A hyperbolic equation for turbulent diffusion

Transcription

1 Nonlinearity 13 (2) Printe in the UK PII: S ()1271-X A hyperbolic equation for turbulent iffusion Sanip Ghosal an Joseph B Keller Center for Turbulence Research, Stanfor University, Stanfor, CA , USA Department of Mathematics an Mechanical Engineering, Stanfor University, Stanfor, CA , USA Receive 21 March 2 Recommene by P Cvitanovic Abstract. A hyperbolic equation, analogous to the telegrapher s equation in one imension, is introuce to escribe turbulent iffusion of a passive aitive in a turbulent flow. The preictions of this equation, an those of the usual avection iffusion equation, are compare with ata on smoke plumes in the atmosphere an on heat flow in a win tunnel. The preictions of the hyperbolic equation fit the ata at all istances from the source, whereas those of the avection iffusion equation fit only at large istances. The hyperbolic equation is erive from an integroifferential equation for the mean concentration which allows it to vary rapily. If the mean concentration varies sufficiently slowly compare with the correlation time of the turbulence, the hyperbolic equation reuces to the avection iffusion equation. However, if the mean concentration varies very rapily, the hyperbolic equation shoul be replace by the integroifferential equation. AMS classification scheme number: Introuction Turbulent iffusion of a passive aitive in an incompressible flui is usually escribe by the parabolic avection iffusion equation, propose by Boussinesq (1877) an Taylor (1915), c [(K + D) c] = g. (1.1) t Here c(x,t) is the concentration, K is the molecular iffusion coefficient, D(x,t) is the turbulent iffusion coefficient, g is the strength of the source of aitive, = t t + u is the avective time erivative an u(x,t)is the mean velocity of the flui. Batchelor an Townsen (1956), p 36, suggest that a escription of the iffusion by some kin of integral equation is more to be expecte. We erive such an equation, an from it we erive the following hyperbolic equation for c: T 2 c t + c ( ) D [(K + D) c] T 2 t t c + D T t c +T k [D u k c] = g + T g t. (1.2) Here T is the correlation time of the turbulent velocity as efine by (4.15). When T is negligible, equation (1.2) reuces to (1.1). We obtain (1.2) by first eriving an Now at: Northwestern University, Mechanical Engineering, 2145 Sherian Roa, Evanston, IL 628, USA // $3. 2 IOP Publishing Lt an LMS Publishing Lt 1855

2 1856 S Ghosal anjbkeller integroifferential equation for c, following Keller (1964, 1966, 1971) an Saffman (1969). From this equation, we obtain (1.2) when c oes not vary appreciably uring the correlation time T. When u, D an T are constant, or slowly varying, equation (1.2) simplifies to T 2 c t + c g (K + D) c = g + T 2 t t. (1.3) This is a three-imensional form of the telegrapher s equation. The one-imensional form of (1.3), with u = an g =, was erive by Golstein (195) for a correlate ranom walk along the x-axis. We shall compare the preictions of (1.1) with experimental ata an with the preictions of (1.2). The experimental ata we use are those of Högstrom (1964) on smoke plumes in the atmosphere an those of Stapountzis et al (1986) on heat iffusion in a win tunnel. We shall fin solutions of (1.1) an (1.2) corresponing to these experiments, an evaluate them using appropriate values of D an T. We shall see that the solutions of (1.1) fit the ata only at points far from the source, whereas the solutions of (1.2) fit the ata fairly well over the whole range of istances. We conclue that (1.2) provies a better escription of turbulent iffusion than oes (1.1) when c varies rapily in space or time. If c varies extremely rapily, the integroifferential equation (4.11) in section 4 shoul be even more accurate than (1.2). There has