DERIVATION OF REYNOLDS AVERAGED NAVIER-STOKES EQUATION AND ANALYSIS OF THIS EQUATION
NAVIER-STOKES EQUATION :-
THE NAVIER-STOKES EQUATION IS A PARTIAL DIFFERENTIATION EQUATION WHICH IS USED TO DETERMINE FLOW IN THE INCOMPRESSIBLE FLUIDS. THESE EQUATIONS HELPS IN DETERMINING THE FLOW QUANTITIES FLOWING IN FLUID AND SOLVES MANY AMBIGIOUS PROBLEMS PREVAILING IN REAL WORLD.
SIGNIFICANCE OF NAVIER-STOKES EQUATIONS:-
THESE EQUATIONS ARE MORE IMPORTANT BECAUSE IT HELPS IN SOLVING THE PROBLEM IN ENGINEERING AND SCIENTIFIC FIELD.
APPLICATION OF NAVIER-STOKES EQUATION:-
1. THEY ARE USED TO MODEL WEATHER, OCEAN CURRENTS, WATER FLOW IN PIPE AND AIR FLOW AROUND A WING.
2. THET ARE USED TO DESIGN AIRCRAFT AND CARS, THE STUDY OF BLOOD FLOW, THE ANALYSIS OF POLLUTION.
3. THEY CAN BE USED TO MODEL AND STUDY MAGNETOHYDRODYNAMICS WHEN COUPLED WITH MAXWELL\'S EQUATION.
DIFFERENT TYPES OF THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS ARE:-
CONTINUITY EQUATION:-
`(delu)/(delx)+(delu)/(dely)+(delu)/(delz)=0.` or `nablau=0`
MOMENTUM EQUATIONS:-
FOR X-MOMENTUM:-
`(delu)/(delt)+(delu^2)/(delx)+(deluv)/(dely)+(deluw)/(delz)=-1/rho(delp)/(delx)+nu(del^2u)/(del^2x)+nu(del^2u)/(del^2y)+nu(del^2u)/(del^2z)` or `(delu)/(delt) +nablarhou^2=-nablap+nunabla^2u`
FOR Y-MOMENTUM:-
`(delv)/(delt)+(deluv)/(delx)+(delv^2)/(dely)+(delvw)/(delz)=-1/rho(delp)/(dely)+nu(del^2v)/(del^2x)+nu(del^2v)/(del^2y)+nu(del^2v)/(del^2z)` or `(delrhov)/(delt) +nablarhov^2=-nablap+nunabla^2v`
FOR Z-MOMENTUM:-
`(delw)/(delt)+(deluw)/(delx)+(delvw)/(dely)+(delw^2)/(delz)=-1/rho(delp)/(delw)+nu(del^2w)/(del^2x)+nu(del^2w)/(del^2y)+nu(del^2w)/(del^2z)` or `(delrhow)/(delt) +nablarhow^2=-nablap+nunabla^2w`
REYNOLDS DECOMPOSITION:-
THE \"REYNOLDS DECOMPOSITION\" IS USED TO ANALYZE THE TURBULENT FLOW THAT ARE STEADY IN THE AVERAGE. IT DECOMPOSES THE TURBULENT FLOW QUANTITIES INTO AVERAGE AND FLUCTUATING COMPONENTS. IN REYNOLDS DECOMPOSITION , IT DEFINES TIME AVERAGED VELOCITY FLOW AS:-
`1/Tint_(t_0-T/2)^(t_0+T/2)f(t)dt=bar(f(t_0))impliesbar(f)(t_0)+f\'(t_0)``
USING REYNOLDS EQUATIONS ON FOLLOWING BELOW EQUATIONS:-
FOR CONTINUITY EQUATION:-
CONTINUITY EQUATION IS GIVEN BY:-
`(delu)/(delx)+(delv)/dy+(delw)/(delz)=0`
APPLYING TIME-AVERAGING ON THE ABOVE EQUATION, WE GET
`(bar(delu))/(delx)+(bar(delv))/dy+(bar(delw))/(delz)=0`
WE WILL DECOMPOSE THE EQUATION INTO AVERAGE AND FLUCTUATING COMPONENTS, WHICH WILL BE AS:-
