11- convection & reynolds number

7/28/2019 11- Convection & Reynolds Number

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Convection

This figure shows a calculation for thermal convection. Colors closer to red are hot areas and colors closer to blueare cold areas. In this figure, a hot, less-dense lower boundary layer sends plumes of hot material upwards, andlikewise, cold material from the top moves downwards. This figure is from a model of convection in the Earth'smantle.

Convection is the movement of molecules within fluids (i.e. liquids, gases) and rheids. It cannot take place insolids, since neither bulk current flows nor significant diffusion can take place in solids.

Convection is one of the major modes of heat transfer and mass transfer. Convective heat and mass transfer takeplace through both diffusion the random Brownian motion of individual particles in the fluid and by

advection, in which matter or heat is transported by the larger-scale motion of currents in the fluid. In the contextof heat and mass transfer, the term "convection" is used to refer to the sum of advective and diffusive transfer.[1]

Note that a common use of the term convection refers specifically to heat transfer by convection, as opposed toconvection in general.

TerminologyThe term "convection" may have slightly different but related usages in different contexts. The broader sense is influid mechanics, where "convection" refers to the motion of fluid (regardless of cause) .[2] However inthermodynamics "convection" often refers specifically to heat transfer by convection.[3]

Additionally, convection includes fluid movement both by bulk motion (advection) and by the motion of individualparticles (diffusion). However in some cases, convection is taken to mean only advective phenomena. For instance,in the transport equation, which describes a number of different transport phenomena, terms are separated into"convective" and "diffusive" effects. A similar differentiation is made in the NavierStokes equations. In such casesthe precise meaning of the term may be clear only from context.

Examples and applications of convectionConvection occurs on a large scale in atmospheres, oceans, and planetary mantles. Fluid movement duringconvection may be invisibly slow, or it may be obvious and rapid, as in a hurricane. On astronomical scales,

convection of gas and dust is thought to occur in the accretion disks ofblack holes, at speeds which may closelyapproach that of light.
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Heat transfer

A heat sink provides a large surface area for convection to efficiently carry away heat.

Convective heat transfer is a mechanism of heat transfer occurring because of bulk motion (observablemovement) of fluids. Heat is the entity of interest being advected (carried), and diffused (dispersed). This can becontrasted with conductive heat transfer, which is the transfer of energy by vibrations at a molecular level througha solid or fluid, and radiative heat transfer, the transfer of energy through electromagnetic waves. Heat is

transferred by convection in numerous examples of naturally occurring fluid flow, such as: wind, oceanic currents,and movements within the Earth's mantle. Convection is also used in engineering practices to provide desiredtemperature changes, as in heating of homes, industrial processes, cooling of equipment, etc.

The rate of convective heat transfer may be improved by the use of a heat sink, often in conjunction with a fan.For instance, a typical computer CPUwill have a purpose-made fan to ensure its operating temperature is keptwithin tolerable limits.

Convection cells

Convection cells in a gravity fieldA convection cell, also known as aBnard cellis a characteristic fluid flow pattern in many convection systems. Arising body of fluid typically loses heat because it encounters a cold surface; because it exchanges heat with colderliquid through direct exchange; or in the example of the Earth's atmosphere, because it radiates heat. Because ofthis heat loss the fluid becomes denser than the fluid underneath it, which is still rising. Since it cannot descendthrough the rising fluid, it moves to one side. At some distance, its downward force overcomes the rising forcebeneath it, and the fluid begins to descend. As it descends, it warms again and the cycle repeats itself.
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Atmospheric circulation

Idealised depiction of the global circulation on Earth

Atmospheric circulation is the large-scale movement of air, and the means (together with the smaller oceancirculation) by which thermal energy is distributed on the surface of the Earth. The large-scale structure of theatmospheric circulation varies from year to year, but the basic climatological structure remains fairly constant.

Latitudinal circulation is the consequence of the fact that incident solar radiation per unit area is highest at theheat equator, and decreases as the latitude increases, reaching its minimum at the poles. It consists of two primaryconvection cells, the Hadley cell and the polar vortex.

