## Discussion

### reynolds number

The Reynolds number (*Re*) is the ratio of inertial resistance to viscous resistance for a flowing fluid. It is named after the British physicist and engineer Osborne Reynolds who is generally regarded as the first to realize its importance in 1883.

Re = |
inertial resistance |
= | ρvℓ |

viscous resistance |
η |

Where…

Re = |
Reynolds number |

ρ = | density of the fluid |

v = |
relative speed of the fluid |

ℓ = | characteristic length of the system |

η = | dynamic viscosity of the fluid |

- The Reynolds number (
*Re*) is the ratio of inertial resistance to viscous resistance for a flowing fluid. - The Reynolds number is a non-dimensional (unitless) factor governing resistance due to viscosity (among other things).
- Reynolds, Osborne." An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels." Royal
*Society, Philosophical Transactions*, 1883. - Reynolds' experiments on flow through pipes. Compare pressure losses due to viscous friction with flow speed through a horizontal pipe. Log-log graph of pressure gradient (Δ
*P*/Δ*d*) vs. flow speed (*v*). The slope of the line of best fit on a log-log graph is the power (*n*) relating the explanatory variable (flow speed) to the response variable (pressure gradient).

- Reynolds, Osborne." An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels." Royal
- Flow Regimes
- For low Reynolds numbers the behavior of a fluid depends mostly on its viscosity and the flow is steady, smooth, viscous, or laminar and
*n*= 1. - For high Reynolds numbers the momentum of the fluid determines its behavior more than the viscosity and the flow is unsteady, churning, roiling, or turbulent and
*n*= 2. - For intermediate Reynolds numbers the flow is transitional — partly laminar and partly turbulent.

- For low Reynolds numbers the behavior of a fluid depends mostly on its viscosity and the flow is steady, smooth, viscous, or laminar and
- Scaling
- The full implications of the Reynolds number were never realized by Reynolds who considered the ratio merely as a criterion for the critical velocity in pipe flow. Lord Rayleigh has shown that it is a non-dimensional factor which governs all problems on fluid flow frictional resistance, and that similar non-dimensional constants exist for many other natural phenomena.
- It is a practice in engineering design that when a large object such as a ship, airplane, or building is to be made, a scale model is constructed and tested so that the performance of the large object can be calculated from the test results of the scale model. Lord Rayleigh showed that the scale model tests gave comparable results only when the non-dimensional factor of the model is equal to that of the large object when working under its design conditions. By equating the non-dimensional factor of the large object to that of the model, the test speed of the model is obtained. This is known as the corresponding speed and the comparison of the two conditions between the large object and the test results of a scale model at its corresponding speed is known as the principle of dynamic similarity.
- From NASA "By 1921 more than a score of wind tunnels had been constructed the world over. But all those of substantial size were operating at normal atmospheric pressures. This meant that the experimental results obtained using scale models in the tunnels were open to question because a special parameter called the Reynolds number did not match those encountered in the actual flights of full-scale aircraft. In other words, the Reynolds number of 1/20-scale models being tested at operational flight velocities would be too low by a factor of 20. Reynolds' classic experiments had shown that airflow conditions could be radically different for model and full-scale aircraft. Since the Reynolds number is also proportional to air density, an obvious solution to the problem of scale effects would be to test 1/20-scale models at a pressure of 20 atmospheres. The Reynolds number would then be the same in the wind tunnel tests and actual full-scale flights."