`(del(bar(baru+uprime)))/(delx)+(del(bar(barv+vprime)))/(dely)+(del(bar(barw+wprime)))/(delz)=0.`
ARRANGING THE EQUATION AS FOLLOW BELOW,
`(delbaru)/(delx)+(delbarv)/(dely)+(delbarw)/(delz)+(delbaruprime)/(delx)+(delbarvprime)/(dely)+(delbarwprime)/(delz)=0.`
WE KNOW THAT THAT TIME AVERAGING OF FLUCTUATING COMPONENT IS ZERO. THEREFORE,
`(delbaru)/(delx)+(delbarv)/(dely)+(delbarw)/(delz)+cancel(delbaruprime)/(delx)+cancel(delbarvprime)/(dely)+cancel(delbarwprime)/(delz)=0.`
HENCE, TIME-AVERAGED EQUATION WILL BE AS:-
`(delbaru)/(delx)+(delbarv)/dy+(delbarw)/(delz)=0.`
MOMENTUM EQUATIONS:-
FOR X-MOMENTUM:-
`(delu)/(delt)+(delu^2)/(delx)+(deluv)/(dely)+(deluw)/(delz)=-1/rho(delp)/(delx)+nu(del^2u)/(del^2x)+nu(del^2u)/(del^2y)+nu(del^2u)/(del^2z).`
APPLYING TIME-AVERAGING ON THE ABOVE EQUATION, WE GET
`bar(delu)/(delt)+bar(delu^2)/(delx)+bar(deluv)/(dely)+bar(deluw)/(delz)=-1/rhobar(delp)/(delx)+nubar(del^2u)/(del^2x)+nubar(del^2u)/(del^2y)+nubar(del^2u)/(del^2z).`
DECOMPOSING THE EQUATION INTO AVERAGE AND FLUCTUATING COMPONENTS, WE GET
`(bar(del(baru+uprime)))/(delt)+bar(del(baru+uprime)(baru+uprime))/(delx)+(bar(del(baru+uprime)(barv+vprime)))/(dely)+bar(del(baru+uprime)(barw+wprime))/(delz)=-1/(rho)bar(del(barp+pprime))/(delx)+nubar(del^2(baru+uprime))/(del^2x)+nubar(del^2(barv+vprime))/(del^2y)+nubar(del^2(barw+wprime))/(del^2z).`
ARRANGING THE EQUATION AS FOLLOW BELOW,
`(delbaru)/(delt)+(delbar(uprime))/(delt) +(delbar(u^2))/(delx)+(delbar(uv))/(dely)+(delbar(uw))/(delz)+(delbar(uprime)^2)/(delx)+(del(bar(uprimevprime)))/(dely)+(del(bar(uprimewprime)))/(delz)=-1/rho(delp)/(delx)+nu((del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2))+nu((del^2bar(uprime))/(delx^2)+(del^2bar(uprime))/(dely^2)+(del^2bar(uprime))/(delz^2)).`
SINCE THE TIME-AVERAGING OF INDEPENDENT FLUCTUATING PART IS ZERO i.e `bar(uprime)=0, bar(vprime)=0 and bar(wprime)=0`
THERFORE,
`(delbaru)/(delt)+cancel((delbar(uprime))/(delt))+(delbar(u^2))/(delx)+(delbar(uv))/(dely)+(delbar(uw))/(delz) =-1/rho(delp)/(delx)+nu((del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2))+nucancel(((del^2bar(uprime))/(delx^2)+(del^2bar(uprime))/(dely^2)+(del^2bar(uprime))/(delz^2)))-(delbar(uprimeuprime))/(delx)-(del(bar(uprimevprime)))/(dely)-(del(bar(uprimewprime)))/(delz).`
HENCE THE REQUIRED TIME AVERAGED EQUATION IS:-
`(delbaru)/(delt)+(delbar(u^2))/(delx)+(del(barubarv))/(dely)+(del(barubarw))/(delz) =-1/rho(delbarp)/(delx)+nu((del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2))-(delbar((uprimeuprime)))/(delx)-(del(bar(uprimevprime)))/(dely)-(del(bar(uprimewprime)))/(delz).`