been a great eal of work on turbulent iffusion (see the treatise of Monin an Yaglom (1975)). One very useful proceure is that of escribing the velocity of a iffusing particle by the Langevin equation. This equation has been use, in particular, by Durbin (198), Sawfor an Hunt (1986) an Thompson (199). These authors have use it an extensions of it, to etermine the variance an two-point correlation of c. Since the Langevin equation inclues a Gaussian process for the forcing function, it leas to arbitrarily large particle velocities. Therefore, the iffusion equation (1.1), which follows from the Langevin equation, leas to iffusion with infinite spee. In the erivation of (1.2), infinite spee is avoie by using the first two moments of the velocity, an not specifying that the force or velocity is Gaussian. Anan an Pope (1985) have shown that for certain choices of the ranom functions in the Langevin equation, its solution provies a goo fit to the plume with in the win tunnel ata of Stapountzis et al (1986). The fit is as goo as that provie by (1.2) or even better. This result epens upon the assumptions mae about the Lagrangian velocity functions in the Langevin equation. The erivation of (1.2) oes not involve such assumptions, but just uses correlation functions of the Eulerian velocity. Another experimental comparison of (1.1) an (1.3) coul be mae by etermining experimentally the ispersion equation for time harmonic plane waves. For the wave c = e i(kx ωt), equation (1.3) with g = an constant coefficients yiels k 2 = (D + K) 1 (T ω 2 +iω). (1.4) Equation (1.1) yiels (1.4) with T =. For ω of the orer of T 1 the ifference between the two preictions shoul be observable. The metho of erivation of the integro-ifferential equation for c in section 4 has been use to erive equations for u an for the two-point correlation function of u (Keller 1966). The resulting equations are similar to those obtaine by the irect interaction approximation of Kraichnan (1959). This erivation provies a ifferent way of consiering closure approximations. In section 2, solutions of (1.1) an (1.2) are obtaine which correspon to the experimental situations. In section 3, these solutions are compare with the ata from the two sets of experiments. The erivations of (1.2) an of more general equations are presente in section 4.

3 A hyperbolic equation for turbulent iffusion Diffusion from point an line sources Let us consier the steay concentrations prouce by a point source of strength g at the origin an by a uniform line source of strength g per unit length along the z-axis. In both cases we assume that the mean flow velocity is along the x-axis an is constant. Thus we write u(x,t)= U ˆx so that = U t x. In the point source case c = c(x, r) where r = (y 2 + z 2 ) 1/2, while in the line source case c = c(x,y). We also let D(x) an T(x)epen upon x, an we neglect K compare with D. When we use these assumptions in (1.2) for the line source, an ivie by U, we obtain [ TU 2 x + x U 1 x D x ( U 1 D + TD ) y 2 T xd x + DT x 2 ] c(x,y) = U 1 g δ(x)δ(y) + g Tδ (x)δ(y). (2.1) Here the prime enotes an x erivative. In the point source case we must replace y 2 c(x,y) by r 1 r r r c(x, r) an δ(y) by δ(r)/πr in (2.1). In the line source case, the integrate concentration at x, c (x), is efine by c (x) = The secon moment, c 2 (x), is efine by c 2 (x) = c(x,y)y. (2.2) c(x,y)y 2 y. (2.3) Upon integrating (2.1) with respect to y, we obtain the following orinary ifferential equation for c (x): [ TU 2 x + x U 1 x D x T x D x + DT x 2 ] c (x) = U 1 g δ(x) + g Tδ (x). (2.4) Then multiplying (2.1) by y 2, integrating over y an using integration by parts, we obtain an orinary ifferential equation for c 2 (x): [ TU 2 x + x U 1 x D x T x D x + DT x 2 ] c2 (x) = 2(U 1 D + TD )c (x). (2.5) For the point source we efine c (x) an c 2 (x) by c (x) = 2π 2c 2 (x) = 2π c(x, r)r r (2.6) r 2 c(x, r)r r. (2.7) With these efinitions, we fin from (2.1), with the replacements inicate just below it, that c an c 2 again satisfy (2.4) an (2.5). By setting T = in (2.4) an (2.5), we obtain the equations for c an c 2 corresponing to the avection iffusion equation (1.1). We shall solve (2.4) an (2.5) for c (x) an c 2 (x), an compare the solutions with corresponing observe values. First we write (2.4) an (2.5) in the forms αc + βc = g U δ(x) + g Tδ (x) (2.8) ac 2 + βc 2 = γc (x). (2.9)