Longitudinal circulation, on the other hand, comes about because water has a higher specific heat capacity thanland and thereby absorbs and releases more heat, but the temperature changes less than land. This effect isnoticeable; it is what brings the sea breeze, air cooled by the water, ashore in the day, and carries the land breeze,air cooled by contact with the ground, out to sea during the night. Longitudinal

Weather

How Foehn is produced

More localized phenomena than global atmospheric movement are also due to convection, including wind andsome of the hydrologic cycle. For example, a foehn wind is a type of down-slope wind which occurs in thedownwind side of a mountain range. It results from the adiabaticwarming of air which has dropped most of itsmoisture on windward slopes. As a consequence of the different adiabatic lapse rates of moist and dry air, the airon the leeward slopes becomes warmer than equivalent elevations on the windward slopes, leading to the wind.
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A thermal column (or thermal) is a vertical section of rising air in the lower altitudes of the Earth's atmosphere.Thermals are created by the uneven heating of the Earth's surface from solar radiation. The Sun warms theground, which in turn warms the air directly above it. The warmer air expands, becoming less dense than thesurrounding air mass. The mass of lighter air rises, and as it does, it cools due to its expansion at lower high-altitude pressures. It stops rising when it has cooled to the same temperature as the surrounding air. Associatedwith a thermal is a downward flow surrounding the thermal column. The downward moving exterior is caused bycolder air being displaced at the top of the thermal.

Another convection-driven weather effect is the sea breeze.

Oceanic circulation

Ocean currents

Solar radiation affects the oceans: warm water from the Equator tends to circulate toward the poles, while coldpolar water heads towards the Equator. Oceanic convection is also frequently driven by density differences due tovarying salinity, known as thermohaline convection, and is of crucial importance in global ocean circulation. In thiscase it is possible for relatively warm, saline water to sink, and colder, fresher water to rise, reversing the normaltransport of heat.

Mantle convection

An oceanic plate is added to by upwelling (left) and consumed at a subduction zone (right).

Mantle convection is the slow creeping motion of Earth's rocky mantle caused by convection currents carryingheat from the interior of the earth to the surface. It is the driving force that causes tectonic plates to move aroundthe Earth's surface.

The Earth's surface is divided into a number oftectonic plates that are continuously being created and consumedat their opposite plate boundaries. Creation (accretion) occurs as mantle is added to the growing edges of a plate.This hot added material cools down by conduction and convection of heat. At the consumption edges of the plate,the material has thermally contracted to become dense, and it sinks under its own weight in the process of
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subduction at an ocean trench. This subducted material sinks to some depth in the Earth's interior where it isprohibited from sinking further. The subducted oceanic crust triggers volcanism.

Stack effectThe Stack effect or chimney effect is the movement of air into and out of buildings, chimneys, flue gas stacks, orother containers due to buoyancy. Buoyancy occurs due to a difference in indoor-to-outdoor air density resultingfrom temperature and moisture differences. The greater the thermal difference and the height of the structure,

the greater the buoyancy force, and thus the stack effect. The stack effect helps drive natural ventilation andinfiltration. Some cooling towers operate on this principle; similarly the solar updraft tower is a proposed deviceto generate electricity based on the stack effect.

Stellar physics

Granulesconvection cells caused by the convection of plasmaon the photosphere of the Sun, with NorthAmerica superimposed on the same scaleThe convection zone of a star is the range of radii in which energy is transported primarily by convection.

Granules on the photosphere of the Sun are convection cells caused by convection of plasma. The rising part of

the granules is located in the center where the plasma is hotter. The outer edge of the granules is darker due tothe cooler descending plasma. A typical granule has a diameter on the order of 1,000 kilometers and lasts 8 to 20minutes before dissipating. Below the photosphere is a layer of "supergranules" up to 30,000 kilometers indiameter with lifespans of up to 24 hours.

The image shows the solar photosphere where granules are visible. North America is superimposed to provide asense of scale.

Convection mechanismsConvection may happen in fluids at all scales larger than a few atoms. There are a variety of circumstances inwhich the forces required for natural and forced convection arise, leading to different types of convection,described below. In broad terms, convection arises because ofbody forces acting within the fluid, such as gravity(buoyancy), or surface forces acting at a boundary of the fluid.