FOR Y-MOMENTUM:-
`(delv)/(delt)+(deluv)/(delx)+(delv^2)/(dely)+(delvw)/(delz)=-1/rho(delp)/(delx)+nu(del^2v)/(del^2x)+nu(del^2v)/(del^2y)+nu(del^2v)/(del^2z).`
APPLYING TIME-AVERAGING ON THE ABOVE EQUATION, WE GET
`bar(delv)/(delt)+bar(deluv)/(delx)+bar(delv^2)/(dely)+bar(delvw)/(delz)=-1/rhobar(delp)/(delx)+nubar(del^2v)/(del^2x)+nubar(del^2v)/(del^2y)+nubar(del^2v)/(del^2z).`
DECOMPOSING THE EQUATION INTO AVERAGE AND FLUCTUATING COMPONENTS, WE GET
`(bar(del(barv+vprime)))/(delt)+bar(del(baru+uprime)(barv+vprime))/(delx)+(bar(del(barv+vprime)(barv+vprime)))/(dely)+bar(del(barv+vprime)(barw+wprime))/(delz)=-1/(rho)bar(del(barp+pprime))/(delx)+nubar(del^2(barv+vprime))/(del^2x)+nubar(del^2(barv+vprime))/(del^2y)+nubar(del^2(barv+vprime))/(del^2z).`
ARRANGING THE EQUATION AS FOLLOW BELOW,
`(delbarv)/(delt)+(delbar(vprime))/(delt) +(delbar(uv))/(delx)+(delbarv^2)/(dely)+(delbar(vw))/(delz)+(delbar((uprimevprime)))/(delx)+(del(bar(vprimevprime)))/(dely)+(del(bar(vprimewprime)))/(delz)=-1/rho(delp)/(delx)+nu((del^2barv)/(delx^2)+(del^2barv)/(dely^2)+(del^2barv)/(delz^2))-nu((del^2bar(vprime))/(delx^2)+(del^2bar(vprime))/(dely^2)+(del^2bar(vprime))/(delz^2)).`
SINCE THE TIME-AVERAGING OF INDEPENDENT FLUCTUATING PART IS ZERO i.e `bar(uprime)=0, bar(vprime)=0 and bar(wprime)=0`
THERFORE,
`(delbaru)/(delt)+cancel((delbar(uprime))/(delt))+(delbar(uv))/(delx)+(delbar(v^2))/(dely)+(delbar(vw))/(delz) =-1/rho(delp)/(delx)+nu((del^2barv)/(delx^2)+(del^2barv)/(dely^2)+(del^2barv)/(delz^2))+nucancel(((del^2bar(vprime))/(delx^2)+(del^2bar(vprime))/(dely^2)+(del^2bar(vprime^2))/(delz^2)))-(delbar(uprimevprime))/(delx)-(delbar(vprime^2))/(dely)-(del(bar(vprimewprime)))/(delz).`
HENCE THE REQUIRED TIME AVERAGED EQUATION IS:-
`(delbarv)/(delt)+(del(barubarv))/(delx)+(delbar(v^2))/(dely)+(del(barvbarw))/(delz) =-1/rho(delbarp)/(delx)+nu((del^2barv)/(delx^2)+(del^2barv)/(dely^2)+(del^2barv)/(delz^2))-(delbar((uprimevprime)))/(delx)-(del(bar(vprimevprime)))/(dely)-(del(bar(vprimewprime)))/(delz)`
FOR Z-MOMENTUM:-
`(delw)/(delt)+(deluw)/(delx)+(delvw)/(dely)+(delw^2)/(delz)=-1/rho(delp)/(delx)+nu(del^2w)/(del^2x)+nu(del^2w)/(del^2y)+nu(del^2w)/(del^2z).`
APPLYING TIME-AVERAGING ON THE ABOVE EQUATION, WE GET
`bar(delw)/(delt)+bar(deluw)/(delx)+bar(delvw)/(dely)+bar(delw^2)/(delz)=-1/rhobar(delp)/(delx)+nubar(del^2w)/(del^2x)+nubar(del^2w)/(del^2y)+nubar(del^2w)/(del^2z).`