Fluid Flow Regimes as a Function of Reynolds Number.

object | lower | upper |
---|---|---|

circular pipes | 2,000 | 2,500 |

flat plates | 300,000 | 500,000 |

Re |
animal | Re |
aircraft | |
---|---|---|---|---|

62,000 | seagull | 2,000,000,000 | boeing 747 | |

50,000 | large fish | 110,000,000 | typical commercial jet | |

3,900 | butterfly | 6,300,000 | cessna | |

1,000 | honeybee | 4,700,000 | light plane | |

300 | african frog tadpole | 1,600,000 | glider | |

120 | housefly | 250,000 | model airplane | |

15 | chalcid wasp | 47,000 | paper airplane | |

0.2 | paramecium | |||

0.025 | dinoflagellate | |||

0.0035 | spermatozoa, sea urchin | |||

0.000,01 | bacterium | |||

Re |
circulatory system | Re |
miscellaneous | |

3,400 | aorta | 250,000,000 | cumulus cloud formation | |

3,300 | vena cava | |||

500 | artery | |||

140 | vein | |||

0.7 | arteriole | |||

0.01 | venule | |||

0.002 | capillary |

### mach number

The Mach number (*Ma*) is the ratio of the speed of an object moving through a fluid (*v*) to the speed of sound in that fluid (*c*). As a ratio of two speeds, it is dimensionless and unitless.

Ma = |
v |

c |

The Mach number is named after the Austrian physicist and philosopher of science Ernst Mach who first predicted that objects traveling faster than the speed of sound have a conical shock wave trailing behind them. English speakers usually pronounce the German "ch" as a "k" so that "Mach" sounds like "mock". German speakers pronounce "ch" as a voiceless post-velar fricative. (Think of the proper Scottish pronounciation of Loch Ness.)

The Mach number also doubles as a unit. An airplane flying through the air with a speed equal to the speed of sound in air at that location has a Mach number of 1. It could also be described as flying at Mach 1. The odd reversal of the usual number-unit sequence to unit-number is unique to the Mach number.

The Mach number is meaningful because it is a comparison of the inertial resistance to the compressional resistance experienced by an object moving through a fluid.

START FIX

No two bodies can occupy the same place at the same time. When a solid object and a fluid are in relative motion — like a bird flying through the air or the wind blowing around a mountain — it is usually the fluid that yields to the solid. Solids are held together by intermolecular forces and atomic bonds. If the cohesive forces between the particles in a solid are considered significant and long lasting, then the cohesive forces in a liquid are weak and short lived. In a gas they are virtually nonexistent. You might think that fluids are a pushover for a moving solid, but this is not always the case.

The molecules that make up even the most tenuous of gases won't be able to get out of the way of a solid body moving at a considerable speed. Meteors quite commonly break up on entering the earth's atmosphere from space. (They also burn up, but that is as much an result of frictional heating as it is of trying to push the air out of the way.) Aircraft are known to have broken up during flight from the buffeting effects of moving air on a weakened or damaged part. Less commonly and more unfortunately, so too have spacecraft.

In 2003 the Space Shuttle Columbia broke apart in the upper atmosphere during its final descent to landing. A piece of foam insulation about the size of a briefcase fell off the external fuel tank while the shuttle was taking off. This punched a fist-sized hole into the leading edge of the orbiter's left wing. The hot plasma produced when the shuttle reenters the earth's atmosphere eventually melted the aluminum frame holding the wing in place. It snapped off and the air rushing by tore the orbiter to pieces. Contact was lost somewhere over northeastern Texas at an altitude of 62,000 m — on the edge of space where the pressure and density of the atmosphere are roughly one-ten thousandth of their values at sea level. Columbia was scheduled to land sixteen minutes later in Florida. Flying this distance in a commercial jet would take something like two hours and forty-five minutes — roughly ten times longer. Columbia was destroyed by an exceptionally tenuous gas while it flying at an exceptionally high speed. At the time of last contact it was traveling at nearly 5,600 m/s.

END FIX

The Mach number squared (*Ma*^{2}) is a ratio of inertial resistance to compressional resistance. It is the non-dimensional factor governing resistance due to longitudinal, compressional wave formation (a.k.a. sound waves).

Ma^{2} = |
F_{inertial} |

F_{compressional} |

The force needed to start or stop an object comes from Newton's second law of motion.

F = _{inertial}ma = |
mv |

t |

When the object being started or stopped is a "piece" of a fluid, the mass (*m*) in Newton's second law is the density of the fluid (ρ) times the volume of the piece (*V*).