4 1858 S Ghosal anjbkeller Here α(x), β(x) an γ(x)are efine by α = UT D/U TD + DT β = 1 D /U TD (2.1) γ = 2(D/U + TD ). We also introuce the integrating factor [ x P(x) = exp β(s) α(s) s ]. (2.11) We seek solutions in which c (x) an c 2 (x) ten to zero as x, an which grow less rapily than exponential as x +. The nature of the solutions epens upon the sign of α(x)/β(x) for large positive x an for large negative x. When α/β < for all x larger than some x, [ ( ) T 1 ] x P(ξ)ξ x < x Pα PαU x= [ ( ) T 1 ] P(ξ)ξ + T() (2.12) x x Pα PαU x= α() x γ(s)c (s) c 2 (x) = P(ξ) s ξ. (2.13) ξ α(s)p(s) c (x) = g When α/β > for all x larger than some x, x< [ ( ) c (x) = g T 1 ] x P(ξ)ξ + T() (2.14) x x Pα PαU x= α() x< c 2 (x) = x ξ γ(s)c (s) P(ξ) α(s)p(s) s ξ x. (2.15) For the atmospheric iffusion experiments to be consiere in section 3, we assume that the turbulence is uniform, so we take D an T to be constants. Then (2.1) shows that α = UT D/U, β = 1 an γ = 2D/U. With these values, the solutions (2.12) (2.15) simplify to the following. When α<, i.e. T<D/U 2, c (x) = c 2 (x) = 2g D U 2 g D U 2 α e x/α x< g U x [ D U α + D Uα x [x + DU ] α ] e x/a x< x. (2.16) (2.17)

5 A hyperbolic equation for turbulent iffusion 1859 When α>, i.e. T>D/U 2, c (x) = g x< [ U 1+ D ] x. c 2 (x) = 2g D U 2 αu e x/α x< [ ( ) D (1 x + U α e x/α ) D ] αu xe x/α x. (2.18) (2.19) For the avection iffusion equation (1.1), T = so (2.16) an (2.17) apply with α = D/U. For the win tunnel experiments to be consiere in section 4, the turbulence is not constant so we shall nee the solutions (2.12) (2.15). 3. Comparison with experiments 3.1. Atmospheric measurements Many measurements of the sprea of smoke plumes ownwin of a point source have been mae. We will use those of Högstrom (1964), who introuce innovations which permitte him to obtain photographic traces of the plume for a much longer time than ha been possible before. As a result, he recore both the initial linear growth of the plume with, an the slower parabolic growth far ownstream. The experiments were performe near the towns of Stusvik an Ågesta, Sween. Since Stusvik is on the coast, the ata from there are complicate by the effect of the topographical iscontinuity on the turbulence. Therefore, we use only the Ågesta ata. The ata for the vertical with σ(x)of the plume, as a function of istance x ownstream from the source, are taken from table 3, p 22 of Högstrom s paper. Results of the three separate experiments liste in table 3 are shown as symbols in figure 1 after they have been mae comparable by multiplication by a factor to remove the effect of stratification in the manner escribe by Högstrom. The mean win profile was measure by means of six anemometers mounte on a mast. Fitting the ata to well known empirical formulae for atmospheric bounary layers yiele the following values for the mean an rms turbulent velocities: U = 5.92 m s 1 an u 2 /U =.12. The theoretical value of σ(x)is efine by the equation σ 2 (x) = c 2(x) c (x). (3.1) The values of c an c 2 are given by (2.16) an (2.17) for the avection iffusion equation (1.1), an by (2.18) an (2.19) for the hyperbolic equation (1.2). To compare the ata with the theory, we first solve (4.7) for y, with u =U, to obtain y(s, t, x) = x + U(s t). We use this in (4.13) for D ij an assume that the correlation function has the form C ij [x,t,x + U(s t),s] = u 2 δ ij f ( ) s t. Then (4.13) shows that D ij = Dδ ij where D is given by ( s t D = u 2 T f T T ) s T. (3.2)