DECOMPOSING THE EQUATION INTO AVERAGE AND FLUCTUATING COMPONENTS, WE GET
`(bar(del(barw+wprime)))/(delt)+bar(del(baru+uprime)(barw+wprime))/(delx)+(bar(del(barv+vprime)(barw+wprime)))/(dely)+bar(del(barw+wprime)(barw+wprime))/(delz)=-1/(rho)bar(del(barp+pprime))/(delx)+nubar(del^2(barw+wprime))/(del^2x)+nubar(del^2(barw+wprime))/(del^2y)+nubar(del^2(barw+wprime))/(del^2z).`
ARRANGING THE EQUATION AS FOLLOW BELOW,
`(delbarw)/(delt)+(delbar(wprime))/(delt) +(delbar(uw))/(delx)+(delbar(vw))/(dely)+(delbar(w^2))/(delz)+(delbar(uprimewprime))/(delx)+(del(bar(vprimewprime)))/(dely)+(del(bar(wprimewprime)))/(delz)=-1/rho(delp)/(delx)+nu((del^2barw)/(delx^2)+(del^2barw)/(dely^2)+(del^2barw)/(delz^2))+nu((del^2bar(wprime))/(delx^2)+(del^2bar(wprime))/(dely^2)+(del^2bar(wprime))/(delz^2)).`
SINCE THE TIME-AVERAGING OF INDEPENDENT FLUCTUATING PART IS ZERO i.e `bar(uprime)=0, bar(vprime)=0 and bar(wprime)=0`
THERFORE,
`(delbarw)/(delt)+cancel((delbar(wprime))/(delt))+(delbar(uw))/(delx)+(delbar(vw))/(dely)+(delbar(w^2))/(delz) =-1/rho(delp)/(delx)+nu((del^2barw)/(delx^2)+(del^2barw)/(dely^2)+(del^2barw)/(delz^2))+nucancel(((del^2bar(wprime))/(delx^2)+(del^2bar(wprime))/(dely^2)+(del^2bar(wprime))/(delz^2)))-(delbar(uprimewprime))/(delx)-(del(bar(vprimewprime)))/(dely)-(del(bar(wprimewprime)))/(delz).`
HENCE THE REQUIRED TIME AVERAGED EQUATION IS:-
`(delbarw)/(delt)+(del(barubarw))/(delx)+(del(barvbarw))/(dely)+(delbar(w^2))/(delz) =-1/rho(delbarp)/(delx)+nu((del^2barw)/(delx^2)+(del^2barw)/(dely^2)+(del^2barw)/(delz^2))-(delbar((uprimewprime)))/(delx)-(del(bar(vprimewprime)))/(dely)-(del(bar(wprimewprime)))/(delz).`
SO IN THIS WAY, WE HAVE OBTAINED THE REYNOLDS-AVERAGED NAVIER-STOKES EQUATION BY USING REYNOLDS DECOMPSITION TECHNIQUE.
FOR REYNOLDS-STRESS:-
WE WILL TAKE THE GENERAL EQUATION FOR MOMENTUM IN NAVIER-STOKES EQUATION WHICH WILL BE AS:-
`(rhodelbaru_i)/(delt)+(rhodelbar(u_iu_j))/(delx_i)=-((delp)/(delx_i))+nu(del^2u_i)/(del^2x_j)-(delrhobar(u_iprimeu_jprime))/(delx_i)`
`(rhodelbaru_i)/(delt)+(rhodelbar(u_iu_j))/(delx_i)=-((delp)/(delx_i))+del/(delx)(nu(delu_i)/(delx_j)-(rhobar(u_iprimeu_jprime))).`
HERE, `-rhobar(u_iprimeu_jprime)`` IS THE REYNOLDS STRESS WHICH IS DERIVED ALONG WITH NORMAL-SHEAR STRESS i.e `nu(delu_i)/(delx_j)`
HERE, REYNOLDS-NORMAL STRESS ARE:- `-rhobar(uprimeuprime)``,`-rhobar(vprimevprime)`or`-rhobar(wprimewprime)`.
