study of the theory of turbulence

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ndian Science Cruiser Vol. 27, No. 3, May 2013

Study of the Theory of Turbulence

N.C. Ghosh

Department of Mathematics

Rabindra Bharati University56A B.T. Road

Kolkata 5!

"mail # nc$snbmms%hotmail.com

Big whirls have lile whirls ha !eed on heir velociy"

lile whirls have lesser whirls, and so on o viscosiy.Th&s ".'. Rechardson e(pressed the ener$y cascadin$ principle in t&rb&lence. )erhaps considerin

comple(ities of analysin$ t&rb&lence one scientist *rote# +f there is a $od and + can meet him, + *ill ask him

somethin$ abo&t *illylilly behavio&rs of t&rb&lent phenomena.

T&rb&lence is one of the oldest and most diffic&lt open problems in physics. Applied Mathematiciansdeals it very caref&lly from the mathematical stand point of vie*.

The story is told of many $iants of modern physics, b&t most pla&sibly of -eisenber$, on his death

bed, he remarked that the t*o $reat &nsolved problems *ere reconcilin$ &ant&m mechanics and $eneral relativityand t&rb&lence. /0o*, +1m optimistic abo&t $ravity.../

Knowledge of the study of Theory Turbulence is essential

Turbulence is a notoriously difficult subject. The goal is for investigation of such complephenomena is to debate with some fundamental questions related to this that are wide ranging, from the initiatioof turbulence through to its asymptotic state at high Reynolds number, including the effects of rotation an

stratification, and the addition of different phases, such as bubbles, particles and polymers.

Kno*led$e of the st&dy of Theory T&rb&lence is essential for analy2in$ many real life sit&ation vi2,

For analyzing atmospheric aspects

3ontin&o&sly movin$ air in the atmospheric re$ion is a b&rnin$ e(ample of t&rb&lence. 4ith o&t th

kno*led$e of t&rb&lent characteristics of air no *ay can say a sin$le *ord perfectly re$ardin$ the atmospheri

re$ion.

For Weather forecasting

Weather forecasting in a rudimentary form is probably one of the oldest sciences. Bac in !"# B$

the Babylonians were able to predict the weather by observing cloud patterns. %ristotle has described weathepatterns in his &eteorologica. $hinese weather prediction traditions date bac to '## B$. (ndia also has a lonhistory of weather prediction. )arly philosophical writings of around '### B$, such as the *pnishadas, contaidiscussions about the processes of cloud formation and rain and the cycle of seasons. +autilyas %rthashastrcontains records of scientific measurements of rainfall and its application to the countrys revenue and relief wor+alidasa has mentioned the onset of monsoon over central (ndia in &eghdoot.

The simplest method of weather forecasting is the use of todays conditions to forecast tomorrows weatherThis method of forecasting strongly depends upon the presence of stagnant weather pattern, lie during th
mailto:[email protected]:[email protected]


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summer season in tropics. -umerical weather prediction or -W is a set of mathematical equations that governthe behaviour of atmosphere. These are combined in a complex mathematical model, and this is applied tobservations of the atmosphere / a turbulent phenomena. There are several variables, which decide the outcome othese predictions. arameters lie the earths si0e, its rotation rate, geography, daily and seasonal variations oincoming solar radiation are more or less constant. 1owever, other parameters lie surface reflectivity, meltingevaporation, cloud, rain, friction and sea temperatures vary through the period of a forecast. %ll these parameterare most complicated due to their turbulent character.

)xperts believe that to2days numerical weather models are still too broad2brush to permit truly 3locaforecasts to be made. 4iven the sheer number of variables that affect local weather, mathematicians believe thaabsolutely accurate forecasts beyond the short term will perhaps remain a chimera as theories for analy0inturbulence is still far from its goal.

For understanding aspects of Cyclone :

3yclone is a storm accompanied by hi$h speed *histlin$ and ho*lin$ *inds. +n meteorology,

cyclone is an area of low atmospheric pressure characteri2ed by in*ard spiraling winds that rotate counte

clockwisein the northern hemisphereand clockwisein the southern hemisphereof the "arth. ince the $eneriterm covers a *ide variety of meteorological phenomena, s&ch as tropical cyclones, etratropical cyclones, an

tornadoes, meteorolo$ists rarely &se it *itho&t additional &alification.

3oldcore cyclones 7most cyclone varieties8 form d&e to the nearby presence of an &pper level tro&$h*hich increases diver$ence aloft over an area that ind&ces &p*ard motion and s&rface lo* press&re. 4armcor

cyclones 7s&ch as tropical cyclones and many mesocyclones8 can have their initial start d&e to a nearby &ppe

tro&$h, b&t after formation of the initial dist&rbance, depend &pon a stormrelative &pper level hi$h to maintain oincrease their stren$th.