6 186 S Ghosal anjbkeller Figure 1. With σ(x)of a smoke plume in the atmosphere as a function of istance x ownstream from the source, both measure in metres. The symbols represent the three experiments liste in table 3 on p 22 of Högstrom (1964) after correcting for the effect of stratification. The full curve is compute from (3.3), which is obtaine from the solution of the avection iffusion equation (1.1). The broken curve is compute from (3.1) using the solutions (2.18) an (2.19) of the hyperbolic equation (1.2). We have set the upper limit t/t = because the turbulence was initiate far upstream, so it is fully evelope. The integral in (3.2) is a numerical constant which we assume to have the value one, so that D = u 2 T. This will be the case, for example, if f(ξ) = e ξ. Then (4.15) shows that T ij = δ ij T provie that f(ξ)ξξ = 1, which is also the case for f(ξ)= e ξ. To etermine D we use (3.1) with c an c 2 given by (2.16) an (2.17) with α = D/U, to obtain σ 2 (x) = c 2(x) c (x) = 2D U ( x + 2D U ). (3.3) A plot of the measure values of σ 2 (x) versus x shows that the last few ata points for large x lie in the asymptotic regime where the slope σ 2 /x is constant. Measurement from the graph gives σ 2 /x = m. Equating this to the theoretical slope 2D/U given by (3.3) yiels D = 1.612U/2 = 4.77 m 2 s 1. We use this value in (3.3) to compute σ 2 (x) for the avection iffusion equation (1.1), an the result is shown by the full curve in figure 1. To evaluate σ 2 (x) for the hyperbolic equation (1.2), we nee the value of T. To get it we use the relation D = u 2 T given by (3.2), with the measure values u 2 /U =.12, U = 5.95 m s 1, an the value D = 4.77 m 2 s 1 etermine above. This yiels T = 9.45 s. Since D/U 2 =.14 s, we see that T D/U 2. The erivative terms in (2.1) are small, so α UT D/U = U(T D/U 2 ) an β 1, an therefore α/β >. Consequently, equations (2.18) an (2.19) apply, an we use them in (3.1) to get σ 2 (x). The result is shown by the broken curve in figure 1. The figure shows that both the avection iffusion equation an the hyperbolic equation give results consistent with the ata at large istances from the source. However, closer to the

7 A hyperbolic equation for turbulent iffusion 1861 source, the hyperbolic equation continues to give fairly accurate results, whereas the preiction of the avection iffusion equation iffers significantly from the experimental ata. This is ue to the ifferent behaviour preicte by the avection iffusion an the hyperbolic equations as x. The telegraph equation with α> gives the correct behaviour σ(x)proportional to x for x small, whereas the avection iffusion equation gives σ(x) 2D/U. Far ownstream, where x, both equations preict σ(x) to be proportional to x in agreement with the ata. The failure of the iffusion approximation close to the source is to be expecte. It is significant that finite correlation time effects, emboie in the hyperbolic equation, lea to much improve agreement with the ata Win tunnel experiments In the experiments of Stapountzis et al (1986), turbulence is generate by passing air through a gri. The passive scalar is the temperature variation prouce by a fine electrically heate wire stretche across the test section of the tunnel. The wire is so thin that its wake oes not alter the turbulence significantly. Molecular iffusion is small; its effect can be accounte for approximately by first measuring the wake in the absence of turbulence, an using this measurement to correct the ata with turbulence. The mean velocity an turbulent intensities are