Lecture 16

.title[
# Lecture 16
]
.subtitle[
## Re: Shape
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/03/14
]

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<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css">

- Review Re

- Consider Shape, drag, and Re

- Sharks, wings, and a crisis

]

]

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# Viscocity and Drag

What is the relationship between fluid motion and pressure?

Along a streamline PE + KE + W + **E dissipated** = constant

Viscosity robs fluid of its momentum. There is a shear stress exerted on the sphere and energy is dissipated by viscosity.

Because of viscosity, velocity cannot increase as much as in the inviscid case.

]

`\((P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0\)`
 
 `\(\small{\frac{\text{Pressure stress}}{\text{Shear stress}} = Re= \frac{\rho uL}{\mu}}\)`
 
<img src="img/paradox2.png" width="650" />

Net forces when `\(\mu\neq0\)`
]

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# Viscocity and Drag

`\((P_2-P_1)/\rho+(u_2^2-u_1^2)/2=0\)`
 
 Because of viscosity, velocity cannot increase as much as in the inviscid case. 
 
 Therefore, we have flow separation.

<img src="https://www.tec-science.com/wp-content/uploads/2021/04/en-gases-liquids-fluid-mechanics-boundary-layer-flow-separation-cylinder-turbulent.jpg" width="550" />
]

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# Drag Forces: Friction and Pressure Stresses

Drag forces arise from pressure and frictional (skin) stresses

`\(D\sim\rho u^2/2\)`

`\(D=C_d\rho S u^2/2\)`

`\(C_d\)`= drag coefficient

`\(S\)`= cross sectional (frontal) area

`\(C_d=\frac{D}{\frac{1}{2}\rho u^2S}\)`

`\(C_d\)` a function of shape and Re

]

.pull-right[
<br>
<br>
<br>
 
<img src="https://ars.els-cdn.com/content/image/3-s2.0-B9780123973085000155-f15-11-9780123973085.jpg" width="650" />

]

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# Shape and drag

]
.footnote[Must specify Re!]

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# Shape and drag

]
.pull-left[.footnote[Must specify Re!]]
.pull-right[.footnote[T. Daniel, UW]]

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# Shape and drag

Fineness: manage your wake
 
fineness ratio=L\D

]

]

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# Shape and drag

Fineness: manage your wake
 
fineness ratio=L\D

Where would you find these fishes? What behaviors would the exhibit? Trade offs?
]

<br>

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# Shape and drag

Drag isn't always bad!
 
Drag can be a propulsive forces!

<img src="https://greenpestservices.net/wp-content/uploads/2015/06/backswimmers.jpg" width="300" />
]

]

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# Sharks, wind, and a crisis

The drag crisis

`\(\small{\frac{\text{Pressure stress}}{\text{Shear stress}} = Re= \frac{\rho uL}{\mu}}\)`

![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg/1200px-Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg.png)
]

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# Sharks, wind, and a crisis

Skin (friction) drag in the boundary layer under laminar flow (low Re)

![](https://www.tec-science.com/wp-content/uploads/2021/04/en-gases-liquids-fluid-mechanics-boundary-layer-flow-separation-cylinder-velocity-profile.jpg)
]

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# Sharks, wind, and a crisis
.center[

Skin (friction) drag in the boundary layer under laminar flow (low Re)

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# Sharks, wind, and a crisis

![](https://www.tec-science.com/wp-content/uploads/2021/04/en-gases-liquids-fluid-mechanics-boundary-layer-flow-separation-laminar-turbulent.jpg)

]

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# Sharks, wind, and a crisis

Put, where have you seen this?

]

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# Sharks, wind, and a crisis

Put, where have you seen this?

![](https://www.tec-science.com/wp-content/uploads/2021/04/gases-liquids-fluid-mechanics-boundary-layer-flow-separation-golf-ball-separation-why-dimples.jpg)

]

# Sharks, wind, and a crisis

Put, where have you seen this?

]
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# Sharks, wind, and a crisis

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# Sharks, wind, and a crisis

![](https://pubs.rsc.org/image/article/2021/RA/d0ra08672j/d0ra08672j-f1_hi-res.gif)

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# Sharks, wind, and a crisis

]

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# Sharks, wind, and a crisis

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# Thanks!

Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).