F = _{inertial} |
ρVv |

t |

The volume of fluid shoved aside when a solid object passes through a fluid is the cross-sectional area of the object (*A*) multiplied by the distance it travels (*vt*).

F = _{inertial} |
ρ(Avt)v |

t |

Simplify.

*F _{inertial}* = ρ

*v*

^{2}

*A*

Simplify even more using the notion of characteristic length (ℓ).

*F _{inertial}* = ρ

*v*

^{2}ℓ

^{2}

The force needed to compress a "piece" of a fluid is the surface area of the piece (*A*) times the bulk modulus of the fluid (*K*).

*F _{compressional}* =

*KA*

This expression can also be simplified using the notion of characteristic length (ℓ).

*F _{compressional}* =

*K*ℓ

^{2}

Combining these two forces gives one expression for the Mach number.

Ma^{2} = |
ρv^{2}ℓ^{2} |

Kℓ^{2} |

The Mach number is equal to the ratio of the speed of flow (*v*) to the speed of sound in a fluid (*c*), which is how it is usually written.

Ma^{2} = |
ρv^{2}ℓ^{2} |
& | c^{2} = |
K |
⇒ | Ma = |
v |

Kℓ^{2} |
ρ | c |

Mach 0.5 corresponds to a flow speed that's half the speed of sound, Mach 2 to a flow speed that's twice the speed of sound, and so on. Fluid flow can be broken up into two general regimes by Mach number: those less than Mach 1 are said to be subsonic, while those greater than Mach 1 are said to be supersonic.

A body moving through a fluid at speeds less than the speed of sound in the fluid is preceded by a region of gradually varying density and pressure. At speeds greater than the speed of sound, such a gradual transition is not possible and a shock wave of nearly discontinuously changing pressure and density is formed. In the case of a supersonic aircraft or a bullet, this shock wave is a double walled cone that forms with the front and back of the object at its vertices (projections in between like wings and stabilizers are placed at the vertices of intermediate shock waves).

Shock waves can also form whenever a fluid is heated so rapidly that the leading edge of its expansion travels at or above the speed of sound in the fluid. Roughly spherical shock waves form when bombs, fireworks, and other pyrotechnic devices explode. A bolt of lightning generates a cylindrical shock wave centered on the bolt's path. The sound of a shock wave produced by a supersonic aircraft is called a sonic boom, while the sound of a shock wave produced by lightning is called thunder.

Mach numbers between 0.8 and 1.5 are said to be transonic. A transonic flow over an aircraft wing will have pockets of subsonic and supersonic flow mixed together, which leads to a loss of stability. The effects of the so called sound barrier also tend to be significant and flight can very easily become hard to control.

When the Mach number in a fluid approaches 5, the behavior of the fluid depends more upon the Reynolds number than the Mach number and the flow is said to be hypersonic. A model of an airplane traveling through any fluid at a certain Mach number will behave much like the real thing traveling through the air at that same Mach number up until one enters the hypersonic regime. Below Mach 5, the shock wave is separated from the object by a small but significant distance. Objects moving faster than Mach 5 start to interact with this shock front.

Fluid Flow Regimes as a Function of Mach Number.

compressible vs. incompressible flow

### froude number

Fr = | v |

√gℓ |

- The Froude number (
*Fr*) is the ratio of the inertial forces to the gravitational forces. - The Froude number is the non-dimensional factor governing resistance due to surface wave formation (among other things).
- Named after William Froude, a Nineteenth Century English scientist who was one of the first to use a towing tank.
- Hydrostatic jump
- The wake in front of a boat. (Is that what it's called?)

### Bond Number

The Bond number is a dimensionless number describing the importance of gravity relative to surface tension in determining the shape of a bubble or droplet.

Bo = |
ρgD^{2} |

σ |

Where…

Bo = |
Bond number |

ρ = | density of the liquid |

g = |
acceleration due to gravity |

D = |
bubble diameter |

σ = | surface temperature |