Nonlinear Lagrangian equations for turbulent motion and buoyancy in inhomogeneous flows

Transcription

1 Nonlinear Lagrangian equations for turbulent motion an buoyancy in inhomogeneous flows Stefan Heinz a) Delft University of Technology, Faculty of Applie Physics, Section Heat Transfer, Lorentzweg 1, 68 CJ Delft, The Netherlans Receive 18 April 1995; accepte 3 August 1996 Linear an nonlinear Lagrangian equations are erive for stochastic processes that appear as solutions of the average hyroynamic equations, since their moments satisfy the bugets given by these equations. These equations inclue the potential temperature, so that non-neutral flows can be escribe. They will be compare with nonlinear an non-markovian equations that are obtaine using concepts of nonequilibrium statistical mechanics. This approach permits the escription of turbulent motion an buoyancy, where memory effects an riving forces with arbitrary colore noise may occur. The equations epen on assumptions that concern the issipation an pressure reistribution. In the approximations of Kolmogorov an Rotta for these terms, the issipation time scale remains open, which can be etermine by the calculation of the prouction issipation ratio of turbulent kinetic energy. The features of these equations are illustrate by the calculation of turbulent states in the space of invariants American Institute of Physics. S I. INTRODUCTION The average hyroynamic equations AHE efine bugets at points of the flow for moments of ifferent orer of the istribution function of the fluctuating hyroynamic variables. Lagrangian moels for the motion of flui particles in high-reynols number turbulent flows may etermine stochastic processes that represent solutions of the AHE, when the issipation an pressure reistribution terms are moelle in these buget equations. This means that the moments of these stochastic processes obey these bugets, e.g., up to secon orer without the nee for any assumptions about the spatial graients of the thir moments the turbulent transport terms. Such Lagrangian moels are use to stuy the realizability of solutions of secon-orer moels, 1, an they are successfully applie to calculate the turbulent iffusion of passive tracers in complex flows. 3 6 The turbulent ispersion can be escribe in non-neutral flows, if the potential temperature is inclue in a stochastic Lagrangian escription of particle motion. 7 Such Lagrangian equations, which are linear in the particle velocities an potential temperature but nonlinear in the particle position, can be erive completely consistent with the AHE up to secon orer, 8 where the issipation an pressure reistribution terms are taken in the approximations of Kolmogorov 9 an Rotta, 10 respectively. But here the problem of nonuniqueness arises which is known for neutral flows, 1,,4,5 which means the consistency with the AHE oes not uniquely etermine the stochastic Lagrangian equations. The assessment of consequences of ifferences in Lagrangian moels satisfying the same AHE poses a ifficult problem, but it may be expecte that these ifferences influence consierably the calculate features of particle motion in complex flows i.e., uner conitions where the effects of inhomogeneities an anisotropy have to a Telephone: ; Fax: ; Electronic mail: heinz@uttwta.tn.tuelft.nl be consiere. Moreover, when the approximations of Kolmogorov an Rotta are use for the issipation an pressure reistribution, a issipation time scale appears in the Lagrangian equations, which has to be estimate. The calculation of this time scale is an important problem, because it etermines the prouction issipation ratio for the turbulent kinetic energy TKE an therefore the local character of the energy transfer. The consistency between stochastic Lagrangian equations an the AHE is consiere at first with respect to linear equations for the particle velocity an potential temperature, where, in particular, the escription of the potential temperature by a stochastic ifferential equation is iscusse. The same problem is consiere for nonlinear Markovian equations in the thir section, where the nonuniqueness problem is iscusse for non-neutral flows. This investigation of the suitability of stochastic processes to appear as solutions of the AHE is compare in the fourth section with another approach, where stochastic ifferential equations are erive by applying concepts of nonequilibrium statistical mechanics. This explains in which way memory effects enter into the Lagrangian equations. As an example, Sawfor s equation 5 for the acceleration of a flui particle is erive, which takes colore noise in the velocity equation into account. The influence of vertical graients of the mean horizontal flow velocity an potential temperature on particle motion is consiere in the fifth section. This is one by consiering the relation between the unknown time scale, which arises by the parametrization of the issipation next section, an the win shear an stratification. Through this relation, the Lagrangian equations that are erive in the secon section epen for balance ratios of prouction an issipation of TKE an heat only upon three flow numbers. These equations reflect nonaverage hyroynamic equations in a scale, where the Lagrangian acceleration correlation is small for time lags much longer than the Kolmogorov time scale 11 Phys. Fluis 9 (3), March /97/9(3)/703/14/$ American Institute of Physics 703