+t is often diffic&lt to predict *here a cyclone *ill strike. 4hen it starts movin$ from oceans to*ards th

land area, a cyclone can chan$e track and hit areas other than those anticipated earlier.

To analyse turbulent condition of fluid during Tsunami

% tsunami 5pronounced tsoo2nah2mee6 is a wave train, or series of waves, generated in a body of

water by an impulsive disturbance that vertically displaces the water column. )arthquaes, landslides, volcaniceruptions, explosions, and even the impact of cosmic bodies, such as meteorites, can generate tsunamis. Tsunamiscan savagely attac coastlines, causing devastating property damage and loss of life.

Tsunamis are unlie wind2generated waves, which many of us may have observed on a local lae orat a coastal beach, in that they are characteri0ed as shallow2water waves, with long periods and wavelengths. Thewind2generated swell one sees at a $alifornia beach, for example, spawned by a storm out in the acific andrhythmically rolling in, one wave after another, might have a period of about 7# seconds and a wave length of 7"#m. % tsunami, on the other hand, can have a wavelength in excess of 7## m and period on the order of one hour.

%s a result of their long wavelengths, tsunamis behave as shallow2water waves. % wave becomes a

shallow2water wave when the ratio between the water depth and its wavelength gets very small. 8hallow2waterwave move at a speed that is equal to the square root of the product of the acceleration of gravity 59.: m;s;s6 andthe water depth 2 letcean, where the typical water depth is about ?### m,a tsunami travels at about @## m;s, or over A## m;hr. Because the rate at which a wave loses its energy isinversely related to its wave length, tsunamis not only propagate at high speeds / turbulent in nature, they canalso travel great, transoceanic distances with limited energy losses.

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http://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Low_pressure_areahttp://en.wikipedia.org/wiki/Spiralhttp://en.wikipedia.org/wiki/Windhttp://en.wikipedia.org/wiki/Counter_clockwisehttp://en.wikipedia.org/wiki/Counter_clockwisehttp://en.wikipedia.org/wiki/Northern_hemispherehttp://en.wikipedia.org/wiki/Clockwisehttp://en.wikipedia.org/wiki/Southern_hemispherehttp://en.wikipedia.org/wiki/List_of_meteorological_phenomenahttp://en.wikipedia.org/wiki/Tropical_cyclonehttp://en.wikipedia.org/wiki/Extratropical_cyclonehttp://en.wikipedia.org/wiki/Tornadohttp://www.geophys.washington.edu/tsunami/general/physics/sounds/tsunami.auhttp://en.wikipedia.org/wiki/Meteorologyhttp://en.wikipedia.org/wiki/Low_pressure_areahttp://en.wikipedia.org/wiki/Spiralhttp://en.wikipedia.org/wiki/Windhttp://en.wikipedia.org/wiki/Counter_clockwisehttp://en.wikipedia.org/wiki/Counter_clockwisehttp://en.wikipedia.org/wiki/Northern_hemispherehttp://en.wikipedia.org/wiki/Clockwisehttp://en.wikipedia.org/wiki/Southern_hemispherehttp://en.wikipedia.org/wiki/List_of_meteorological_phenomenahttp://en.wikipedia.org/wiki/Tropical_cyclonehttp://en.wikipedia.org/wiki/Extratropical_cyclonehttp://en.wikipedia.org/wiki/Tornadohttp://www.geophys.washington.edu/tsunami/general/physics/sounds/tsunami.au


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Turbulence in Gas turbine

+n modern society &se of $as t&rbine in ind&stries:en$ine is all most a common feat&re. tron$ t&rb&lent

condition of fl&id inside operates $as t&rbine.

For !erfect "isualization of Turbulence in !lume of Chimney

Use of chimney, thro&$h *hich $asses are been e;ected, is a common in h&man life. )l&me of $as come

o&t from chimney. The pl&me is a $ood e(ample of t&rb&lent character of fl&id.

For #rtificial $aining :

Artificial rainin$ is sometimes called clo&d seedin$.


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(athematical Concepts for #nalysing Turbulence


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dropped into the fl&id form nice layers aro&nd the bo&ndaries of the flo*, *hich is *hy the flo* is calledlaminar, and these laminG are stable.

