U \): the velocity of the object relative to the fluid
Drag force calculation involves understanding how changes in velocity or size affect the resistance faced by the object. When these variables are adjusted appropriately, especially in a dynamic similarity scenario, drag forces on scaled models can accurately represent those experienced by the full-sized object.
scale modeling in aerodynamics
Scale modeling in aerodynamics plays a vital role in testing and predicting the behavior of full-sized objects like airplanes and cars. Models help engineers understand airflow and forces without the need for full-sized prototypes. A key aspect of scale modeling is ensuring dynamic similarity, particularly ensuring that Reynolds numbers match between model and prototype. To achieve dynamic similarity, engineers may need to adjust the model's speed or the test fluid's properties. In practical terms:
  • The size and velocity of the scale model must be adjusted to achieve the same flow conditions as the prototype.
  • Maintaining proportional relationships in terms of characteristics like force and speed is essential.
  • Due to dynamic similarity, drag forces and other aerodynamic properties can be extrapolated from the model to the prototype.
Scale modeling offers a cost-effective way of predicting performance and troubleshooting potential issues before manufacturing full-sized objects.

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Most popular questions from this chapter

The speed of propagation \(C\) of a capillary wave in deep water is known to be a function only of density \(\rho,\) wavelength \(\lambda,\) and surface tension \(Y\). Find the proper functional relationship, completing it with a dimensionless constant. For a given density and wavelength, how does the propagation speed change if the surface tension is doubled?

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