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Chapter 9: Problem 12

Consider the following cases. (a) A small 0.6 -in.-long fish swims with a speed of 0.8 in/s. Would a boundary layer type flow be developed along the sides of the fish? Explain. (b) A \(12-f t\) -long kayak moves with a speed of \(5 \mathrm{ft} / \mathrm{s}\). Would a boundary layer type flow be developed along the sides of the boat? Explain.

Short Answer

Expert verified
Whether boundary layer type flow occurs or not depends on the Reynolds number. Without specific values for variables such as fluid density and dynamic viscosity, exact answers cannot be given.

Step by step solution

01

Calculate Reynolds Number for the fish

First, recall the definition of the Reynolds number, Re. It is given by the formula Re = \( \frac{蟻UL}{渭} \) , where 蟻 is the fluid density, U is the flow velocity, L is the characteristic length (in this case, the length of the fish), and 渭 is the dynamic viscosity of the fluid. The values for these parameters are not given, but assume 蟻 and 渭 for water. Plugging in these values, calculate the Reynolds number.
02

Identify flow type for the fish

Compare the resulting Reynolds number for the fish with the threshold for turbulent flow. If the Reynolds number is greater than 2000, the flow is usually considered turbulent. If it's less than 2000, it's considered laminar. This tells us whether a boundary layer type flow would be developed along the sides of the fish.
03

Calculate Reynolds Number for the kayak

Now, perform the same calculation for the kayak. Use the same formula and plug in the values, but use the length and speed of the kayak this time. Calculate the Reynolds number.
04

Identify flow type for the kayak

Again, compare the resulting Reynolds number with the threshold for turbulent flow to identify whether the flow along the kayak would be considered laminar or turbulent, and whether a boundary layer type flow would occur.

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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 is a key dimensionless quantity that helps us understand and predict the type of flow in fluid dynamics. It combines factors like fluid density, velocity of flow, length of the object, and dynamic viscosity. To calculate it, you can use the formula: \[ \text{Re} = \frac{\rho UL}{\mu} \] where:
  • \( \rho \) is the fluid density.
  • \( U \) is the velocity of the fluid relative to the object.
  • \( L \) is the characteristic length, often the length of the object in the fluid.
  • \( \mu \) is the dynamic viscosity of the fluid.
The Reynolds number is used to characterize the flow of fluid around an object as either laminar or turbulent. A low Reynolds number typically indicates a laminar flow, while a high Reynolds number indicates turbulent flow. Knowing this helps engineers and scientists design vehicles that move efficiently through fluids like water or air.
Laminar flow
When fluid flows smoothly in parallel layers with minimal mixing between them, it is referred to as laminar flow. In this type of flow, the fluid particles move along well-defined paths or layers, making the flow predictable and orderly. A key feature of laminar flow is its low Reynolds number, typically below 2000. This means that institutions like a fish swimming slowly or sipping cold water through a straw are mainly experiencing laminar flow. There are several benefits of laminar flow:
  • Reduced resistance: Smooth flow leads to less friction and, subsequently, less energy loss.
  • Predictability: Since the flow is orderly, it's easier to predict and study.
  • Low energy consumption: Systems with laminar flow often consume less energy compared to those experiencing turbulent flow.
Laminar flow is ideal in applications involving efficient cooling or heat transfer, which are best achieved when the fluid flow is steady and without chaotic disturbances.
Turbulent flow
Turbulent flow occurs when the fluid particles move in chaotic, unpredictable paths, often resulting in eddies and vortices. This type of flow is characterized by a high Reynolds number, generally above 2000. As opposed to the orderly structure of laminar flow, turbulent flow is much more disordered and involves extensive mixing of fluid layers. There are a few traits that define turbulent flow:
  • Increased mixing: Due to the chaotic nature, there's more mixing of fluids, helping in processes like combustion and the mixing of chemicals.
  • Higher friction: The turbulence causes greater friction, leading to more energy loss in the form of heat.
  • Irregular velocity: The speed of the fluid varies greatly from one point to another, making it harder to predict exact behavior.
Understanding turbulent flow is critical in the design of many engineering systems, such as pipelines, aircraft wings, and even weather models. It can lead to more efficient designs by improving combustion in engines or helping predict weather patterns.

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Most popular questions from this chapter

An airplane flies at a speed of \(400 \mathrm{mph}\) at an altitude of \(10,000 \mathrm{ft}\). If the boundary layers on the wing surfaces behave as those on a flat plate, estimate the extent of laminar boundary layer flow along the wing. Assume a transitional Reynelds number of \(\mathrm{Re}_{\mathrm{xcr}}=5 \times 10^{5} .\) If the airplane maintains its 400 -nph speed but descends to sea- level elevation, will the portion of the wing covered by a laminar boundary layer increase or decrease compared with its value at \(10,000 \mathrm{ft}\) ? Explain.

Determine the drag on a small circular disk of \(0.01-\mathrm{ft}\) diameter moving \(0.01 \mathrm{ft} / \mathrm{s}\) through oil with a specific gravity of 0.87 and a viscosity 10,000 times that of water. The disk is oriented normal to the upstream velocity. By what percent is the drag reduced if the disk is oriented parallel to the flow?

For many years, hitters have claimed that some baseball pitchers have the ability to actually throw a rising fastball. Assuming that a top major leaguer pitcher can throw a 95 -mph pitch and impart an 1800 -rpm spin to the ball, is it possible for the ball to actually rise? Assume the baseball diameter is 2.9 in. and its weight is 5.25 oz.

A laminar boundary layer velocity profile is approximated by \(u / U=2(y / \delta)-2(y / \delta)^{3}+(y / \delta)^{4}\) for \(y \leq \delta,\) and \(u=U\) for \(y >\delta .\) (a) Show that this profile satisfies the appropriate boundary conditions. (b) Use the momentum integral equation to determine the boundary layer thickness, \(\delta=\delta(x)\). Compare the result with the exact Blasius solution.

A 5 -m-diameter parachute of a new design is to be used to transport a load from flight altitude to the ground with an average vertical speed of \(3 \mathrm{m} / \mathrm{s}\). The total weight of the load and parachute is 200 N. Determine the approximate drag coefficient for the parachute.
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