4ith t&rn &p the Reynolds n&mber, the nonlinearities become important, and the flo* $ets &$lier it is no

lon$er steady, b&t erratic 7probably chaoticin the strict sense8, and the nice re$&lar streamlines and their laminG$et snarled and then completely conf&sedC eddies and vortices form and spin and dissolve *itho&t m&ch obvio&s

pattern, and the develop their o*n eddies in t&rnC odd str&ct&res *ith names like /von KHrmHn streets/ appear

T&rb&lence yea, /f&lly developed t&rb&lence/, even is *hen this decay into conf&sion is complete, *hen

there are eddies and motions on all len$th scales, from the lar$est possible in the fl&id on do*n to the socalled/dissipation scale,/ *hich is 7ro&$hlyI8 the minim&m eddy si2e, as set by the mechanical properties of the fl&id

7its viscosity and the like8. 4hen faced *ith this conf&sion, if not *ell before, one $ive &p and t&rn to statisticsC

Th&s one can make some nice observations, and even come &p *ith t*o *ellconfirmed empirical la*s abo&tthese statistics, and endless $raphs.

A sol&tion to the problem of t&rb&lence *o&ld be, more or less, a valid derivation from the 0aviertokes

e&ation 7and statements abo&t the appropriate conditions8 of o&r meas&red statistics. )hysicists are very far from

this at present. 3&rrent closest approach stems from the *ork of Kolmo$orov, *ho, by means of some statisticalhypotheses abo&t smallscale motion, *as able to acco&nt for the empirical la*s. Unfort&nately, no one has

mana$ed to coa( the hypotheses from the 0aviertokes e&ation and the hypotheses hold e(actly only in the

limit of infinite Reynolds n&mber, i.e. they are not tr&e of any act&al fl&id.

All sorts of thin$s, incl&din$ more or less direct sim&lations of flo*s by co&sins of cell&lar a&tomatacalled/lattice $asses/. Ene approach &ses the vorticity 7the c&rl of the velocity field, *hich tells &s abo&t ho* the fl&id

s*irls8, since it t&rns o&t to be possible to identify some 7more or less8 simple ob;ects in the flo*, called vorte(

lines or vorte( t&bes, *ork o&t ho* they interact, and then &se statistical mechanicsto calc&late vario&s emer$entpropertiesand tolerate ne$ative temperat&res 7*hich are not impossible, and act&ally hotter than infinity8 $ives

the Kolmo$orov la*s. This co&ld1ve been c&stomtailored for philosophical and methodolo$ical biases as do all

the leaps in the appro(imation scheme &sed.

3onsiderin$ velocity, press&re and density correlation in a homo$eneo&s and isotropic flo* fieldCinvesti$ation may be carried o&t for conse&ent local chan$es of velocity, temperat&re, press&re, density etc.


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x(s)N a. limf n(s)andy(s)N b. limf n7s8 OOOOOOOO OOOOO78 nPQ nPQ

*heref(s)N Fs(1s), s So,S and coordinates of the vertices of the rectan$le enclosin$ the crosssectional area is$iven by 7!, !8, 7a, !8, 7a, b8 and 7o, b8.


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&ince simulated vortex in a dynamic fluid thus resolved by solvin' the * (") may be termed as fu++y vortex andchaotic occurrence of such fu++y vortices in a dynamic fluid may be called fu++y turbulence.

Conclusion : A short acco&nt on t&rb&lent research methodolo$y has been done here. Development andshortcomin$s are briefly stated. 4ith the advent of s&percomp&ters considerable pro$ress has been made on

t&rb&lence research based on n&merical methods for last t*enty years. Also, a n&mber of e(perimental *orks

have been done by the e(perimentalists to vis&alise the vario&s aspects of the t&rb&lent phenomena. Details of thee(perimental *orks not mentioned here as a&thor has disc&ssed these in another article. Based on s&ch *orks

optimists predict that f&lly developed t&rb&lence *ill be &nderstood in a fe* years time. B&t many more years

may be needed to tr&ly &nderstand all of the comple(ity of t&rb&lent flo* a problem that has been challen$in$physicists, mathematicians and en$ineers for at least half a millenni&mJ6_L.

: $eferences :

. Bro*der , 8 # The Theory of -omo$eneo&s T&rb&lence. 3ambrid$e University)ress. 3ambrid$e. U.K.

!. ?in, 3.3. Reid, 4.-. 76>8 # T&rb&lent .

_


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6. Batchelor,'.K. and # The 0at&re of T&rb&lant Motion at ?ar$e To*nsend, A.A7F8 4ave 0&mbers, )roc. Roy. oc. ?ondon.

er, A, .

_. Kraichnan, R.-. 76_ b8 # +ntermittency in the =ery mall cales ofT&rb&lence. )hy. . Bro*n, '.?. Rashko, A. 7_F8 # @. .>. !_

>.


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F9. 6.

5>. Ben2i, R. =er$assola,M. 78 # .55. Ar$o&l, 5.

56. "verson et al 7!8 # )hysics ?etter A F5.

5_. 'hosh, 0.3. 768 # ome Aspects on Data Analysis


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@. +ntell. 6_.

study of the theory of turbulence

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