The causes of convection are generally described as one of either "natural" ("free") or "forced", although othermechanisms also exist (disscussed below). However the distinction between natural and forced convection isparticularly important for convective heat transfer.
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Natural convectionNatural convection, or free convection, occurs due to temperature differences which affect the density, and thusrelative buoyancy, of the fluid. Heavier (more dense) components will fall while lighter (less dense) componentsrise, leading to bulk fluid movement. Natural convection can only occur, therefore, in a gravitational field. Acommon example of natural convection is a pot of boiling water in which the hot and less-dense water on thebottom layer moves upwards in plumes, and the cool and more dense water near the top of the pot likewise sinks.

Natural convection will be more likely and/or more rapid with a greater variation in density between the twofluids, a larger acceleration due to gravity that drives the convection, and/or a larger distance through theconvecting medium. Convection will be less likely and/or less rapid with more rapid diffusion (thereby diffusingaway the gradient that is causing the convection) and/or a more viscous (sticky) fluid.

The onset of natural convection can be determined by the Rayleigh number (Ra).Note that differences in buoyancy within a fluid can arise for reasons other than temperature variations, in whichcase the fluid motion is called gravitational convection (see below).Forced convectionIn forced convection, also called heat advection, fluid movement results from external surface forces such as a fanor pump. Forced convection is typically used to increase the rate of heat exchange. Many types of mixing alsoutilize forced convection to distribute one substance within another. Forced convection also occurs as a by-product to other processes, such as the action of a propeller in a fluid or aerodynamic heating. Fluid radiatorsystems, and also heating and cooling of parts of the body by blood circulation, are other familiar examples offorced convection.

Forced convection may produce results more quickly than free convection. For instance, a convection ovenworksby forced convection, as a fan which rapidly circulates hot air forces heat into food faster than would naturallyhappen due to simple heating without the fan.

Gravitational or buoyant convectionGravitational convection is a type of natural convection induced by buoyancy variations resulting from materialproperties other than temperature. Typically this is caused by a variable composition of the fluid. If the varyingproperty is a concentration gradient, it is known as solutal convection.[4]For example, gravitational convection canbe seen in the diffusion of a source of dry salt downward into wet soil due to the buoyancy of fresh water insaline.[5]

Variable salinity in water and variable water content in air masses are frequent causes of convection in the oceansand atmosphere which do not involve heat, or else involve additional compositional density factors other than thedensity changes from thermal expansion (seethermohaline circulation). Similarly, variable composition within theEarth's interior which has not yet achieved maximal stability and minimal energy (in other words, with densest

parts deepest) continues to cause a fraction of the convection of fluid rock and molten metal within the Earth'sinterior (see below).

As buoyant convection is due to the effects of gravity, it does not occur in microgravity environments.
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Convection of a fluid

Granular convectionVibration-induced convection occurs in powders and granulated materials in containers subject to vibration wherean axis of vibration is parallel to the force of gravity. When the container accelerates upward, the bottom of thecontainer pushes the entire contents upward. In contrast, when the container accelerates downward, the sides ofthe container push the adjacent material downward by friction, but the material more remote from the sides is

less affected. The net result is a slow circulation of particles downward at the sides, and upward in the middle.

If the container contains particles of different sizes, the downward-moving region at the sides is often narrowerthan the largest particles. Thus, larger particles tend to become sorted to the top of such a mixture. This is onepossible explanation of the Brazil nut effect.

Thermomagnetic convectionThermomagnetic convection can occur when an external magnetic field is imposed on a ferrofluidwith varyingmagnetic susceptibility. In the presence of a temperature gradient this results in a nonuniform magnetic bodyforce, which leads to fluid movement. A ferrofluid is a liquid which becomes strongly magnetized in the presenceof a magnetic field.

This form of heat transfer can be useful for cases where conventional convection fails to provide adequate heattransfer, e.g., in miniature microscale devices or under reduced gravity conditions.

Capillary actionCapillary action is a phenomenon where liquid spontaneously rises in a narrow space such as a thin tube, or inporous materials. This effect can cause liquids to flow against the force of gravity. It occurs because of inter-molecular attractive forces between the liquid and solid surrounding surfaces; If the diameter of the tube issufficiently small, then the combination of surface tension and forces of adhesion between the liquid and containeract to lift the liquid.

Marangoni effectThe Marangoni effect is the convection of fluid along an interface between dissimilar substances because ofvariations in surface tension. Surface tension can vary because of inhomogeneous composition of the substances,and/or the temperature-dependence of surface tension forces. In the latter case the effect is known as thermo-capillary convection.A well-known phenomenon exhibiting this type of convection is the "tears of wine".

