Dynamic Similarity

http://www.typefreediabetes.com/Articles.asp?ID=150

upgDt

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http://www.eosnap.com/?tag=strait-of-gibraltar

Dimensionless Science

Ideas needed to perform Dimensional Analysis

upgDt

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2 All terms in equation must have same dimensions

Terms in equation can be expressed with 3 basic dimensions: Mass, Length, Time

LTMLTMpLMTLu 23

Dimensions of these variables can be arranged in “Dimensional Matrix”

10121311

1101

TLM

up

10121311

1101

TLM

up

Dimensional Matrix

The rank r of a matrix is the size of the largest square submatrix with non-zero determinant

r = 3Most problems in fluid mechanics

Buckingham’s PI theorem: “n variables can be combined to form exactly (n-r) independent non-dimensional variables.”

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2

2

2

2

2

21)zu

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xp

zuw

yuv

xuu

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2

2

2

2

2

21)zu

yu

xu

xp

zuw

yuv

xuu

tux

Re2

2

2

2

ULLU

LUxu

xuuinertial vs viscous

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Reynolds Number

Low Re

High Re

Flow is laminar when Re < 1000

Flow is in transition to turbulence when 100 < Re < 105 to 106

Flow is turbulent when Re > 106, unless the fluid is stratified

UL

Re

Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s

610

100001.0Re 910

Is friction negligible in the ocean?

2

2

2

2

2

21)zu

yu

xu

xp

zuw

yuv

xuu

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FrgHULgHLU

xp

xuu

2

21inertial vs pressure

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Froude Number

1 gHUFr

1 gHUFr

http://www.yourlocalweb.co.uk/greater-manchester/city-of-manchester/higher-blackley/pictures/

2

2

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21)zu

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xuu

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2

21 pressure vs inertial

TtHgpHzLxUu

Euler Number

http://www.freefoto.com/browse/1222-02-0?ffid=1222-02-0

Great Fountain Geyser, Yellowstone National Park, USA

2UpEu

http://ya.astroleague.org/?p=1445

2UpEu

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2

2

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21)zu

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zuw

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StTULLUTU

xuu

tu

2local vs inertial

TtHgpHzLxUu

Strouhal Number

UAfSt

f = frequency of motion; A = amplitude of motion – for flying or swimming organisms (fin or wing)

Taylor at al. (2003, Nature 425, 707-711(16 )doi:10.1038/nature02000)

http://www.nature.com/nature/journal/v425/n6959/fig_tab/nature02000_F1.html#figure-title

“Left panels, root-flapping motion; right panels, heaving motion. Amplitude, twice wing chord; static angle of attack, 15°; flow speed, 1.5 ms-1; smoke wire visualizations made at end of downstroke.

a, For St < 0.10, flow separates at the sharp leading-edge, but no discrete vortex forms.

b, For 0.10 < St < 0.25, a leading-edge vortex forms but is shed before the downstroke ends.

c, For 0.25 < St < 0.45, the leading-edge vortex is shed as the downstroke ends.

d, For St > 0.45, trailing edge separation produces a characteristic mushroom-shaped wake.

At higher St, the wing collides with shed vorticity on the upstroke, giving an energetically inefficient mode.”

UAfSt

AfAfU 3.33.0

Dynamic Similarity

http://www.typefreediabetes.com/Articles.asp?ID=150

upgDt

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http://www.eosnap.com/?tag=strait-of-gibraltar

UTLStUpEugHUFr

UL

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Re