2 an the three flow number play the role of the molecular constants as parameters. II. LINEAR MARKOVIAN EQUATIONS Let us first consier linear Markovian equations for the escription of flui particle motion an the change of the potential temperature of particles. Whereas there are goo reasons to treat the velocity of a flui particle as a Markov process in high-reynols number flows because of the structure of the acceleration correlation, 3,4,11 the escription of the potential temperature by a stochastic process is not comparably investigate. To escribe the important part of buoyancy effects on the turbulent ispersion, it was suppose by Zanetti an Al-Maani 1 an Cogan 13 that the potential temperature also satisfies a stochastic ifferential equation. Van Dop investigate this approach in a more funamental way. 7 He pointe out the conceptional problem of assigning a potential temperature to a flui particle. Nevertheless, this approach is continue here, which means a stochastic ifferential equation is assume for the potential temperature of a particle. Accoring to inertial subrange theory, the structure function an the autocorrelation of the temperature woul be expecte as proportional to the time lag an exponential, respectively. 7 The reprouction of these inertial subrange relations can be foun, if the potential temperature is escribe by a stochastic ifferential equation. As shown below, this approach guarantees, e.g., that the AHE are fulfille up to secon orer. These equations are able to explain well the basic features of buoyancy effects in the turbulent ispersion. 7,14 The available experimental evience tens to inicate that the probability ensity function of an inert ynamically passive scalar convecte by homogeneous turbulence evolves asymptotically towar a Gaussian shape. 15 This is supporte by results of irect numerical simulations. 16 At least for an approximately neutral stratification with a negligible influence of the potential temperature on the turbulence, this quantity may be escribe by a linear stochastic ifferential equation proviing a Gaussian istribution. The iea of a flui particle with a potential temperature represents a reference picture for the real process in corresponence with the AHE an permits the escription of the basic characteristics of buoyancy processes. This is equivalent to approaches employe in the theory of turbulent mixing, where moels are use that are not irectly relate to a physical process, but are a goo characterization of essential features of the consiere phenomena. 15,17,18 Turbulent motion an buoyancy are escribe by the same structure of equation in this way. This is particularly avantageous for the comparison with the AHE, as iscusse below. The turbulent flow is regare as a whole of flui particles, each having a constant mass. The time-epenent total mass of the flui results from particle mass times the time-epenent total number of flui particles. Neglecting chemical reactions, each particle is characterize at the time t by its position x L (t), velocity U L (t) vectors with components x L I (t) an U L I (t), where I1,,3 an the subscript L enotes a Lagrangian quantity an potential temperature L (t). Particle ensity an volume change in time accoring to a state equation. With respect to comparisons with the AHE, it is avantageous to combine the particle velocity U L (t) an potential temperature L (t) to the fourimensional state vector Z L (t)u L (t), L (t). Assuming x L (t), Z L (t) as a Markov process an only linear fluctuations of the state Z L (t), these quantities change accoring to 19,0 t x L I tz I L t, 1a t Z L i ta i G ij Z j L Z j W j E b ij, 1b t where the small superscripts run from 1 to 4 in contrast to capitals which run from 1 to 3 an summation over repeate superscripts is assume. The first two terms in 1b give the systematic particle motion with unknown coefficients a i an G ij, where the ensemble average is enote by. The ensemble averages of Eulerian quantities subscript E are estimate at fixe positions x, which are replace by xx L (t) in the equations. Consequently, these equations are linear in the state Z L but may be nonlinear in the position x L. The last term of 1b escribes the influence of a stochastic force, characterize by the white noise W j /t an a matrix b with elements b ij. Here, W j /t is a Gaussian process having a vanishing mean an uncorrelate values to ifferent times, W i /t0 an, W i /t(t) W j /t(t) ij (tt), ij enotes the Kronecker elta, an tt is the elta function. Instea of consiering the equations 1a an 1b for the stochastic transport of particles an their changing properties, the equivalent Fokker Planck equation can be consiere for the probability ensity to fin given values of particle properties at given locations an times. The Lagrangian joint mass ensity function will be enote by F L. This function is similar to the corresponing probability ensity function, but normalize to the mean concentration cx,t of consiere particles which are emitte, e.g., by a source, Z F L Z,x,tcx,t. The transport equation of F L can be erive by ifferent methos 17 an is given in corresponence with the stochastic ifferential equations 1a an 1b by the equation F L t x I Z I F L Z i a i G ij Z j Z j E F L Z i Z j B ij F L, 3 where B ij 1/b ik b kj is introuce. This coefficient B ij is given by 7 C0q B 1 0 C 0 q C 0 q C 1 Z 4 E Z 4 E, Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz

3 if Kolmogorov s theory 9 is aopte for a high-reynols number flow an the time scale for the issipation of the potential temperature variance is assume to be of the same orer as that for the issipation of TKE, which means the unknown constants C 0 an C 1 are consiere as being of the same orer which is iscusse below. In4, q I (Z E Z I E )(Z I E Z I E ) is twice the TKE with I1,, 3, an q / is the unknown time scale of issipation of TKE relating q an the mean issipation rate of TKE. By F L, the statistical properties of an ensemble of observe particles are etermine at a fixe point. The corresponing mass ensity function of all flui particles is enote by the Eulerian function F, which is normalize to the average flui ensity, Z FZ,x,tx,t. 5 This mass ensity also has to fulfill the transport equation 3. This relation of F with the unknown coefficients a i an G ij can be use for eriving consistency constraints between these coefficients an Eulerian means an variances of the velocity an potential temperature fiels. The transport equations for the mean values of the win an potential temperature fiels can be erive by replacing F L by F in 3, multiplying this relation with Z i, an integrating over Z. Then, a i is etermine by DZ i E ViL x L a i, 6 where the abbreviation D/ /t /x K Z K E is use an the matrix of secon moments of the couple win an potential temperature fiel is written by u Vu1u1 1 u u 1 u 3 u 1 u u 1 u u u u 3 u u 3 u 1 u 3 u u 3 u 3 u 3, 7 u 1 u u 3 with z k Z E k Z E k for the fluctuations. Accoringly, by multiplication of 3 with F instea of F L with Z i Z j an integration over Z, the transport equations for the secon moments can be erive, which rea as DV ij R ij P ij G ik V kj G jk V ki C 0q C 0q C 1 V 44 i4 j4. 