Weissenberg effectThe Weissenberg effect is a phenomenon that occurs when a spinning rod is placed into a solution of liquidpolymer. Instead of being thrown outward, entanglements cause the polymer chains to be drawn towards the rodinstead of being thrown outward as would happen with an ordinary fluid (i.e., water).

CombustionIn a zero-gravity environment, there can be no buoyancy forces, and thus no natural (free) convection possible, soflames in many circumstances without gravity, smother in their own waste gases. However, flames may bemaintained with any type of forced convection (breeze); or (in high oxygen environments in "still" gasenvironments) entirely from the minimal forced convection that occurs as heat-induced expansion(not buoyancy)
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of gases allows for ventilation of the flame, as waste gases move outward and cool, and fresh high-oxygen gasmoves in to take up the low pressure zones created when flame-exhaust water condenses.[6]

Mathematical models of convectionMathematically, convection can be described by the convectiondiffusion equation or the generic scalar transportequation.

Quantifying natural versus forced convectionIn cases of mixed convection (natural and free occurring together) one would often like to know how much of theconvection is due to external constraints, such as the fluid velocity in the pump, and how much is due to naturalconvection occurring in the system.

The relative magnitudes of the Grashof and Reynolds number squared determine which form of convection

dominates. if forced convection may be neglected, whereas if natural convection may beneglected. If the ratio is approximately one both forced and natural convection need to be taken into account.
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References1. ^ Frank P. Incropera; David P. De Witt and D. P. Dewitt (1990). Fundamentals of Heat and Mass Transfer(3rd ed.).

John Wiley & Sons. ISBN 0-471-51729-1.2. ^Munson, Bruce R.. Fundamentals of Fluid Mechanics. John Wiley & Sons. ISBN 047185526X.3. ^ engel, Yunus A.; Boles, Michael A.. Thermodynamics:An Engineering Approach. McGraw-Hill Education.

ISBN 007121688X.4. ^ "CiteSeerX Pattern Formation in Solutal Convection: Vermiculated Rolls and Isolated Cells".

Citeseerx.ist.psu.edu. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.15.8288. Retrieved 2010-09-12.5. ^Raats, P. A. C. (1969). "Steady Gravitational Convection Induced by a Line Source of Salt in a Soil". Soil Science

Society of AmericaProceedings33 (4): 483. doi:10.2136/sssaj1969.03615995003300040005x.6. ^ Does a candle burn in zero-g?
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Dimensionless quantityIn dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantitywithout an associatedphysical dimension. It is thus a "pure" number, and as such always has a dimension of 1.[1]Dimensionless quantitiesare widely used in mathematics, physics, engineering, economics, and in everyday life (such as in counting).

Numerous well-known quantities, such as, e, and, are dimensionless.

Dimensionless quantities are often defined as products or ratios of quantities that are not dimensionless, butwhose dimensions cancel out when their powers are multiplied. This is the case, for instance, with the engineeringstrain, a measure of deformation. It is defined as change in length over initial length but, since these quantitiesboth have dimensions L(length), the result is a dimensionless quantity.

Properties Even though a dimensionless quantity has no physical dimension associated with it, it can still have

dimensionless units. To show the quantity being measured (for example mass fraction or mole fraction), itis sometimes helpful to use the same units in both the numerator and denominator (kg/kg or mol/mol).The quantity may also be given as a ratio of two different units that have the same dimension (for

instance, light years over meters). This may be the case when calculating slopes in graphs, or whenmaking unit conversions. Such notation does not indicate the presence of physical dimensions, and ispurely a notational convention. Other common dimensionless units are % (= 0.01), (= 0.001), ppm(= 106), ppb (= 109), ppt (= 1012) and angle units (degrees, radians, grad). Units of amount such as thedozen and the gross are also dimensionless.

