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Dimensional Analysis in Fluid Mechanics: Unlocking Dynamic Similarity

Dive into the world of Fluid Mechanics with Dimensional Analysis. Understand the significance of the Dimensional Matrix and Buckingham’s PI theorem in solving complex problems. Explore the transition from laminar to turbulent flow using Reynolds Number and Froude Number. Witness the applications in real-life scenarios like the Great Fountain Geyser in Yellowstone. Discover the essence of inertial vs. pressure forces and the Strouhal Number in analyzing motion patterns.

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Dimensional Analysis in Fluid Mechanics: Unlocking Dynamic Similarity

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  1. Dynamic Similarity Dimensionless Science http://www.typefreediabetes.com/Articles.asp?ID=150 http://www.eosnap.com/?tag=strait-of-gibraltar

  2. Ideas needed to perform Dimensional Analysis All terms in equation must have same dimensions Terms in equation can be expressed with 3 basic dimensions: Mass, Length, Time Dimensions of these variables can be arranged in “Dimensional Matrix”

  3. Dimensional Matrix The rank r of a matrix is the size of the largest square submatrix with non-zero determinant r = 3 Most problems in fluid mechanics Buckingham’s PI theorem: “n variables can be combined to form exactly (n-r) independent non-dimensional variables.”

  4. inertial vs viscous Reynolds Number

  5. 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 Low Re High Re

  6. Consider an oceanic flow where U = 0.1 m/s; L = 10 km; kinematic viscosity = 10-6 m2/s Is friction negligible in the ocean?

  7. Froude Number inertial vs pressure

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

  9. Euler Number pressure vs inertial

  10. Great Fountain Geyser, Yellowstone National Park, USA http://www.freefoto.com/browse/1222-02-0?ffid=1222-02-0

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

  12. Strouhal Number local vs inertial f = frequency of motion; A = amplitude of motion – for flying or swimming organisms (fin or wing)

  13. Taylor at al. (2003, Nature 425, 707-711(16 ) doi:10.1038/nature02000) “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.” http://www.nature.com/nature/journal/v425/n6959/fig_tab/nature02000_F1.html#figure-title

  14. Dynamic Similarity http://www.typefreediabetes.com/Articles.asp?ID=150 http://www.eosnap.com/?tag=strait-of-gibraltar

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