The Physics
Opus in profectus

Flow Regimes

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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  =  ρv
viscous resistance η


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

Flow regimes at different Reynolds numbers

Fluid Flow Regimes as a Function of Reynolds Number.

Critical Reynolds numbers
object lower upper
circular pipes 2,000 2,500
flat plates 300,000 500,000
Selected Reynolds numbers
Re animal
62,000 seagull
50,000 large fish
3,900 butterfly
1,000 honeybee
300 african frog tadpole
120 housefly
15 chalcid wasp
0.2 paramecium
0.025 dinoflagellate
0.0035 spermatozoa, sea urchin
0.00001 bacterium
Re circulatory system
3,400 aorta
3,300 vena cava
500 artery
140 vein
0.7 arteriole
0.01 venule
0.002 capillary
Re aircraft
2,000,000,000 boeing 747
110,000,000 typical commercial jet
6,300,000 cessna
4,700,000 light plane
1,600,000 glider
250,000 model airplane
47,000 paper airplane
Re miscellaneous
250,000,000 cumulus cloud formation

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

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.

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. 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.

The Mach number squared (Ma2) 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).

Ma2 =  Finertial

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

Finertial = ma =  mv

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

Finertial =  ρVv

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

Finertial =  ρ(Avt)v


Finertial = ρv2A

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

Finertial = ρv22

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).

Fcompressional = KA

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

Fcompressional = K2

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

Ma2 =  ρv22

The Mach number is then 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.

Ma2 =  ρv22  &  c2 =  K  ⇒  Ma =  v
K2 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 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.

Flow regimes at different Mach numbers

Fluid Flow Regimes as a Function of Mach Number.

compressible vs. incompressible flow

Froude number

Fr = v

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 =  ρgD2


Bo =  Bond number
ρ =  density of the liquid
g =  acceleration due to gravity
D =  bubble diameter
σ =  surface temperature