The dimensionless ratio of two quantities with the same dimensions has the same value regardless ofthe units used to calculate them. For instance, if body A exerts a force of magnitude Fon body B, and Bexerts a force of magnitude fon A, then the ratio F/fwill always be equal to 1, regardless of the actualunits used to measure Fand f. This is a fundamental property of dimensionless proportions and followsfrom the assumption that the laws of physics are independent of the system of units used in theirexpression. In this case, if the ratio F/fwas not always equal to 1, but changed if we switched from SI toCGS, for instance, that would mean that Newton's Third Law's truth or falsity would depend on thesystem of units used, which would contradict this fundamental hypothesis. The assumption that the lawsof physics are not contingent upon a specific unit system is also closely related to the Buckingham theorem. A formulation of this theorem is that any physical law can be expressed as an identity (alwaystrue equation) involving only dimensionless combinations (ratios or products) of the variables linked bythe law (e. g., pressure and volume are linked by Boyle's Law they are inversely proportional). If thedimensionless combinations' values changed with the systems of units, then the equation would not be anidentity, and Buckingham's theorem would not hold.

Buckingham theoremAnother consequence of the Buckingham theorem of dimensional analysis is that the functional dependencebetween a certain number (say, n) ofvariables can be reduced by the number (say, k) ofindependent dimensionsoccurring in those variables to give a set ofp= n kindependent, dimensionless quantities. For the purposes ofthe experimenter, different systems which share the same description by dimensionless quantity are equivalent.

Example
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The power consumption of a stirrerwith a given shape is a function of the density and the viscosity of the fluid tobe stirred, the size of the stirrer given by its diameter, and the speed of the stirrer. Therefore, we have n= 5variables representing our example.

Those n= 5 variables are built up from k= 3 dimensions which are:

Length: L(m) Time: T(s) Mass: M(kg).

According to the -theorem, the n= 5 variables can be reduced by the k= 3 dimensions to form p= n k= 5 3= 2 independent dimensionless numbers which are, in case of the stirrer:

Reynolds number (a dimensionless number describing the fluid flow regime) Power number (describing the stirrer and also involves the density of the fluid)

Standards effortsThe CIPM Consultative Committee for Units contemplated defining the unit of 1 as the 'uno', but the idea was

dropped.[2][3][4]

ExamplesConsider this example: Sarah says, "Out of every 10 apples I gather, 1 is rotten.". The rotten-to-gathered ratio is(1 apple) / (10 apples) = 0.1 = 10%, which is a dimensionless quantity. Another more typical example in physics andengineering is the measure of plane angles. An angle is measured as the ratio of the length of a circle's arcsubtended by an angle whose vertex is the centre of the circle to some other length. The ratio, length divided bylength, is dimensionless. When using radians as the unit, the length that is compared is the length of the radius ofthe circle. When using degree as the units, the arc's length is compared to 1/360 of the circumference of the circle.
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Reynolds numberA vortex street around a cylinder. This occurs around cylinders, independently of the fluid, the cylinder size andthe fluid speed, provided that there is a Reynolds number of between 250 and 200,000. Picture courtesy, Cesareode La Rosa Siqueira.

In fluid mechanics, the Reynolds numberRe is a dimensionless number that gives a measure of the ratio ofinertial forces to viscous forces and consequently quantifies the relative importance of these two types of forcesfor given flow conditions.

The concept was introduced by George Gabriel Stokes in 1851,[1]but the Reynolds number is named after OsborneReynolds (18421912), who popularized its use in 1883.[2][3]

Reynolds numbers frequently arise when performing dimensional analysis of fluid dynamics problems, and as suchcan be used to determine dynamic similitude between different experimental cases.

They are also used to characterize different flow regimes, such as laminar or turbulent flow:

laminar flowoccurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth,constant fluid motion;

while turbulent flowoccurs at high Reynolds numbers and is dominated by inertial forces, which tend to producechaotic eddies, vortices and other flow instabilities.

DefinitionReynolds number can be defined for a number of different situations where a fluid is in relative motion to asurface (the definition of the Reynolds number is not to be confused with the Reynolds Equation or lubricationequation). These definitions generally include the fluid properties of density and viscosity, plus a velocity and acharacteristic lengthor characteristic dimension. This dimension is a matter of convention for example a radius

or diameter are equally valid for spheres or circles, but one is chosen by convention. For aircraft or ships, thelength or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is generallyused today. Other shapes (such as rectangular pipes or non-spherical objects) have an equivalent diameterdefined.For fluids of variable density (e.g. compressible gases) or variable viscosity (non-Newtonian fluids) special rulesapply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels.

