Flow Regimes

Discussion

reynolds number

Re =  ρ(Δvx,z)(Δy)  =  ρvD  =  ρv
η η η
  • Flow Regimes
  • Scaling

  • [slide]

    Fluid Flow Regimes as a Function of Reynolds Number.

    object lower upper
    circular pipes 2,000 2,500
    flat plates 300,000 500,000
    Critical Reynolds numbers
    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      
    Selected Reynolds numbers

    mach number

    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 isulation 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 5600 m/s.

    The Mach number (Ma) is a ratio of inertia to compressibility. It is the non-dimensional factor governing resistance due to longitudinal (compressional) wave formation.

    Ma2 =  ρv22  &  c2 =  B  ⇒  Ma =  v
    B2 ρ c

    The ratio of the speed of flow (v) to the speed of sound in a fluid (c) is known as the Mach number.

    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 shockwave 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 shockwave centered on the bolt's path. The sound of a shockwave 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.


    [slide]

    Fluid Flow Regimes as a Function of Mach Number.

    compressible vs. incompressible flow

    froude number

    Fr = v
    g