AND, REYNOLDS-SHEAR STRESS ARE:- `-rhobar(uprimevprime)``-rhobar(uprimewprime)` or `-rhobar(vprimewprime)`.
SIGNIFICANCE OF REYNOLDS-STRESS:-
THIS REYNOLDS-STRESSES ARE NOT STRESSES BUT INSTEAD ARE THE EXTRA ADDITIONAL MOMENTUM FLUXES WHICH ORIGINATE DUE TO UNSTEADY TURBULENT MOTIONS. THESE REYNOLDS-STRESSES WORK AS A EFFECTIVE EXTRA STRESS IN THE FLUID MOTION.
THESE REYNOLDS STRESSES WHICH DEVELOPS DUE TO TURBULENT MOTION IN FLUID REFLECTS EFFECT ON THE STRESS TENSOR. THE IMPACT OF REYNOLDS STRESS ON STRESS TENSOR HAS BEEN EXPLAINED BELOW:-
THE REYNOLDS AVERAGED NAVIER-STOKES EQUATION FOR STEADY STATE CAN ALSO BE WRITTEN AS:-
`(rhodelbar(u_iu_j))/(delx_i)=1/rho((delbarsigma_(ij)-rhodelbar(u_iprimeu_jprime)))/(delx_j)`
THUS THE TOTAL EFFECTIVE STRESS-TENSOR IN THE TURBULENT FLOW DENOTED BY `sigma_(ij)^star` IS GIVEN BY :-
`sigma_(ij)^star=bar(sigma_(ij))-rhobar(u_iprimeu_jprime`. THIS WAY IT LEADS TO AN INCREASE OR DECREASE IN STRESS TENSOR BY DEPENDING ON THE TYPE OF RELATION BETWEEN `u_i and u_j` WHICH DECIDES WHETHER IT WOULD BE 0,+VE OR-VE FOR `bar(u_iprimeu_jprime)`.
APART FROM IT, IT DETERMINES THE EFFECT OF IT'S MAGNITUDE ON THE FLOW OF FLUID IN TERMS OF STRESS DUE TO FLUCTUATION ARISE BECAUSE OF TURBULENT MOTION. FOR BETTER UNDERSTANDING, PLOT GIVEN BELOW HELPS IN UNDERSTANDING THE CONCEPT OF REYNOLDS STRESS IN MUCH INTUTIVE WAY.
HERE, THE PLOT FOR REYNOLDS STRESS IN A TURBULENT BOUNDARY LAYER EXPLAINS THE MAGNITUDE OF REYNOLDS STRESS ON THE ACCOUNT OF DISTANCE FROM WALL. THE PLOT SHOW THAT FOR REYNOLDS NORMAL STRESS , THE RELATIVE MAGNITUDE IS TOO LESS WHICH IS AROUND 0.004 IF WE COMPARE THE REYNOLDS NORMAL STRESS TERMS TO PRIMARY INERTIAL TERMS i.e `(uprime^2)/U^2 , (vprime^2)/U^2 , (wprime^2)/U^2`. SO IT GIVES US INFORMATION THAT RENOLDS NORMAL STRESS CAN BE NEGLECTED IN TURBULENT MODEL SINCE RENOLDS NORMAL STRESS ARE TOO SMALL COMPARED TO THE PRIMARY INERTIAL TERMS.
