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Chapter 6: Problem 85

The performance of a new blimp is to be tested using a scale model in a wind tunnel. The full-scale blimp is designed to move at \(54 \mathrm{~km} / \mathrm{h}\) in standard air. If a 1: 12 model is used in a wind tunnel with air at \(15^{\circ} \mathrm{C}\) and an airspeed of \(80 \mathrm{~m} / \mathrm{s},\) what must the air pressure in the wind tunnel be to achieve dynamic similarity? If dynamic similarity is achieved and a drag force of \(200 \mathrm{~N}\) is measured in the model, what is the corresponding drag force in the prototype?

Short Answer

Expert verified
The air pressure needed ensures dynami similarity, and the prototype drag force is greater than 200N.

Step by step solution

01

Understand Dynamic Similarity

Dynamic similarity involves maintaining similar flow conditions between the model and prototype. This means the Reynolds number (Re) of both the model and prototype should be the same. The Reynolds number is given by \( \mathrm{Re} = \frac{ \rho V L }{ \mu } \), where \( \rho \) is the density, \( V \) is the velocity, \( L \) is the characteristic length, and \( \mu \) is the dynamic viscosity.
02

Calculate Reynolds Number for the Model

For the model, the characteristic length \(L_m\) is the scale ratio times the prototype length \(L_p\). Let \( V_m = 80\, \mathrm{m/s} \), and the air's density \( \rho_m \) and dynamic viscosity \( \mu_m \) are determined at \( 15^{\circ} \mathrm{C} \). Substitute these into the Reynolds number equation to calculate \( \mathrm{Re_m} \).
03

Calculate Reynolds Number for the Prototype

For the prototype, \( V_p = 54\, \mathrm{km/h} = 15\, \mathrm{m/s} \) and the density \( \rho_p \) and dynamic viscosity \( \mu_p \) are determined at standard air conditions. Substitute these into the Reynolds number equation to calculate \( \mathrm{Re_p} \).
04

Solve for Pressure to Achieve Dynamic Similarity

Since \( \mathrm{Re_m} = \mathrm{Re_p} \), equate the expressions for the Reynolds numbers and solve for the unknown pressure in the wind tunnel \( \rho_m \), considering the conversion factors. Adjust the pressure in the wind tunnel until both Reynolds numbers are equal.
05

Calculate Drag Force on the Prototype

Once dynamic similarity is achieved with the appropriate pressure, use the drag force scaling relationship. The drag force on the prototype \( F_p \) relates to the drag force on the model \( F_m \) by \( F_p = F_m \left( \frac{L_p}{L_m} \right)^2 \). Substitute \( F_m = 200\, \mathrm{N} \) and the given scale ratio to find \( F_p \).

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Key Concepts

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

Reynolds number
The Reynolds number (\( \mathrm{Re} \)) is a dimensionless value that helps us understand flow dynamics. It鈥檚 calculated using the formula:\[\mathrm{Re} = \frac{ \rho V L }{ \mu }\]Where:
  • \( \rho \) is the density of the fluid
  • \( V \) is the velocity of the fluid
  • \( L \) is the characteristic length
  • \( \mu \) is the dynamic viscosity
The Reynolds number is crucial for achieving dynamic similarity between model tests and full-scale prototypes. By maintaining the same Reynolds number, engineers can predict how the full-scale version will perform based on model testing. It allows us to maintain similar flow characteristics, like turbulence and drag effects, regardless of the size of the model.
scale model testing
Scale model testing is a method used to predict the real-world behavior of large structures, like aircraft or blimps, by testing smaller versions. This process helps in understanding phenomena like under controlled environments.
  • It reduces costs since building a full-scale model is expensive and time-consuming.
  • It allows testing different designs quickly to find optimal performance.
  • Also, it ensures safety by identifying potential risks without affecting the actual prototype.
  • The results need to be interpreted correctly, considering the differences between the model and prototype environment.
When a 1:12 scale model blimp was tested, the findings allowed for adjustments in the design based on drag and flow conditions. This testing in a wind tunnel provided critical data that engineers use to improve design efficiency.
aerodynamic drag
Aerodynamic drag is the resistance a body experiences as it moves through a fluid, such as air or water. It is a crucial factor when designing any object that moves through air, including vehicles and aircraft. Drag force can greatly affect the speed, fuel efficiency, and stability of a vehicle.
There are two main types of aerodynamic drag:
  • Form drag: Caused by the shape of the object and how the pressure of the fluid impacts it.
  • Skin friction drag: Caused by the friction between the fluid and the surface of the object.
In scale model testing, once a model's drag force is measured under controlled conditions, engineers can translate that measurement using the scale factor. For example, if the drag force of a model is 200 N, scaling up the computed results allows for predictions of full-scale behavior. This helps engineers make design tweaks to reduce drag, resulting in more efficient real-world performance.
wind tunnel experiments
Wind tunnel experiments are an essential part of aviation and vehicle testing. They allow engineers to simulate conditions that an object would encounter in its intended environment. By doing so, it鈥檚 possible to measure forces like lift and drag on the scale model with great accuracy.
The wind tunnel provides a controlled environment to test:
  • Different velocities and pressures
  • Various angles of attack
  • Temperature changes
For the blimp model, a wind tunnel was used to achieve the necessary conditions for dynamic similarity. The air velocity was set to 80 m/s, while pressure was adjusted to match the Reynolds number of the full-size blimp, thus ensuring accurate results. These experiments are essential because they help in recognizing issues that might not have been apparent in theoretical analysis or smaller tests. Understanding the aerodynamic forces in the wind tunnel allows developers to refine designs before investing in full-scale production.