substantially constant across the tunnel except in the bounary layers. The experiment is set up so that the turbulent thermal wake never reaches these layers. Therefore, the theory of turbulent iffusion from a line source in a uniform mean flow can be expecte to hol. The rigorously controlle laboratory environment permits the theoretical iealizations to be realize more accurately than is possible in the atmospheric iffusion experiments. The following properties of the win tunnel turbulence were measure: ( x u 2 (x) =.15 M + x ) 1.43 U 2 (3.4) M λ(x) ( x M =.73 M + x ).47. (3.5) M Here U = 4.35 m s 1 is the mean spee of the air in the win tunnel, M =.254 m (1 inch) is the wire spacing in the gri generating the turbulence, x = 19.3M is the istance from the source, at x =, to the gri, an λ(x) is the integral scale, measure using the Taylor hypothesis of frozen turbulence. The turbulence exhibits a slight anisotropy so that the turbulent intensities an scales in the streamwise an cross-stream irections are not precisely equal. Here we have taken the values corresponing to the cross stream irection for the following reason: in the anisotropic case D is a tensor with principal irections along the coorinate axes an in this case, if the erivations of (2.4) an (2.5) are carrie through, the component D yy woul take the place of D. To etermine D(x) an T(x) we procee as before to obtain D(x) = u 2 (x) T(x). We expect T(x)to be expressible in terms of λ(x) an u 2 (x), but we have not foun any measurements relating them. Therefore, we make the plausible assumption that T(x)is given by T(x)= Cλ(x)/ u 2 (x). (3.6) Here C is a imensionless parameter which is chosen to make the theory fit the ata at the point furthest ownstream. The proper value of C was.7 for the avection iffusion equation an 1.95 for the hyperbolic equation. By using (3.5) in (3.6), an in D = u 2 T, we obtain the

8 1862 S Ghosal anjbkeller Figure 2. With σ(x) of a thermal plume in a win tunnel as a function of the istance x from the source, both measure in metres. The open circles are base upon the ata of Stapountzis et al (1986). The full curve is obtaine from (3.1) using the solutions (2.12) an (2.13) of the avection iffusion equation (1.1). The broken curve is obtaine from (3.1) using the solutions (2.14) an (2.15) of the hyperbolic equation (1.2). following values for D(x) an T(x): D(x) =.27C(MU)( x M ).25 (3.7) T(x)=.182C ( ) M ( x ) 1.19 U M (3.8) Now we can use (3.7) for D(x), with T =, in (2.12) an (2.13) to compute c (x) an c 2 (x) for the avection iffusion equation. Then we use the results in (3.1) to obtain σ 2 (x). The integrals are evaluate numerically an the result is shown by the full curve in figure 2. Next we use both (3.7) an (3.8) in (2.14) an (2.15) to obtain c an c 2 for the hyperbolic equation (1.2). Then (3.1) gives σ 2 (x), which is shown as the broken curve in figure 2. The value of α(x)/β(x) was checke an foun to be positive. It can be seen that the hyperbolic equation gives a much better fit to the ata points in figure 2 than oes the avection iffusion equation, as was the case for the atmospheric iffusion experiments. The avection iffusion equation preicts a plume with that is too wie close to the source. However, the agreement of the preiction of the hyperbolic equation with the experiment in the region near the source is not perfect. This might be ue to the failure of the hyperbolic equation ue to too rapi variation of c(x, t) close to the source, in which case the full integroifferential equation nees to be use. The iscrepancy might also be ue to experimental errors; the experimenters report that the region very close to the wire may be ominate by effects such as the wake of the heating wire itself or the increase viscosity of air ue to the elevate temperature.