8 The graients of triple correlations are enote by R ij z K z i z j /x K an the prouction is written as P ij z K z i Z j E /x K z K z j Z i E /x K. These equations 6 an 8 will be compare with the corresponing Eulerian buget equation of first an secon orer. Using the Boussinesq approximation an the incompressibility constraint, Z K E /x K 0, the conservation equations of momentum an potential temperature rea as D Z E i i Z E 4 Z E x K x K x K x K i4 p 1 x K Ki g1z 4 E Z 4 E i3, 9 where D //t/x K Z K E, is the kinematic viscosity, is the coefficient of molecular heat transfer, is the thermal expansion coefficient, p is the pressure, an g is the acceleration ue to gravity. Consequently, a i is etermine by the average right-han sie of 9, a i Z i E Z 4 E x K x K x K x K i4 p 1 x K Ki g i3. 10 By moelling the issipation accoring to Kolmogorov s theory 9 an supposing a return-to isotropy pressure reistribution accoring to Rotta, 10 the transport equations of secon orer can be erive from the conservation equations. These equations rea 1 as ij DV ij R ij P ij k 1 4 ik k 1k 3 i4 k4 g i3 k4 V kj k 1 4 jk k 1k 3 j4 k4 g j3 k4 V ki k q Z E L x K Li Kj Lj Ki q k 1 3 ij k 1/3 q k 3 k 4 k 1 V 44 i4 j4, 11 where the closure parameters k 1, k, k 3, an k 4 are introuce. The parameters k 1 k an k 3 arise from the return-toisotropy theory of Rotta. Because of its simplicity, this approximation is well suite for the illustration of the approach. Whereas the Kolmogorov approximation z I /x K z J /x K 1 3 IJ for the issipation of TKE is wiely accepte, some remarks are neee to the formally corresponing assumption 1, for z 4 /x K z 4 /x K k 4 q V 44 for the issipation of the potential temperature variance. In investigations of ecaying gri turbulence it was shown that the closure parameter k 4 introuce in this way cannot be consiere as a universal constant. In epenence on initial conitions, a range of 0.6k was foun for this quantity, an variations were observe over the length of the win tunnel. 3 5 When a time scale V 44 / is introuce analogous to q /, this closure supposes the proportionality of both time scales, /k 4. As state by Pope, 17 this assume pro- Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz 705

4 portionality can be expecte to be less unrealistic for shear flows in which potential temperature an win fiels share a common history an common bounary conitions. This argument supports the parametrization 4 of the coefficient B which requires the assumption of time scales of the same orer for the issipation of TKE an the potential temperature variance, i.e. C 0 an C 1 of orer unity, but this proportionality cannot be expecte generally. 3 We see by comparing the transport equations 8 an 11 for the variances V ij to estimate the unknown coefficients G ij, that in this way only conclusions to the symmetric component [(GV) ij (GV) ji ]of(gv) ij can be rawn. This comparison reveals that the equations 11 are consistent with the buget equations 8 for the variances, if 1 G ij k 1 4 ij k 1k 3 i4 j4 g i3 j4 k q 1 Aim q 4 k 1 3 ZE L x K L i Km L j Km C 0 im k 1/3C 0 q k 3 k 4 k 1 C 1 V 44 4 i4 m4 V 1 mj, 1 where A is any antisymmetric matrix. The simplest choice of G ij is obtaine by setting A im 0, k 0 Sec. V, C 0 (k 1 )/3, an C 1 k 3 k 4 k 1. This brings for the matrix G, k G 1 0 k k 1 4g k 3 k 1, which means G becomes inepenent of the variances V. The nonuniqueness of the estimation of the coefficient matrix G by the appearance of the unknown antisymmetric matrix A is well known for neutral flows. 1, This problem will be iscusse in the next section with respect to nonlinear equations. Instea, let us come back to the justification of the escription of the potential temperature by a stochastic ifferential equation at the beginning of this section. When C 1 0 is set such that no stochastic forcing appears in the Lagrangian equation 1b for the potential temperature, we fin from 1 nonvanishing contributions for the coefficients G 4K with K1,, 3 cause by the last term on the righthan sie of 1. This means that the change of potential temperature is influence by velocity fluctuations in general an consequently also by stochastic influences. Such stochastic forces explicitly appear by setting C 1 k 3 k 4 k 1, but the avantage of this choice is the possibility to escribe G ij by the simple expression 13, which is inepenent of the variances V ij. Then, G ij epens only over the time scale on the state of the flow. The estimation of is consiere in the fifth section, where its relations with win shear an temperature stratification are investigate. III. NONLINEAR MARKOVIAN EQUATIONS Lagrangian equations can be foun that are consistent with the AHE up to secon orer, as shown in the previous section. These equations are linear in the particle state, but may be nonlinear in the particle position. It remains open uner which conitions nonlinear velocity an potential temperature fluctuations have to be consiere in the Lagrangian equations an how the consistency between the Lagrangian equations an the AHE can be guarantee with respect to higher moments. This is investigate now by assuming a Markovian process x L (t),z L (t) as above. Through these equations, the relation emerges more istinct between the above consiere linear equations an more complicate non-markovian equations, which are erive in the next section. Aitionally, the nonuniqueness problem appears here for non-neutral flows in a more general formulation as before. The particle motion an the potential temperature are escribe by the nonlinear Ito stochastic ifferential equation, 19,0 t x L I tz I L t, 14a t Z L i t i Z L,x L,tb ij Z L,x L,t Wj t. t 14b The transport equation for F L that replaces 3 for the nonlinear equation system 14a an, 14b reas as F L t x K Z K F L Z i i Z,x,tF L Z i Z j B ij Z,x,tF L, 15 where again B ij 1/b ik b kj, which is etermine by 4 as above in the linear moel, so that the systematic transport coefficient remains as a unknown function to be estimate. In the previous section this coefficient was assume to be a linear function in ZZ E an the rift parameter a as well as the coefficient G of linear fluctuations ha been estimate by the consistency constraint with the AHE up to secon orer in the approximations of Kolmogorov an Rotta. The 706 Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz

5 hyroynamic relations for higher moments can be inclue into this consistency investigation by consiering an equation for the mass ensity function F, from which all moments can be calculate. The most general transport equation for F was erive from Pope, 17 which has the structure of 15 on the left-han sie with F L F an a right-han sie, which can be written as h i Z,x,t F/Z i. The function h i Z,x,t represents a conitional ensemble average at fixe Z of the graients of the mean pressure an pressure fluctuations an of the molecular stress tensor an gravity acceleration. This structure of the equation for F is simply a consequence of the hyroynamic conservation equations for mass, momentum, an heat. Thus, by aopting the approximation of Kolmogorov, the nonlinear equation 15 correspons with the general equation structure of the mass ensity function F, an the coefficient is a function to be etermine in epenence on the gravity acceleration, the mean pressure graient, an in particular the correlations of pressure fluctuations. Consequently, the moel 15 is consistent with the infinite hierarchy of AHE in the approximation of Kolmogorov, if F appears as a possible solution of 15. This conition was propose by Thomson 4 an provies a relationship between the systematic transport coefficient an the Eulerian mass ensity F. It may be written as i F 1 Z j B ij F i, where a function is introuce that satisfies i Z i F t x K Z K F, 16a 16b an i 0 for Z. By the relations 16a, 16b, an 4, the coefficients an B are etermine an the nonlinear Lagrangian equations are foun to be consistent with the AHE in the Kolmogorov approximation. However, for inhomogeneous flows not much is known about the mass ensity function F, which is require for these estimation of an B. Only partial information is available for real flows in terms of the moments of lower orer of this istribution function. How this information can be applie to construct F as a maximum missing information ensity function was investigate, e.g., by Du et al. 6 But even for given F, the rotation of, rot, remains open accoring to 16b, asthe antisymmetric component of G ik V kj. To get more insight into this fining, let us consier as an example the mass ensity function F in a simple approximation as local Gaussian in the state Z, 7 which means F et V exp 1 Zk Z k E V 1 kl Z 1 Z 1 E, 17 where et V is the eterminant of V. Taking B ij in corresponence with 4 as inepenent of Z, we obtain for the equation 14b, t Z L i i ij ta NL G NL Z j L Z j E H ijk Z j L Z j E Z k L Z k E b ij Wj, 18 t where 16a, an 16b are use, Z K E /x K 0 is applie, an the subscript NL of the first two terms on the right-han sie enotes quantities in this nonlinear approach, in contrast to the corresponing ones in the linear approach. Here, i a NL ij G NL i DZ E 1 ViK x K 1 jl V ik V x K V 1 lj, B ik 1 DVik i Z E k x L V L 1 A ik NL 1 Z E i x L V L k V 1 kj, H ijk Vin x M V 1 nk Mj Vin x M V 1 nj Mk 1 Vnl im V x M V 1 lj V 1 nk, 19a 19b 19c ik are introuce, where A NL is any antisymmetric matrix, is any scalar quantity, an a constant ensity is assume for simplicity. The expression 19c shows that nonlinear terms appear proportional to spatial graients of the variances V ij in epenence on an unknown parameter. The equation 18 is a generalization of the linear equations 1a an 1b. The latter one is euce by averaging the quaratic terms in the nonlinear equation 18, so that the thir term on the right-han sie becomes H ijk V jk. The create linear term is equal to a i i in the linear moel, which means a NL H ijk V jk a i is fulfille inepenent of. Instea ij of consiering the matrix G NL itself, let us consier G ik NL V kj separate into the sum of its symmetric part 1 G NL V ij G NL V ji an antisymmetric part 1 G NL V ij G NL V ji. These two summans follow from 19b: G ik NL V kj G NL jk V ki DVij Z E i x K V Kj Z E j x K V Ki B ij, 0a inepenent of as well as G ik NL V kj G jk NL V ki ij A NL. 0b The symmetric component of G ik NL V kj is given by the same expression as the symmetric component of G ik V kj which is etermine by 8 of the linear moel, up to the graients of thir moments R ij that vanish here. The antisymmetric component of G ik NL V kj cannot be erive within this approach. This fact as well as the unknown reflect the uncertainty of rot for the consiere Gaussian turbulence. This nonu- Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz 707

6 niqueness was consiere by Thomson, 4 Sawfor, 5 an Borgas an Sawfor 8 with respect to neutral flows. The appearance of shows that this problem has not only geometrical aspects. If the thir term on the right-han sie of 18 is interprete as a fluctuating rift it contributes in the average i to a NL, we see that ifferent intensive fluctuations of this rift may occur. However, the rift is not affecte by the istribution parameter in the ensemble average, since i a NL H ijk V jk a i. For homogeneous an stationary turbulence, we fin by means of 18 an 19a an 19c that Z i L (t)/t Z j L (t) 1 ij A NL, where Z i E 0 an Z i L (t) Z j L (t) V ij are applie. This relation reveals that correlations between states an state changes of ifferent components are taken into account by the matrix A NL. As shown by Sawfor, Borgas, an Guest, 8,9 ifferent assumptions to rot prouce significantly ifferent results in the inertial subrange of neutral bounary layer flows. The erivation of conitions to estimate such open istribution parameters was stuie by Borgas an Sawfor consiering two-particle ispersion. 