IN CONTRAST, WHEN REYNOLDS SHEAR STRESS IS COMPARED TO THE PRIMARY INERTIAL STRESS. IT GAVE US VERY TOO VALUE OF MAGNITUDE WHICH WAS AROUND 0.001 BUT WHEN THESE REYNOLDS SHEAR STRESSES WAS COMPARED WITH PRIMARY INERTIAL STRESS AT DISTANCE FROM WALL OF SURFACE LAYER , WE FOUND THAT THE RELATIVE MAGNITUDE COULD BE TOO HIGH DUE TO TURBULENCE AT THE WALL OF SURFACE LAYER AS AN ADDITIONAL FACTOR ALSO MULTIPLIES WITH IT. THE DEVELOPED RELATIVE STRESS IS GIVEN BY:-
`bar(uprimevprime)/(nu(delbaru)/(dely))=(bar(uprimevprime))/(barU^2)(Udelta)/nu` WHERE `(Udelta)/nu` IS AN ADDITIONAL FACTOR KNOWN AS Y+ FOR WALL FUNCTION WHICH HELPS IN DETERMINING THE TYPE OF FLOW AT SURFACE OF FLOW.
NOW THE OBTAINED REYNOLDS-AVERAGED NAVIER-STOKES EQUATION ARE:-
1. `(delbaru)/(delx)+(delbarv)/dy+(delbarw)/(delz)=0`.
2.`(delbaru)/(delt)+(delbar(u^2))/(delx)+(del(barubarv))/(dely)+(del(barubarw))/(delz) =-1/rho(delp)/(delx)+nu((del^2baru)/(delx^2)+(del^2baru)/(dely^2)+(del^2baru)/(delz^2))-(delbar((uprimeuprime)))/(delx)-(del(bar(uprimevprime)))/(dely)-(del(bar(uprimewprime)))/(delz)`.
3. `(delbarv)/(delt)+(del(barubarv))/(delx)+(delbar(v^2))/(dely)+(del(barvbarw))/(delz) =-1/rho(delp)/(delx)+nu((del^2barv)/(delx^2)+(del^2barv)/(dely^2)+(del^2barv)/(delz^2))-(delbar((uprimevprime)))/(delx)-(del(bar(vprimevprime)))/(dely)-(del(bar(vprimewprime)))/(delz)`.
4. `(delbarw)/(delt)+(del(barubarw))/(delx)+(del(barvbarw))/(dely)+(delbar(w^2))/(delz) =-1/rho(delp)/(delx)+nu((del^2barw)/(delx^2)+(del^2barw)/(dely^2)+(del^2barw)/(delz^2))-(delbar((uprimewprime)))/(delx)-(del(bar(vprimewprime)))/(dely)-(del(bar(wprimewprime)))/(delz)`.
TURBULENT VISCOSITY:-
IT IS A TYPE OF VISCOSITY WHICH ORGINATES DUE TO TURBULENCE IN THE FLOW OF FLUID. THIS TYPE OF VISCOSITY BASICALLY ORGINATES DUE TO THE ORGINATION OF FLUCUATION IN THE FLOW DUE TO TURBULENCE. IN ORDER TO KNOW TURBULENT VISCOSITY, HERE WE DEAL WITH EDDIES WHICH IS NOTHING BUT THE SMALL RAPID WHIRLPOOL DEVELOPED DUE TO FLUCTUATION.
ALTHOUGH THIS INDIVIDUAL EDDIES DO NOT HAVE SIGNIFICANT EFFECT ON THE FLOW OF FLUID BUT WHEN CONSTITUTE AS A WHOLE IN FLOW DOMAIN , IT CAN REFLECT BACK IN THE FLUID MOTION AS A FORM REYNOLDS SHEAR STRESS BACKED BY THIS EDDIES IN THE VISCOSITY.
THIS EDDIES ARE SOLE RESPONSIBLE FOR DEVELOPMENT OF VISCOSITY. THE CHARACTERISTICS OF EDDIES ARE DOMAINTED BY THE SIZE OF EDDY AND MOLECULAR INTERCATION OF MOLECULES. IF THE EDDIES HAS LARGE SIZE THEN TURBULENCE IS DOMINATED BY SIZE OTHERWISE IF SIZE IS VERY SMALL OF EDDY i.e IT WORKS AS MINI-MOLECULES THEN IT IS DOMINATED BY THE MOLECULAR INTERCATION. THIS EDDIES EFFECTS THE TURBULENT TRANSPORT PROPERTIES LIKE DIFFUSIVITY OF MOMENTUM , MASS AND HEAT.