[4]

where:

is the mean velocity of the object relative to the fluid (SI units: m/s) Lis a characteristic linear dimension, (travelled length of the fluid; hydraulic diameterwhen dealing with

river systems) (m)

is the dynamic viscosity of the fluid (Pas or Ns/m or kg/(ms)) is the kinematic viscosity ( = /) (m/s) is the density of the fluid (kg/m)
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Note that multiplying the Reynolds number, by yields which is the ratio,

.[5]

Significance

Flow in PipeFor flow in a pipe or tube, the Reynolds number is generally defined as:[6]

where:

DH is the hydraulic diameter of the pipe (m). Q is the volumetric flow rate (m3/s) A is the pipe cross-sectionalarea (m).

Flow in a non-circular duct (annulus)For shapes such as squares, rectangular or annular ducts (where the height and width are comparable) thecharacteristic dimension for internal flow situations is taken to be the hydraulic diameter,DH, defined as 4 timesthe cross-sectional area (of the fluid), divided by the wetted perimeter. The wetted perimeter for a channel is thetotal perimeter of all channel walls that are in contact with the flow.[7]This means the length of the water exposedto air is NOT included in the wetted perimeter

For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter, as can be shownmathematically.

For an annular duct, such as the outer channel in a tube-in-tube heat exchanger, the hydraulic diameter can beshown algebraically to reduce to

DH,annulus = Do Di

where

Do is the inside diameter of the outside pipe, andDi is the outside diameter of the inside pipe.
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For calculations involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter ofa circular duct, with reasonable accuracy.

Flow in a Wide DuctFor a fluid moving between two plane parallel surfaces (where the width is much greater than the space betweenthe plates) then the characteristic dimension is twice the distance between the plates.[8]

Flow in an Open ChannelFor flow of liquid with a free surface, thehydraulic radiusmust be determined. This is the cross-sectional area ofthe channel divided by the wetted perimeter. For a semi-circular channel, it is half the radius. For a rectangularchannel, the hydraulic radius is the cross-sectional area divided by the wetted perimeter. Some texts then use acharacteristic dimension that is 4 times the hydraulic radius (chosen because it gives the same value of Re for theonset of turbulence as in pipe flow),[9]while others use the hydraulic radius as the characteristic length-scale withconsequently different values of Re for transition and turbulent flow.

Object in a fluidThe Reynolds number for an object in a fluid, called the particle Reynolds number and often denoted Rep, isimportant when considering the nature of flow around that grain, whether or not vortex sheddingwill occur, andits fall velocity.

Sphere in a fluidFor a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocityis that of the sphere relative to the fluid some distance away from the sphere (such that the motion of the spheredoes not disturb that reference parcel of fluid). The density and viscosity are those belonging to the fluid .[10]Notethat purely laminar flow only exists up to Re = 0.1 under this definition.

Under the condition of low Re, the relationship between force and speed of motion is given by Stokes' law.[11]

Oblong object in a fluidThe equation for an oblong object is identical to that of a sphere, with the object being approximated as anellipsoid and the axis of length being chosen as the characteristic length scale. Such considerations are importantin natural streams, for example, where there are few perfectly spherical grains. For grains in which measurementof each axis is impractical (e.g., because they are too small), sieve diameters are used instead as the characteristicparticle length-scale. Both approximations alter the values of the critical Reynolds number.

Fall velocityThe particle Reynolds number is important in determining the fall velocity of a particle. When the particle

Reynolds number indicates laminar flow, Stokes' law can be used to calculate its fall velocity. When the particleReynolds number indicates turbulent flow, a turbulent drag law must be constructed to model the appropriatesettling velocity.

Packed BedFor flow of fluid through a bed of approximately spherical particles of diameter D in contact, if the voidage(fraction of the bed not filled with particles) is and the superficial velocity V (i.e. the velocity through the bed asif the particles were not there - the actual velocity will be higher) then a Reynolds number can be defined as:
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Laminar conditions apply up to Re = 10, fully turbulent from 2000.[10]

Stirred VesselIn a cylindrical vessel stirred by a central rotating paddle, turbine or propellor, the characteristic dimension is thediameter of the agitator D. The velocity is NDwhere Nis the rotational speed (revolutions per second). Then theReynolds number is:

The system is fully turbulent for values of Re above 10 000.[12]

Transition Reynolds number[citation needed]In boundary layer flow over a flat plate, experiments can confirm that, after a certain length of flow, alaminar boundary layer will become unstable and become turbulent. This instability occurs across different scales

and with different fluids, usually when , where xis the distance from the leading edge of theflat plate, and the flow velocity is the freestreamvelocity of the fluid outside the boundary layer.