9 A hyperbolic equation for turbulent iffusion Derivation of an equation for the mean concentration. Let c(x,t)be the concentration of a passive aitive at the point x at time t in an incompressible flui in turbulent motion with velocity u(x,t). We suppose that c satisfies the avection iffusion equation [ t + u i i i K ij j ]c = g(x,t). (4.1) Here K ij is the molecular iffusion coefficient tensor, i = / x i, g is the source strength istribution an repeate inices are summe. We assume that u an g are ranom functions, an then c will be ranom also. We enote the average of any function f by f an its fluctuating part by f = f f, which we will also write as f = Pf in terms of a projection operator P. Our goal is to etermine c. We begin by averaging (4.1) an writing = t t + u i i, to obtain [ t ik ij j ] c + u i ic = g. (4.2) Then we subtract (4.2) from (4.1) to obtain the following equation for c : [ ] t ik ij j + Pu i i c = g u i i c. (4.3) We write the solution of (4.3) as [ ] 1 c = t ik ij j + Pu i i (g u j j c ) + c (x, ). (4.4) Next we substitute (4.4) into (4.2) to obtain an equation for c : { [ ] } 1 t ik ij j u i i t kk km m + Pu k k u j j c [ ] 1 = g u k k t ik ij j + Pu i i g u i (x,t) i c (x, ). (4.5) To estimate the terms in this operator we write c /t max(t 1, u L 1 ) c an u k k c u 2 1/2 L 1 c. Here T an L are the time an length scales on which c varies, an we assume that the time an space erivatives o not cancel. The ratio of the secon of these two terms to the first one is u k k c ( ) / c /t u 2 1/2 T min L, 1 u. When c is inepenent of time then T =, an this ratio becomes u 2 1/2 / u. In the atmospheric measurements escribe in section 3.1, u = U an this ratio is.12, as is state there at the en of the secon paragraph. In the win tunnel experiments escribe in section 3.2, u = U an (3.4) shows that this ratio satisfies u 2 1/2 /U [.15(x /M) 1.43] 1/2 = [ ].15(19.3) /2.2. Thus in both cases the term u k k is small compare with /t. When this is the case we shall neglect the u k term. The resulting operator /t kk km m is then the iffusion operator, which can be inverte by using its Green function. For simplicity, we shall assume that the molecular iffusive part k K km m is small compare with the avective part /t, so we shall also neglect the molecular iffusive part. With these assumptions, we just have to invert the first-orer operator /t, i.e. to solve ϕ(x,t)= f(x,t) t ϕ(x, ) =. (4.6)

10 1864 S Ghosal anjbkeller To o so, we introuce the characteristic curve or particle path y(s, t, x) which passes through x at s = t. It is efine by y = u(y,s) y(t, t, x) = x. (4.7) s In terms of y the solution ϕ of (4.6) is given by ( ) 1 t f = f [y(s, t, x), s] s. (4.8) t By using (4.8) for the inverse operator in (4.5) we obtain t (/t i K ij j ) c u i (x,t) x i u j [y(s, t, x), s] y j c[y(s, t, x), s] s t = g u i (x,t) x i g [y(s, t, x), s]s u i (x,t) ic (x, ). (4.9) We recall that i u i =. Therefore, we can rewrite the term in (4.9) involving u i an u j by introucing the correlation function C ij (x,t,y,s)= u i (x,t)u j (y,s). (4.1) When u i is inepenent of g an of c (x, ), for example, when g an c (x, ) vanish, the last two terms in (4.9) vanish. Then we can write (4.9) in the form t (/t i K ij j ) c xi C ij [x,t,y(s, t, x), s] yj c[y(s, t, x), s] s = g. (4.11) This is our main equation for c. It involves the velocity correlation at points along the path of a particle moving with the mean velocity. The initial value of c is etermine by the prescribe initial value of c. We note that c (x,t)is given in terms of c by (4.4), in which the inverse operator can be approximate by that in (4.8). As a first application of (4.11) we treat the case in which the correlation time an correlation length of the velocity fluctuations are small compare with the time scale