T : Mean temperature du. dz dt

Transcription

1 ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI Volume, Issue 8, Feruary 4 A spectral moel for the turulent inetic energy in a stratifie shear flow Dejit Dutta Physics & Applie Mathematics Unit, Inian Statistical Institute, Kolata - 7 8, Inia. Astract: In this paper we attempt to erive an exact solution for the equation governing the inetic energy spectrum, forware y Vinnicheno et al.[] in a staly stratifie shear flow. A simple empirical moelis evelope for the turulent inetic energy spectrum E( of wealystratifie turulent shear flow.first a scheme for the existing moel ase on the Pao s (965 turulence energy cascae process haseen iscusse thoroughly. hen we put forwar the physical hypothesis for the spectral energy transfer hypothesis ue to Ouhov in its moifie form. he reuce spectral equation is solve in its exact form.he heat flux spectrum an the prouction term ue to mean velocity shear are moelle approximatelyin each of the cases. In the next section we treat the spectral case in which the moel is reuce to itssimple form, as otaine for the non-stratifie turulence phenomena. Spectral characteristics of suchturulent energy spectrum are investigate in the wave an numer ranges. Effectof anisotropy ue to shear an stratification is also emonstrate through simple computations. In thelast section we iscusse the results that lea to some conclusive remars. Nomenclature : Wave numer E( : urulent energy spectrum : Rate of total energy issipation : Coefficient inematic viscosity W( : Represent the transfer of energy through the hierarchy of eies ( : Spectrum of turulent friction stress H( : Spectrum of vertical heat flux ( g/ : Buoyancy parameter g : Acceleration ue to gravity u : Mean velocity uw:, Longituinal an vertical velocity fluctuationcomponents : Mean temperature u : Mean velocity graient z : Mean temperature graient z : Ratio etween the coefficients of turulent mixing forheat an momentum P: Kolmogoroff constant L: A length comparale with the scale of flow as awhole u z z : A imensional parameter /4 / : A efine wave numer / /4 : Kolmogoroff wave numer I. INRODUCION Atmospheric motions are in general turulent Kellogg s [] oservations of smoe puffs also suggests that the atmosphere is at least wealy turulent everywhere. he turulent motions in the atmosphere exist in many ifferent scales. he motions in the large eies may e approximate quasi-horizontal [], whereas the vertical velocities are appreciale in the smaller eies. In the case of homogeneous turulent shear,chen (cf. Hinze [4] has propose a ifferent mechanism for the prouction of energy y the mean shear. Accoring to chen, two cases may arise. In the first case, the vorticity of the asic stream is small compare with the vorticity of the turulent motion. In the secon case the vorticity of the asic stream is comparale with that of the turulent motion. he intersections etween the two vorticities are terme wea an strong respectively, in the first an secon case.gisina [5] has extene the aove iea of chen to the case of thermally stratifie turulent shear flow. Panchev an Syraov[6, 7] have investigate the spectral characteristics of thermally stratifie turulent flow respectively, with no shear an shear, ase on the physically iea aout the natural realisation of the egree of interaction etween mean an turulent fiels. In the present paper, we examine the spectral characteristics of the turulent energy for the small scale motions in an inertial su range, efine y an,taing into account the effect of anisotropy ue to shear an stratification. In moelling, the spectral transfer of energy, spectral flux of turulent friction stress an spectral heat flux we tae some extene view of the semi-empirical concept aout the mechanism of turulence an epen as well on imensional reasoning s. We confine ourselves to the case of wea interaction etween mean an turulent fiels, an postulate in turn that the spectral characteristics of turulent energy epen on, an, a parameter, efine y u z z. 96