8 They consiere the reuction of open parameters uner the constraint that one-particle statistics must follow from two-particle moels. The parameters can be estimate by this reuction proceure up to one parameter, which remains open analogous to the appearance of here the matrix A NL is omitte by the assume isotropic turbulence. Through the equations presente here, the effect of the choice of these open parameters on the moelling of non-neutral flows can be stuie. The escription of the interaction between the turbulent an the buoyant motion may be change, e.g., by the choice of A 4k NL (V 1 ) kj 0 or A 4k (V 1 ) kj 0 appearing in 19b an in 1, respectively. In this case, the change of the potential temperature is influence by velocity fluctuations. The investigation of this effect can provie a better insight into the effect of the choice of these open parameters. Consequently, taking reference to the Kolmogorov approximation 4 it is foun that the consistency between the Lagrangian escription an the infinite hierarchy of AHE is guarantee, if the relations 16a an 16b are fulfille. Aopting the approach of Du et al. 6 to construct the neee istribution ensity F, the non-uniqueness problem has to be solve, an in particular the total time erivative of F is require to solve 16b. For local Gaussian turbulence the latter problem is reuce accoring to 19a 19c to the estimation of the graients of mean values an variances. With DV ij / in the Rotta appproximation for the pressure reistribution Sec. II, the calculation of the time scale is again neee Sec. V. IV. NONLINEAR NON-MARKOVIAN EQUATIONS Up to now the suitability of ifferent stochastic processes was consiere to present solutions of the AHE, which means to have moments that obey the AHE. But these conitions tolerate a variety of ifferent processes. Another approach to the escription of motion an properties of particles consists in the erivation of equations from the microscopic ynamics. This is an important aim of statistical mechanics. It can be achieve, e.g., by the projection operator technique, 30 3 where the ynamics of observables is extracte from the couple ynamics of all particles. A feature of this metho is that the obtaine equations are given as a superposition of systematic terms which have in general a non-markovian character an of a term that shows properties of a stochastic force. The coefficients appearing in these equations are given as ensemble averages of microscopic quantities, which can be calculate using moels for the Liouville operator. However, in most cases the calculation of these averages is very complicate, an fluctuation issipation theorems are more useful, which relate coefficients characterizing the intensity of stochastic forces with those characterizing the systematic motion. Whereas for equilibrium systems a well-establishe theory exists, there are many ifferent attempts to exten this approach to the escription of nonequilibrium processes. The estimation of coefficients that appear in these equations requires assumptions on the nonequilibrium probability ensity function. These approaches are limite to the consieration of isolate systems or parts of isolate systems reaching an equilibrium state, or assumptions on the kin of the nonequilibrium state are use. Instea, an approach is applie here, 33 where an ientity replaces the Liouville equation of statistical mechanics an an initial istribution function plays the role of the nonequilibrium istribution function. Assumption on the initial state are not neee, which may be in a strong nonequilibrium. In this way, a nonlinear stochastic ifferential equation can be erive using formally the approach of nonequilibrium statistical mechanics. It is shown how correlations of stochastic forces lea to memory effects in the systematic terms, which is escribe by a fluctuation issipation theorem. Let us start from the abstract equation of motion i1 4, I1 3 t x L I tz I L t, 1a t Z L i tlz i L t, 1b corresponing with the Liouville equation in statistical mechanics L being the Liouville operator, whereas Eq. 1b appears here as an ientity. Using the projection operator technique, the right-han sie of 1b can be written such that this equation shows similar properties like a stochastic ifferential equation. Instea of this evolution equation for the process Z L (t), the ynamics of the generating function v,tvz L (t) for all polynomials of Z L can be consiere, t v,tlv,t, from which 1b is erive by multiplication with v i an integration over v. Accoring to, the Taylor series of reas v,t exp(lt) 0 v, where 0 vv, t0. The equation 1b can be written with this series as Z i L /t vv i L exp(lt) 0 (v). This expression is the starting point for the applie proceure. Let us consier an ensemble of particles with fixe initial positions x L 0x 0 for all particles an the space of proucts of functions A i Z L (t i ) with 708 Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz

7 0t i, where any element A(t 1,t,...) of this space may be written as A(t 1,t,...)A 1 Z L t 1 A Z L t. The projection operator P is efine on this space by PAt 1,t,... vg 0 1 vat 1,t,... 0 v 0 v, 3 which projects any function A(t 1,t,...) onto the 0. The istribution function of fluctuating initial values Z L 0Z 0 which is often use below is abbreviate by g 0 vvz 0. This function replaces the nonequilibrium istribution function of statistical mechanics. The efinition 3 contains A(t 1,t,...) 0 v, which is an average over the consiere ensemble, which means a conitional expectation value to be taken for x L 0x 0 as all the other averages in this section. The ensemble average A(t 1,t,...) 0 v is etermine by At 1,t,... 0 v v vvz 0 vz L t 1 vz L t A 1 v A v. 4 More frequent than P, the operator Q1P is use, which is characterize by QQQ, Q 0 0, an QA B QB A for any elements A an B of the consiere function space. Applying the usual ientity of the projection operator technique, expltexpqlt 0 t t expltpl expqltt, 5 which can be prove by ifferentiation, Z L i /t vv i L exp(lt) 0 (v) can be written as t Z Lt i v g 1 0 vm i v,0v,t t 0 t v g 1 0 v Mi v,tt t v,t f i t, 6 where M i v,tw w i L exp(qlt) 0 w 0 v an f i (t)w w i QL exp(qlt) 0 w are introuce. The integration of f i (t) 0 vqf i (t) 0 vf i (t) Q 0 v 0 over v leas to f i (t)0, so that this quantity can be interprete as a stochastic force an the equation 6 as a nonlinear stochastic ifferential equation. The expressions for the coefficients appearing in 6 are given in contrast to 14b. Writing the time epenence of the state vector as Z L (t)z L x L (t),t to enable the comparisons consiere below, the coefficient M i v,0 appearing in the first term on the right-han