THE CONCEPT OF TURBULENT VISCOSITY IS DERIVED THROUGH THE MIXING-LENGTH THEORY GIVEN BY THE PRANDTL IN 1925 I THE FOLLOWING WAY:-
THE \"MIXING -LENGTH\" THEORY MODELS THE TURBULENT REYNOLDS-STRESS IN SIMILAR WAY AS MOLECULAR MOTION CAUSED THE LAMINAR STRESS IN LAMINAR FLOW. HERE ,THE RANDOM MOTION OF COHERENT REGION LIKE\"EDDIES\" OF TURBULENT FLOW IS ASSUMED TO BE EQUIVALENT TO THE RANDOM MOTION OF MOLECULAR MOTION OF MOLECULES IN THE LAMINAR CASE. AS WE CONSIDERED THE FREE PATH LENGTH BETWEEN THE MOLECULAR COLLISIONS IN LAMINAR CASE, HERE WE ASSUMES THAT THERE IS SOME MIXING LENGTH (l) OF EDDIES OVER WHICH IT EXCHANGES IT\'S TRANSPORT PROPERTIES LIKE EXCHANGE OF MOMENTUM.
HERE WE HAVE CONSIDERED THE KINMETIC EDDY VISCOSITY AS\"`lprime`\", WHERE vprime IS THE TYPICAL VERTICAL VELOCITY FLUCTUATION. IN OTHER WORDS,`lamdararrl` AND `ararrvprime`. FROM CONTINUITY `uprime+vprime=0`. WE HAVE ALSO ASSUME THAT THERE IS NO DIRECTION FOR EDDY LENGTH SCALES.AND `vprime` IS ORDER OF `uprime`.AND `uprime` IS ESTIMATED AS `l(delu)/(dely)`. SINCE WE HAVE CONSIDERED `uprime` AS ORDER OF `vprime`, SO THE MAGINITUDE OF `uprime`WILL BE EQUIVALENT TO TRANSVERSE VELOCITY`vprime`. THAT GIVES A TURBULENT EDDY VISCOSITY AS,
`nu_T=l^2abs((delu)/(dely)`
IF WE COMPARE TURBULENT VISCOSITY TO MOLECULAR VISCOSITY, WE KNOW THAT THE IN LAMINAR VISCOSITY, THE VISCOSITY ARISES DUE TO THE MOLECULAR INTERACTION BETWEEN MOLECULES IN THE FORM OF MOLECULAR COLLISION WHEREAS THE VISCOSITY IN TURBULENT ARISES DUE TO DEVELOPMENT OF EDDIES WHICH CAUSES TURBULENT VISCOSITY IN FLOW.
IN LAMINAR FLOW , THERE IS TRANFER OF MOMENTUM FROM ONE MOLECULE TO THE ANOTHER MOLECULE IN FREE PATH LENGTH(`lamda`) WHEN THERE IS MOLECULAR COLLISION BETWEEN MOLECULES AT MOLECULAR LEVEL. HERE, THE LAMINAR VISCOSITY IS GIVEN BY :-
`nu=alamda((delu)/(dely))`
SO HERE RELATION PROVES THAT THERE IS VISCOSITY IN THE FLUID FLOW DUE TO MOELCULAR COLLISIONS.
SO HERE WE HAVE FIND OUT THE THE RELATION THAT,IF THERE IS LAMINAR FLOW, THEN THE VISCOSITY DUE TO LAMINAR WILL DOMINATE.
i.e LAMINAR VISCOSITY>>TURBULENT VISCOSITY. IN THAT CASE, THE DISSIPATION OF ENERGY DUE TO TURBULENCE WILL BE LESS OR NEGLIGIBLE.
OTHERWISE,IF THERE IS TURBULENT FLOW, THEN THE VISCOSITY DUE TO TURBULENCE WILL DOMINATE.