2 ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI II. OBUKHOVHYPOHESIS AND IS MODIFICAION We integrate first the spectral equation relations of the energy transfer spectrum W(, tan the turulent inetic energy spectrum E(, t etween the limits of, e.g., from to, we otain (Panchev, 97. t ' ' ' ' ' ' E(, t W (, t E(, t ( Where L. hat is in effect, we consier the universal equilirium range lan for. E(,t is the equilirium spectrum. Accoring to Vinnicheno, ' ' ' ' E(, t E(, t ( ' an E(, t t ( ' ' ' E(, t t =- =constant hus when, we may write ' ' W (, t E(, t (4 We now nee a relation etween W(, t an E(, t ase on some intuitive physicalpicture of the energy transfer mechanism across the wave numers. It was Ouhov[8]who first propose an assumption which is rea as Volume, Issue 8, Feruary 4 In other wors, it was assume that the mean Reynols stresses of the macro-componentcan e approximately expresse in terms of the values of E( an alone.in that case, imensional analysis shows that they shoul e proportional to E(,an hence / (, E ' ( ', '. ( (7 W t E t E Accoring to this form, the Reynols stress interacting with the mean rate of shear or Vorticity of the wave numers smaller than is entirely restricte to. From the physical point of view, the assumption that W( is inepenent of the values of E( with ' > is not very satisfactory. We rewrite the aove form as: / ( E ( ' ( ', ',( E E W E E t (8 which is propose y Ellison after historically first approximation for the energy transfer function W ( E( ' '. ' E( ', t ' One may choose Kolmogorov s spectrum E( P / (9 E P / in orer to otain / 5/ ( W ( ', t ' [ ' E( ', t '] E( K ", t " ( 5 Where is an universal constant.ouhov otaine a E(, t 5/ reuce form fore(, t as ut at large values of, the solution ecreases at slower rate than the 5/ continuation of curve until at a certain cut off value of, then E(, t is still finite. Since some of the consequences of Ouhov s theory are physically unliely, Ellison (96[9] introuce a moification of this theory y assuming that the Reynols stresses of the microcomponent were largely etermine y the extreme largescale isturances in this component (with wave numers approaching such that / (, ' ( ', '. ( ', ' (6 F t E t E t as a solution of the spectral equation w(= in the inertial range. Here P is the imensionless constant. Moreover, in the inertial surange, where (is vali, we have [6] : / / ( ( w ' E ( ' ' ( where ( is the mean square of the vorticity. A w sustitution of ( into (8 leas to the approximation for F( suggeste y Pao [,]: W P E / 5/ ( ( ( III. MODELLING OF RANFER,PRODUCION AND BUOYANCY ERMS We concentrate our attention on the smallscale motion on the atmosphere, which areapproximately in local energetic equilirium.for the equilirium region 97

3 L ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI,the equation for the energy alance is reucile(cf.vinnichenco et al. [] to u ' E( ' ' ( ' ' w( H( ' ' z ( he first term on the right han sie of eqn.( escries the issipation of turulent energy. he secon term escries the prouction of turulent energy ue to mean velocity shear. he thir term represents the inertial transfer of turulent inetic energy, while the last term escries the contriution of the uoyancy force towars turulent energy. For W(,we tae the Moifie Ouhov form propose y Ellison [9], which is ase on Onsger s [] spectral jump concept, as W ( EE( ' E( ' ' /. Expressionfor W( as given y ( may e interprete as: Volume, Issue 8, Feruary 4 / 5/ / / ' E( ' ' P E( P E( Where u z can e written as N Rf, z WhereN is the Brunt Vaisala flux Richarson numer. (8.It is to e notice that frequency an Rfis the IV. EXAC SOLUION FOR HE URBULEN ENERGY SPECRUM We may rewrite equation (8 in the form / 5/ / 5/ ' E( ' ' P [ E( ] P [ E( ] (9 W(.( Vorticity (4 / / Since the vorticity is approximate y [], the coefficient of ey viscosity (or turulent inematic viscosity taes the form: ( (5 / / P E As we are consiering of wee interaction etween the main an turulent motions,the secon term on the righthan sie of ( can e moelle for sufficiently largewave numer as u u ( ' ' z z (6 where has the expression given y (5. For the same reason we may moel heat flux or uoyancy term as H( ' ' * z (7 Where * = (cf. Vinnicheno et. al.[]an Lumley an Panofsy [4]an has the imension of the z square of the vorticity. Sustituting (,(6 an (7 in equation (, we otain an fin its exact solution. Differentiating (9 with respect to an simplifying weotain 4 P E(.Integration of equation ( yiels / 7/ / 5/ [ E( ] 5/ / / ( 4 ( WhereQis the constant of integration. We now introuce a wave numer, given y 5/ / 7/ / / / E( Q.exp ( P.( /4 / ( In view of (, equation ( can e written as 5/ 4 / 7/ / / E( Q.exp ( P. [ ] = / ( / / P / P Q e. ( ( Aove equation gets simplifie to / P P / / E( Q e (4 98