sie of 6 is etermine by v i M v,0 i 0v v i i LZ 0 L 0 v t k x K v v,t t0, 7 an M i v,0 0 for v. The coefficient of the secon term of 6 can be written as M i v,t/t Lf i (t) 0 v. L becomes an anti-hermitian operator, if for arbitrary elements A an B of the consiere function space LA BLB AA B/t0, which means for homogeneous an stationary turbulence. In this case we fin Lf i (t) 0 LQf i (t) 0 f i (t) QL 0 f i (t) f k (0) 0 v/v k, an M i v,t/t can be converte into t M i v,t v k f i t f k 0 0 v. 8 This result represents a fluctuation issipation theorem, because the correlation of stochastic forces is relate with the memory function M i v,t that characterizes the issipation process. Let us consier the consequences of elta an exponential correlate forces. In the first case, the force is suppose to be proportional to a vectorial Wiener process, f i (t)b ik Z L,x L,tW k (t)/t, an the Markov limit of 6 for t0 is given by B 0 BZ 0,x 0,0 t Z Lt i v v,tg 1 0 v M i v,0 v k B ik Wk ik 0 0 b t. 9 t A comparison of this equation with the nonlinear Markov equation 14b with the relations 16a an 16b shows, that for the consiere homogeneous an stationary turbulence Fg 0, i M i an B ik FB 0 ik 0. The equation 7 for M i represents the relation 16b for this case. As state by Linenberg an West Chapter 1..., 3 a epenence of fluctuations b ik Z L,x L,tW k (t)/t from the present an perhaps even the past state Z L is expecte in general, but it is a question to be answere if such a state epenence can be inclue in a stochastic ynamic escription. By the above results 9, only an average contribution of a state epenence of fluctuations can contribute to the consiere systematic motion, since only B 0 ik 0 acts here. The first term on the right-han sie of 9 represents a mean rift. By averaging the equation 9 at t0, we fin that Z L i (t)/t(t 0) v M i (v,0), since the other terms o not contribute. For the consiere homogeneous an stationary turbulence we neglect M i v,0, which is justifie by 7. The quantity Z L is then prouce by the white noise term an issipate by the secon term on the right-han sie of 9. The latter term becomes a linear function in Z L (t), if B 0 ij is suppose to be uncorrelate with 0 an the istribution function ensity of initial states is taken as Gaussian, g 0 vexp 1 1 ) kl v k v l with the ispersion matrix. In this case we have Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz 709

8 v g 0 1 vv,t B 0 ik 1 kl Z L 1 t. v k B 0 ik 0 30 Let us consier now the correlation function following from this nonlinear Markovian equation. Multiplying 9 for M i i v,00 with Z 0 an averaging leas for t0 to t Z L i t Z j 0 vv,t Z i 0 g 1 0 v v k B ik 0 0, 31 since the stochastic force f i (t)b ik W k (t)/t is uncorrelate with Z i 0, which follows from the above consiere properties of f i by v v j f i (t) 0 v f i (t) Z j 0 0. Assuming B 0 to be uncorrelate with 0 v, which means ij B ij 0 0 vb ij 0 0 v, an aopting a Gaussian g 0 vexp ( 1 1 ) kl v k v l as before, this relation 31 becomes t Z L i t Z j 0 B ik 0 1 kl Z 1 L t Z j 0, 3 proviing with ij Z i 0 Z j 0 the usual exponential correlation function for homogeneous an stationary turbulence for t0, Z L i t Z 0 j Z 0 i Z 0 k expb 0 1 t kj. 33 The limit t 0 of the erivative of Z L i (t) Z 0 j is then etermine by lim t 0 t Z L i t Z j 0 B 0 ij. 34 It is important to note that this result is inepenent of the ij applie assumptions of uncorrelate B 0 an 0 v an a Gaussian g 0 v, which can be seen from 31 for the correlation function of the nonlinear process at t0 by applying partial integration. This result 34 is a shortcoming of the Markov theory, because the erivative of Z i L (t) Z j 0 must vanish in the limit t This behavior is foun if processes in the orer of the Kolmogorov microscale are taken into account, which is consiere now. In orer to o this let us consier as a secon example an exponential function for the correlation of stochastic forces an as above a Gaussian initial istribution function g 0 expv /(), where one component of the velocity is consiere for simplicity with as the ispersion parameter. The stochastic force f is riven by white noise, f t f f W t, 35 where f enotes the ispersion f f. The force vanishes in the ensemble average an its correlation is given by assume stationarity by f (t) f (ts) f exps. 35 The correlation time of this colore noise f is 1. In the limit 1 0, the stochastic force becomes again elta correlate as assume above, since f (t) f (ts) f / (s). The equation for the stochastic particle velocity reas with these assumptions for g 0 an f (t) f (ts) as t Z Lt t f t expttzl t f t, 0 36 where 8 is use an M i v,0,x 0 is neglecte in the equilibrium, as iscusse above. An equation for the acceleration is obtaine, if 36 is ifferentiate by time an f/t is substitute by 35, t Z L t Z L t f t Z Lt f W t, t 37 which is again riven by white noise. This equation correspons with Sawfors equation for the flui particle acceleration, 5,34,36 if 1 an f 1, with 1 1/T L an Re* 1/ /T L. Here Re* is a number that is proportional to the Reynols number Re, T L is the Lagrangian integral time scale in the limit Re, an 1 is proportional to the Kolmogorov microscale. 34 This moel explains well the Reynols number effects in moels of the turbulent ispersion. For Re we fin 1 0, so that the forces f become uncorrelate. But instea of 36, the velocity is moelle by Sawfor by t Z Lt 1 T L Z Lt f t, 38 so that 37 has to be solve with another initial conition for the acceleration. Accoringly, we fin the solution Z L (t) in these two approaches by Z L tz L 0 exp 1 t 1 exp t 1 exp 1 t, f 0 exp 1 texp t t W 1 s 0 s s exp 1 tsexp ts. with 36 with In both moels, for asymptotically large times the velocity autocorrelation function is obtaine to be Z L (t)z L (ts) / exp( 1 s 1 exp( s/ 1, which arises from the thir term on the right-han sie of 39. However, with Z L 0 the moel 36 provies the corresponing expression for the ecay of the initial correlation, which means Z L 0Z L (t)/ exp( 1 t 1 exp( t/ 1 in contrast to 38, which leas to Z L 0Z L (t)/exp( 1 t. This consistent escription of the ecay of correlations becomes important, if other processes have to be resolve over times of orer 1 i.e., of the orer of the Kolmogorov microscale. This may occur in cases where chemical transformations have to be consiere near sources. By aopting the equation 36, we fin for the correlation of initial values for small times 710 Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz

9 Z L 0Z L (t)/1 1 / t, such that the erivative of this function vanishes in the limit t 0 in contrast to 34. V. THE PRODUCTION DISSIPATION RELATION FOR THE TKE As state above, the Lagrangian equations can be chosen, so that the AHE are guarantee up to secon orer for the moments of the stochastic processes, or for the whole infinite hierarchy of equations, respectively. But in these equations an unknown time scale may appear arising, e.g., from the applie approximations, as iscusse in Sec. II. The estimation of this time scale for inhomogeneous turbulence is an important problem an neee, for instance, for the solution of the equation 1a an 1b with the relations 4, 10, an 13 for the coefficients an for the solution of the secon-orer equations This time scale etermines the ij ratio of the total prouction P tot P ij g i3 V 3j g j3 V 3i in 11 to the issipation given by the terms proportional to 1 ij on the right-han sie. For 0, this input P tot is immeiately issipate an the terms on the left-han sie of 11 vanish, which means the fluxes are in balance with the graients of the mean fiels. For, the issipation terms can be neglecte an we have DV ij / R ij ij P tot, which means the variances V ij change as long as the spatial transport is in balance with the input P tot. Consequently, let us ij consier the relation between the prouction issipation ratio for TKE an, in orer to calculate this time scale by assumptions on that scalar prouction issipation ratio. This ratio expresses uner stationary conitions the amount of spatial transport of TKE, which plays, e.g., an essential role uner convective conitions. 38,39 Hence, the eviation of the prouction issipation ratio for TKE from unity cause by these transports has to be assesse, which is simpler than fining irectly for inhomogeneous conitions in corresponence with the graients of the mean win an potential temperature fiels. This prouction issipation relation will be erive now for a vertical stratifie flow for simplicity. Let us suppose a mean horizontal win U into the x 1 irection an a mean potential temperature, which epen only on the vertical coorinate x 3. The mean vertical win W is assume as constant, so that Z E [U(x 3 ),0,W,(x 3 )]. The secon-orer equations 11 provie for the TKE buget, Dq RKK P, 40 where PP KK /gv 34 is the prouction of TKE an the mean issipation rate of kinetic energy is given by q /, where the time scale is assume as time inepenent. With this expression for P, the prouction issipation ratio pp/ is then p q TVˆ 13 Vˆ 34, 41 where the normalize time scale T U/x 3 is introuce. The variances Vˆ 34 an Vˆ 13 are elements of the matrix Vˆ, which is introuce by u 1 u 1 u 1 u u 1 u 3 gu 1 u Vˆ u 1 u u u u 3 gu u 3 u 1 u 3 u u 3 u 3 gu 3 gu 1 gu gu 3 g, 4 where all elements have the imension of the TKE. The relation 41 requires the calculation of Vˆ 34 an Vˆ 13, which can be obtaine by the secon-orer equation system 11. In orer to o this the consieration of moifie graients Rˆ of thir moments is avantageous, which have the same imension as Vˆ an are given by R 11 R 1 R 13 gr 14 R Rˆ 1 R R 3 gr 4 R 31 R 3 R 33 gr gr 41 gr 4 gr 43 g R 43 If the time epenence is observe in tt/, which is normalize to the time-inepenent, the operator D/( )[/t/x 3 W] is applie an the graient Richarson number Rig /x 3 /(U/x 3 ) is use, we obtain from the secon-orer equations 11 the system of couple equations, D KK Vˆ Rˆ k1/ 1 0 T 0 0 Vˆ 14 Rˆ 14 Ri T 13 k 3 / T Vˆ 34 Rˆ k 3 / Ri T 1 0 Vˆ 33 Rˆ k 1 / 0 k 1 /6Vˆ , Rˆ RiT 0 k 4 0 Vˆ 44 Rˆ T q Vˆ 44 q 44a Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz 711

10 which inclues the variances Vˆ 34 an Vˆ 13. The other components of the matrix Vˆ obey the equations DVˆ 11 Rˆ 11 k 1 Vˆ 11 T Vˆ 13 k 1 q, 6 44b where q an Vˆ 13 are given as solutions of 44a, Vˆ is etermine by Vˆ q Vˆ 33 Vˆ 11, an the remaining components of Vˆ satisfy D Vˆ 1 Vˆ 3 4Rˆ 1 3 Vˆ Rˆ 4 k 1 / T 0 0 k 1 / 1 Vˆ 3 Vˆ 4. 0 Ri T k 3 / Vˆ 1 44c For the consiere vertical stratifie flow, the horizontal components with i1, an j1,,3,4 on the left-han sies of the secon-orer equation system 44a 44c will be neglecte. The term Dq /Rˆ KK in 44a can be replace by q (p1) accoring to 40, an DVˆ 33 /Rˆ 33 can be replace in this equation system also by this expression, i.e. DVˆ 33 /Rˆ 33 q (p1), because Vˆ 11 an Vˆ are assume to be not contributing to the buget of q in this approximation. The equations system 44a can be solve simply, if the time erivatives an graients of triple correlations are consiere as inhomogeneities. In this way Vˆ 34 an Vˆ 13 can be calculate, where a quantity p appears, with (p 1) q (DVˆ 44 /Rˆ 44 )/k 4 DVˆ 34 /Rˆ 34. The meaning of p becomes clear, if the bugets for the variance V 44 of heat fluctuations an the vertical heat flux V 34 are consiere in analogy to the buget equation 40. The secon-orer equations 11 give for these quantities DV 44 DV 34 R 44 P, R 34 P w w, 45a 45b where P V 34 /x 3 an P w (V 33 /x 3 V 34 U/x 3 )/ are the prouction terms an k 4 V 44 / as well as w k 3 V 34 /(4) are the issipation terms. The parameter p is etermine by these expressions by the relation p 1 k 4 4 gv44 P 1 k 3 4 gv34 P w w Hence, eviations of p from unity are cause by eviations of P / an P w / w from unity, so that p escribes the prouction issipation ratio of heat in analogy to p escribing this ratio for the TKE. Inserting the variances Vˆ 34 an Vˆ 13 calculate by 44a in epenence on p, p, T, an Ri into 41 leas then to the relation p 4 k 3 p 1 T T 0 Ri T Ri 0 T 0 18 T 0 k 1 p1 Pr stp 1Pr ut Pr 0 14 T 0 Ri c Pr 0 Ri 0 k 1 Pr 0 p1, 47 for the prouction issipation ratio p. Pr st k 3 /k 1 k 1 k 3 Ri c /Ri 0 4RiT /k 1 k 3 4RiT is the turbulent Prantl number Pr t Ri TVˆ 13 /Vˆ 34 for a prouction issipation ratio p 1 for heat, Pr ut (k 1 k 3 )/[(k 1 )/6(p1)]/k 1 k 3 4RiT is a contribution relate with an unbalance ratio p of heat, an Pr 0 k 3 /k 1 follows from Pr st for a neutral stratification, which means Pr 0 Pr st Ri0. Aitionally, in the erive relation 47 the parameters T 0 3k 1 /[4(k 1 )], Ri 0 3k 1 k 3 k 4 / (4T 0 [k 4 (k 1 4)3k 1 ]), an Ri c (k 3 k 4 )/k 4 Ri 0 /Pr 0 appear. This relation 47 quantifies the expectation that the prouction P of TKE normalize to the issipation is etermine by the quaratic win shear T ( U/x 3 ) an the unstable temperature stratification Ri T g /x 3, which appear as factors on the right-han sie of 47. For balance prouction issipation ratios pp 1, this relation is given by the simple expression p1 T T 0 Pr st Ri Pr 0 Ri This relation shows that T 0 is the value of T for a neutral stratification Ri0 for pp 1. The relation 48 is characterize by the parameters Pr 0,Ri 0 as well as Ri c, which limits the applicability of 48 as an equation for T by the conition RiRi c. These numbers can be interprete as flow numbers. 