For flow in a pipe of diameter D, experimental observations show that for 'fully developed' flow (Note:[13]), laminarflow occurs when ReD < 2300 and turbulent flow occurs when ReD > 4000.

[14] In the interval between 2300and 4000, laminar and turbulent flows are possible ('transition' flows), depending on other factors, such as piperoughness and flow uniformity). This result is generalised to non-circular channels using the hydraulic diameter,allowing a transition Reynolds number to be calculated for other shapes of channel.

These transition Reynolds numbers are also called critical Reynolds numbers, and were studied by OsborneReynolds around 1895 [see Rott].

Reynolds number in pipe friction
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Pressure drops seen for fully-developed flow of fluids through pipes can be predicted using the Moody diagramwhich plots the DarcyWeisbach friction factor fagainst Reynolds number Re and relative roughness / D. Thediagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases. Thenature of pipe flow is strongly dependent on whether the flow is laminar or turbulent

The similarity of flowsIn order for two flows to be similar they must have the same geometry, and have equal Reynolds numbers andEuler numbers. When comparing fluid behaviour at corresponding points in a model and a full-scale flow, thefollowing holds:

quantities marked with 'm' concern the flow around the model and the others the actual flow. This allowsengineers to perform experiments with reduced models inwater channels or wind tunnels, and correlate the datato the actual flows, saving on costs during experimentation and on lab time. Note that true dynamic similitudemay require matching other dimensionless numbers as well, such as the Mach number used in compressible flows,or the Froude number that governs open-channel flows. Some flows involve more dimensionless parameters thancan be practically satisfied with the available apparatus and fluids (for example air or water), so one is forced todecide which parameters are most important. For experimental flow modeling to be useful, it requires a fairamount of experience and judgement of the engineer.

Typical values of Reynolds number[15][16]

Ciliate ~ 1 x 101 Smallest Fish ~ 1 Blood flow in brain ~ 1 102 Blood flow in aorta ~ 1 103

Onset of turbulent flow ~ 2.3 103 to 5.0 104 for pipe flow to 106 for boundary layers
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Typical pitch in Major League Baseball ~ 2 105 Person swimming ~ 4 106 Fastest Fish ~ 1 x 106 Blue Whale ~ 3 108 A large ship (RMS Queen Elizabeth 2) ~ 5 109

Reynolds number sets the smallest scales of turbulent motionIn a turbulent flow, there is a range of scales of the time-varying fluid motion. The size of the largest scales offluid motion (sometimes called eddies) are set by the overall geometry of the flow. For instance, in an industrialsmoke stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of thesmallest scales is set by the Reynolds number. As the Reynolds number increases, smaller and smaller scales of theflow are visible. In a smoke stack, the smoke may appear to have many very small velocity perturbations or eddies,in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in theflow. The higher the Reynolds number, the greater the range of scales. The largest eddies will always be the samesize; the smallest eddies are determined by the Reynolds number.

What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are notimportant at large scales of the flow. With a strong predominance of inertial forces over viscous forces, the largestscales of fluid motion are undampedthere is not enough viscosity to dissipate their motions. The kinetic energymust "cascade" from these large scales to progressively smaller scales until a level is reached for which the scale issmall enough for viscosity to become important (that is, viscous forces become of the order of inertial ones). It isat these small scales where the dissipation of energy by viscous action finally takes place. The Reynolds numberindicates at what scale this viscous dissipation occurs. Therefore, since the largest eddies are dictated by the flowgeometry and the smallest scales are dictated by the viscosity, the Reynolds number can be understood as theratio of the largest scales of the turbulent motion to the smallest scales.

Example of the importance of the Reynolds numberIf an airplane wing needs testing, one can make a scaled down model of the wing and test it in a wind tunnelusing the same Reynolds number that the actual airplane is subjected to. If for example the scale model has linear

dimensions one quarter of full size, the flow velocity of the model would have to be multiplied by a factor of 4 toobtain similar flow behavior.