4 ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI with the help of wave numer, as efine y the relation (. o etermine the constant Q we choose Kolmogoroffwave numer as such /4 /4 (5 which implies the conition /4 (6 Following Pesin an Baw [5], constant Q is now etermine as / P P / / Q E( e (7 Sustituting the value of Q in (4 we get after some simplification, / P[ ] E( e E( P 8 Scaling the wave numer an energy spectrum E( as E ( an E, equation (8 can e written E ( in the form P ( / E e P ( P (. When 5/ = we have E e,the moel is reuce to its simple form, as otaine for the non-stratifie turulencephenomena[]. 4. Case: Now uner the assumption, eqn. ( reuces to 5/ 4 / 7/ / / E( Q.exp ( P [ ] ( [Neglecting the higher powers than in the expansion]. Putting the value of in aove equation it can e reaily otaine Volume, Issue 8, Feruary 4 5 / / 8/ ( P E Q exp( exp( P exp( ( In a similar fashion, following Pesin an Baw s prescription, constant Q can eetermineas 5 / / 8/ ( P Q E exp( exp( P exp( ( Now using the aove form of Q an scaling the wave numer an energy spectrumfunction, eqn.( can e reformulate as 5 P 8/ E exp[ ( ]exp[ P( ]exp[ ( ] ( V. DISCUSSION OF HE RESULS AND CONCLUDING REMARKS In this paper we have expresse the energy transfer spectrum (section as a prouct of ey viscosity coefficient times the (Vorticity. Monin an Yaglom (975 showe in their analysis that the mean Reynols stresses of the micro component may approximately e proportional to ey viscosity coefficient E(.his expression accoring to them was ueto Ellison (96 an the mean square of the vorticity may e calculate from the results ' E( ', t ' / = ( / [, 6] which leas to the Pao s formula for the energy transfer spectrum. he solution for the turulence energy spectrumin the wave numer range where in the effect of stratification is no more an we otaine 7 Heisenerg power law of the isotropic case. he energy spectrum function as escrie y (9 has een plotte in (Fig.. an it is clear that from this graph that it represents the istriution of ecay of energy spectra over an the range. he energy spectrum function as preicte y equation ( is plotte for a value of the non-imensional parameter, satisfying the conition (6. For comparison with the case of homogeneous an isotropic turulence, we also plot the energy spectrum from equation for =. he value of the constant P is chosen to e.7. It is oservale (Fig. that for <, which implies the inertial surange < < as efine with conition (6 satisfie, the spectral ensity of turulent energy of wealy-stratifie shear flow is less than that of isotropic turulent flow. It may e notice that the effect of the non-imensional parameter on the shape of the spectrum is significant in < 99