37 As state by Derbyshire, 40 in a free stratifie shear layer of uniform shear an stratification at high Reynols an Peclet numbers, turbulence is expecte to grow or ecay on a time scale of orer (U/x 3 ) 1, epening on the graient Richarson number Ri. The crossover point at which turbulence neither grows nor ecays efines a turbulent critical Richarson number. As shown below, the normalize time scale T U/x 3 an for a finite win shear also becomes infinite for Ri Ri c. Hence, for Ri Ri c the ecay of turbulence by issipation becomes weaker because of an it iminishes at RiRi c, so that Ri c efines the critical graient Richarson number. Here Pr 0 Pr st Ri0 was foun as turbulent Prantl number Pr st for a neutral stratification an Ri 0 is a characteristic graient Richarson number, as can be seen from 48. The first term on the right-han sie of 48, Pr st T /Pr 0 T 0, is positive for a positive Pr st, such that the conition (g /x 3 ) T 0 Ri 0 arises from 48. This relation represents uner unstable stratification a constraint for the issipation that has to be large enough the time scale has to be small enough such that convective processes spatial transports of TKE are exclue an the relation 48 can be fulfille. Consequently, Ri 0 characterizes the onset of convective processes uner unstable stratification, in corresponence to Ri c, which 71 Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz

11 TABLE I. The secon-orer closure parameters k 1, k, k 3, an k 4 estimate by ifferent authors an the flow numbers Pr 0,Ri c, an Ri 0 as well as C 0 an C 1, calculate by their relations with the closure parameters. k 1 k k 3 k 4 Pr 0 Ri c Ri 0 C 0 C 1 Wichmann an Schaller Mellor an Yamaa Zeman an Lumley Anré et al Wyngar et al Yamaa characterizes the onset of turbulence uner stable stratification. 37 These flow numbers Ri c,pr 0, an Ri 0 can be estimate by means of their relations with the secon-orer closure parameters k 1, k 3, an k 4. It is a common feature of the estimations of k 1, k 3, an k 4 that the fit of these parameters is relate with the consieration of properties of neutral stratifie flows, or the closure parameters are fitte to the characteristics of ifferent flows. 1 These estimations provie very ifferent ata for k 1, k 3, an k 4, as given in Table I. Aitionally, the values of the flow numbers Pr 0,Ri c, an Ri 0, an parameters C 0 (k 1 )/3 an C 1 k 3 k 4 k 1 Sec. II are shown in epenence on k 1, k 3, an k 4. The obtaine ata for the flow numbers Pr 0,Ri c, an Ri 0, as well as the avantage of consiering these numbers, are iscusse in relation to the solution of the secon-orer equations The values for C 0 in Table I are somewhat smaller than those estimate by Du et al. 45 for this quantity. By comparisons with water channel ispersion an win tunnel measurements they erive C an iscusse the wie range of estimates of C 0 to be foun in the literature e.g., the relation to the much higher values erive for this quantity by Sawfor 5,34 an Pope 6. With the exception of the value obtaine from the ata of Zeman an Lumley, C 0 is foun here in a range 1C 0.33, which agrees qualitatively with the finings of Du et al. Measurements of C 1 seem to be unavailable. It is remarkable that negative values of C 1 are obtaine with the ata of Wichmann an Schaller 1 an Wyngaar et al. 43 This means, e.g., that in these cases more complicate coefficients G ij of the linear Lagrangian moel 1b have to be chosen in epenence on the variances, which cannot be escribe by the simple choice 13. The equation 47 can be consiere as a quaratic equation for for given prouction issipation rates p an p an graients U/x 3 an /x 3. The normalize time scale T U/x 3 can be estimate by the equation T 1 A BB 4AC, 49 where the quantities A, B, an C are functions of the graient Richarson number Ri an the prouction issipation ratios for TKE an heat, p an p, respectively. They are foun to be 4T 0 A4 Ri RiRi c p1 Ri 0 k 1 Pr 0 Ri 0 RiRi c Pr 0 Ri 0 Ri c Pr 0, B4 RiT 0 pk 1 Pr 0 RiRi 0 Ri 0 4T 0 50a p1 RiRi Pr 0 Ri c Pr 0 Ri 0 Ri 0 Pr 0 0 p 1 16T 0 Ri1 Pr k 1 Pr 0, 0 pp 1 Ck 1 Pr 0 T 0 4 k 1 Pr 0. 50b 50c These quantities epen only on the flow numbers Pr 0,Ri c, an Ri 0, since the closure parameter k 1 can be expresse by these numbers, k 1 Pr 0 3Ri c 4Ri 0 3Ri 0 /Pr 0 ]/ (Pr 0 Ri 0 ). Here RiRi c appears as a conition for the solution of 49 for pp 1. When the win shear vanishes, we fin from 47 a relation between /x 3 an p an p, g x 3 Ri 0 T p4/k 3 p T 0 /k 1 Ri c Pr 0 Ri 0 /Pr 0 p1. 51 We note that real solutions of only exist uner unstable stratification. The values for the prouction issipation ratios for TKE an heat cannot be expecte to be near unity, if consierable transports of these quantities occur, as for instance uner convective conitions. 37 Since is only well efine by 47, if the issipation is at least of the same orer as the spatial transport, only ranges 0p,p are consiere in Figs. 1 an, where the Ri epenence of T is shown for Ri c 0.3, Pr 0 1, an Ri Here T becomes infinite at a critical number Ri that epens upon p an p an is equal to Ri c 0.3 for pp 1. For increasing instability of stratification Ri becomes smaller, T U/x 3 also becomes smaller. By assuming the win shear U/x 3 as only a little influence by the changing stratification, we fin that the time scale q / of the TKE issipation becomes Phys. Fluis, Vol. 9, No. 3, March 1997 Stefan Heinz 713