Alternatively, tests could be conducted in a water tank instead of in air (provided the compressibility effects of airare not significant). As the kinematic viscosity of water is around 13 times less than that of air at 15 C, in thiscase the scale model would need to be about one thirteenth the size in all dimensions to maintain the sameReynolds number, assuming the full-scale flow velocity was used.

The results of the laboratory model will be similar to those of the actual plane wing results. Thus there is no needto bring a full scale plane into the lab and actually test it. This is an example of "dynamic similarity".

Reynolds number is important in the calculation of a body's drag characteristics. A notable example is that of the

flow around a cylinder.[17]Above roughly 3106 Re the drag coefficient drops considerably. This is important whencalculating the optimal cruise speeds for low drag (and therefore long range) profiles for airplanes.

Reynolds number in physiologyPoiseuille's law on blood circulation in the body is dependent on laminar flow. In turbulent flow the flow rate isproportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressuregradient in laminar flow.
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Using the definition of the Reynolds number we can see that a large diameter with rapid flow, where the densityof the blood is high, tends towards turbulence. Rapid changes in vessel diameter may lead to turbulent flow, forinstance when a narrower vessel widens to a larger one. Furthermore, an atheroma may be the cause of turbulentflow, and as such detecting turbulence with a stethoscope may be a sign of such a condition.

Reynolds number in viscous fluids

Creeping flow past a sphere: streamlines, drag force Fd and force by gravity Fg.

Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. TheReynolds number is very small and Stokes' Law can be used to measure the viscosity of the fluid. Spheres areallowed to fall through the fluid and they reach the terminal velocity quickly, from which the viscosity can bedetermined.

The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who exude viscoussolutions from their skin to aid flow over their bodies while swimming. It has been used in yacht racing by ownerswho want to gain a speed advantage by pumping a polymer solution such as low molecular weightpolyoxyethylene in water, over the wetted surface of the hull. It is however, a problem for mixing of polymers,because turbulence is needed to distribute fine filler (for example) through the material. Inventions such as the

"cavity transfer mixer" have been developed to produce multiple folds into a moving melt so as to improve mixingefficiency. The device can be fitted onto extruders to aid mixing.

References and notes1. ^ Stokes, George (1851). "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums". Transactions of the

Cambridge Philosophical Society9: 8106.2. ^ Reynolds, Osborne (1883). "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". Philosophical Transactions of the Royal Society174(0): 935982. doi:10.1098/rstl.1883.0029. JSTOR 109431.

3. ^ Rott, N. (1990). "Note on the history of the Reynolds number". Annual Review of Fluid Mechanics 22 (1): 111.doi:10.1146/annurev.fl.22.010190.000245.

4. ^ Reynolds Number5. ^Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. pp. 211215.6. ^ Reynolds Number (engineeringtoolbox.com)7. ^Holman, J. P.. Heat Transfer. McGraw Hill.[Full citation needed]8. ^Fox, R. W.; McDonald, A. T.; Pritchard, Phillip J. (2004). Introduction to Fluid Mechanics(6th ed.). Hoboken: John Wiley and

Sons. p. 348. ISBN 0471202312.9. ^V. L. Streeter (1962)Fluid Mechanics, 3rd edn (McGraw-Hill)10. ^abM. Rhodes (1989) Introduction to Particle TechnologyWiley ISBN 0-471-98482-5at Google Books11. ^Dusenbery, David B. (2009). Living at Micro Scale, p.49. Harvard University Press, Cambridge, Mass. ISBN 978-0-674-03116-6.
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12. ^R. K. Sinnott Coulson & Richardson's Chemical Engineering, Volume 6: Chemical Engineering Design, 4th ed (Butterworth-Heinemann) ISBN 0 7506 6538 6 page 473

13. ^Full development of the flow occurs as the flow enters the pipe, the boundary layer thickens and then stabilises after severaldiameters distance into the pipe.

14. ^J.P Holman Heat transfer, McGraw-Hill, 2002, p.20715. ^Patel, V. C.; Rodi, W.; Scheuerer, G. (1985). "Turbulence Models for Near-Wall and Low Reynolds Number Flows A Review".

AIAA Journal23 (9): 13081319. doi:10.2514/3.9086.16. ^Dusenbery, David B. (2009). Living at Micro Scale. Cambridge, Mass.: Harvard University Press. p. 136. ISBN 9780674031166.17. ^ Cylinder Drag from Eric Weisstein's World of Physics
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