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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?

An orifice plate is a flat plate with a central opening that is sometimes used to measure the volume flow rate in a pipe. The pressure drop across an orifice plate can be assumed to be a function of the diameter of the opening in the plate, the diameter of the pipe, the velocity of flow in the pipe, and the density and viscosity of the fluid. Use dimensional analysis to determine the functional relationship between the pressure drop across the plate and the influencing variables. Express this functional relationship in both a dimensional and nondimensional form. Identify any named conventional dimensionless groups that appear in the nondimensional relationship.

In a particular two-dimensional flow field of an incompressible fluid in the \(x z\) plane, the \(z\) component of the momentum equation is given by $$\rho u \frac{\partial w}{\partial x}=\mu\left(\frac{\partial^{2} w}{\partial x^{2}}+\frac{\partial^{2} w}{\partial z^{2}}\right)-\rho g$$ where \(u\) and \(w\) are the \(x\) and \(z\) components of the velocity, respectively, \(\rho\) and \(\mu\) are the density and dynamic viscosity of the fluid, respectively, and \(g\) is the gravity constant. The relevant scales are the length scale, \(L,\) and the velocity scale, \(V\). Express Equation 6.30 in normalized form, using the Reynolds number, Re, defined as \(\mathrm{Re}=\rho V L / \mu,\) and the Froude number, Fr, defined as \(\mathrm{Fr}=V / \sqrt{g L},\) in the final expression. What is the asymptotic form of the governing equation as the Reynolds number becomes large?

A prototype water pump with an impeller diameter of \(470 \mathrm{~mm}\) is designed to operate at a rotational speed of \(950 \mathrm{rpm},\) and at this speed, the pump delivers a volume flow rate of \(1.7 \mathrm{~m}^{3} / \mathrm{s}\). The temperature of the water in the prototype pump is \(20^{\circ} \mathrm{C}\). The performance of a 1: 6 scale model of the pump is tested using standard air as the fluid. The model pump has a rotational speed of \(1750 \mathrm{rpm},\) and the power required to drive the model pump is \(95 \mathrm{~W}\). The Reynolds number in the prototype and the model are both sufficiently high that Reynolds similarity is not a requirement. (a) Determine the volume flow rate of air in the model that corresponds to the design volume flow rate of water in the prototype. (b) Determine the power requirement of the prototype that corresponds to the measured power requirement in the model.

An atomizer is a common term used to describe a spray nozzle that produces small droplets of a liquid that is drawn into the nozzle by the low pressure that exists in the nozzle. Consider a nozzle of diameter \(D\) that generates droplets of diameter \(d\) when the velocity at the nozzle exit is \(V\). The relevant liquid properties are the density, \(\rho\), the viscosity, \(\mu,\) and the surface tension, \(\sigma .\) With the objective of predicting the size of the droplets generated by an atomizer, express the relationship between the relevant variables in nondimensional form where, to the extent possible, dimensionless groups representing force ratios are used with each of the fluid properties.
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