5 ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI <an this correspons to the case of staly stratifie atmosphere. hat is some of the turulent energy esies usual energy transfer is converte into potential energy. Volume, Issue 8, Feruary 4 transfer hypothesis of Ouhov as moifie y Ellison is fully capale of explaining the energy transfer process in homogeneous turulence. It is important to note that Chara orty an Mazumer (986[8] solve the staly stratifie turulent flow y Fig : Plot of log as a function of log in the wave numer range he two curves ( =.75 an = intersect at = an cross over for >. In the issipation range >, the present moel suggeststhat the spectral ensity of turulent energy of wealy-stratifie shear flow althoughslightly excees that of the isotropic turulent flow ut seems to e significant (Fig.. More precisely,we recover the solution for the turulence energy spectrumin the wave numer range,where in the effect of stratification is no 7 more an we otaine Heisenerg power law of the isotropic case [4].We showe that yplotting log E vs log for the range of wave numer >> that at low wave numer spectral ensity oes not excee 5/, whereas at intermeiate wave numers, E exhiits approximately a 5/ falls off. At high wave numers, E exhiits a 7 ehaviour an at higher wave numers it ecays more faster. It may e conclue onthe aove analysis that the physical energy u z z as a small non zero parameter. He etermine inetic energy spectra for the case << an >>, where is the Kolmogoroffwave numer. We have iscusse that energy is fe in this type of turulence into energy containing range of the spectrum y mean velocity shear an extracte y the uoyancy forces of staly stratification. Gargett et al.(984 argue that for Small scale of turulence local isotropy may e achieve espite the presence of macroscopic properties of the mean flow, incluing its mean staility N. he influence of the uoyancy forces is not necessarily restricte to wave L numers of orer,wherel isa length comparale with the scale of flow as a whole. Gargett et al. (984 [9] also stuie the one-imensional spectrum of the staly stratifie turulent flow in relation to the value /4 /4 of I successfully where an / / N two respective wave numers epicting the staly stratifie turulent flows. he results with I = 6 case was emonstrate well. o sum up, the present analysis clearly suggests that the energy transfer function ue to moifie Ouhov form may e employe successfully for preiction of the energy ensity spectrum in the case of staly stratifie turulent flow. Fig : Non Dimensional energy spectra

6 ISSN: ISO 9:8 Certifie International Journal of Engineering an Innovative echnology (IJEI Volume, Issue 8, Feruary 4 ACKNOWLEDGEMEN he author is grateful to Professor HimariPaiMazumar of Inian Statistical Institute, Kolata (Inia for useful iscussions an suggesting improvements of this manuscript. REFERENCES [] N.K. Vinnicheno et al., urulence in the free Atmosphere, Consultant Bureau New Yor (98 Page.,. [] W.W. Kellogg, J. Meteor, (956 P.4. [] R.R. Long, Applie MechanicsReviews, 5(97 P. 97. [4] J.O. Hinze, urulence, McGraw-Hill, New Yor (975 P [5] F. A. Gisina, Izv, Aa, NauSSSR, fiz. Atmos. Oeana, (966 P. 84. [6] S. Panchev an D. Syraov, ellusxxiii, 6 (97 P.5. [7] D. Syraov, Compt. Ren. Aca. Bul.Sci., 4 (97 P. 75. [8] A.M. Ouhov, Compt. Ren. Aca. Sci. U.S.S: R,, (94 P.9. [9]. H. Ellison, Mecanique e la turulence, Symp., Merceille, France, (96. []. Karman an C.C. Lin, Av. Appl.Mech. (95, No.. [] Y.H. Pao,Phys. Fluis, 8 (965 P.6. [] L. Onsger, Phys. Rev. 68 (945 P. 86. [] S. Panchev, Ranom function an urulence, Pergamom Press, Oxfor USA (97, P. 8. [4] J.L. Lumley an H. A. Panofsy, he structure of Atmospheric urulence.inter science Pulishers, New Yor (964 P.65. [5] R.L. Pesin an P.H.S. Baw, echnical Report No. 8- MAE-FNYO 9- (968. [6] S. Panchev an D. Kesich,(Physique es fluie (969. [7] A.S. Monin an A. M. Yaglom, Statistical Flui Mechanics, MI Press, (975 P. 95,P. 4. [8] A. K. Charaorty an H. P. Mazumar, Inian J. ech., 4 (986 P.549. [9] A.E. Gargett,.R. Osorn, P.W. Nasmyth, J. Flui Mech. 44 (984, P.. AUHOR BIOGRAPHY Dejit Dutta got his Ph.D egree from University of Calcutta, Kolata, INDIA, 4 (Fe. He was the full time research fellow of Inian Statistical Institute, Kolata, INDIA. He is now a visiting scientist in the epartment of Physics an Applie Mathematics Unit, Inian Statistical Institute, Kolata, INDIA. His research areas are super symmetric Quantum mechanics, Non Hermitian Quantum Physics, Plasma Physics, Non-linear Integrale moels an urulence. He is a life memer of Calcutta Mathematical Society, Salt Lae, Kolata.