an length scale on which c varies. Then we can replace yj c [y(s, t, x)s] by xj c(x,t) in the integran in (4.11) an write the resulting equation in the form [ t + u i i i [ Kij + D ij (x,t) ] j ] c(x,t) = g. (4.12) Here D ij (x, t) is efine by the integral D ij (x,t)= t C ij [x,t,y(s, t, x), s] s. (4.13) Thus in this case c satisfies an avection iffusion equation, an the turbulent iffusion coefficient is given by (4.13). This expression for D ij is similar to that erive by Taylor (192), but here y is etermine by the mean velocity rather than by the total velocity, an the upper limit is finite. When K ij = Kδ ij an D ij (x, t) = Dδ ij, equation (4.12) becomes (1.1) in which the angular brackets are omitte. Next we consier somewhat larger correlation times an lengths. We expan yi c[y(s, t, x), s] to first orer in t s, making use of (4.7), to obtain yj c[y(s, t, x), s] = xj c(x,t) (t s)[ u k (x,t) xj xk c(x,t)+ xj t c(x,t) ]+. (4.14)

11 A hyperbolic equation for turbulent iffusion 1865 We use (4.14) in the integran in (4.11) an we get two integrals. The first integral is D ij (x,t) an the secon we call T ij (x,t)d ij (x,t)with no summation: T ij (x,t)d ij (x,t)= t C ij [x,t,y(s, t, x), s](t s)s. (4.15) Thus T ij is the correlation time corresponing to C ij, an (4.11) becomes ( ) t ik ij j c i D ij j c + i T ij D ij t j c = g. (4.16) To eliminate the thir erivatives of c from (4.16), we assume that T ij = T(x,t) is inepenent of i an j. Then we use (4.12), neglecting K ij, to obtain i D ij j c = t c g. We use this in the thir erivative terms in (4.16) to replace them by secon erivatives. After a bit of calculation we obtain the secon-orer equation [ ( ) ] 2 T + t t i(k ij + D ij ) j c [ ( ) ] + ( i T)D ij t Dij j T i j + T k D ij ( i u k ) j c = g + T t t g. (4.17) Here erivatives in parentheses apply only to the quantities within the parentheses. When K ij = Kδ ij an D ij = Dδ ij, equation (4.17) reuces to (1.2) in which the angular brackets have been omitte. When the flow is not incompressible u i i c in (4.1) must be replace by i (u i c), an the analysis yiels slightly ifferent results. When molecular iffusion is not negligible, a more complicate inverse operator must be use in (4.5). This inverse appears in the resulting form of (4.11). References Anan M S an Pope S B 1985 Diffusion behin a line source in gri turbulence Turbulent Shear Flows vol 4, e L J S Brabury et al (Berlin: Springer) p 46 Batchelor G K an Townsen A A 1956 Turbulent iffusion Surveys in Mechanics e G K Batchelor anrmdavies (Cambrige: Cambrige University Press) pp Boussinesq J 1877 Essai sur la théorie es eaux courantes Mém. prés. par iv. savants á l Aca. Sci. Paris vol 23 pp 1 68 Durbin P A 198 A stochastic moel of two-particle ispersion an concentration fluctuations in homogenous turbulence J. Flui Mech. 1 p 279 Golstein S 195 On iffusion by iscontinuous movements an on the telegraph equation Q. J. Mech. Appl. Mech. 4 pp Högström U 1964 An experimental stuy on atmospheric iffusion Tellus Keller J B 1964 Stochastic equations an wave propagation in ranom meia Proc. AMS Symp. on Applie Mathematics vol 16 (New York: McGraw-Hill) pp Stochastic equations an the theory of turbulence Notes on the 1966 Summer Stuy Program in Geophysical Flui Dynamics, Woos Hole Oceanographic Institution no 66 46, vol I, pp Turbulent iffusion Notes on the 1971 Summer Stuy Program in Geophysical Flui Dynamics, Woos Hole Oceanographic Institution no 71 63, vol I, pp 63 4 Kraichnan R H 1959 The structure of isotropic turbulence at very high Reynols numbers J. Flui Mech Monin A S an Yaglom A M 1975 Statistical Flui Mechanics of Turbulence (Cambrige, MA: MIT Press) Saffman P G 1969 Application of the Wiener Hermite expansion to the iffusion of a passive scalar in a homogeneous turbulent flow Phys. Fluis

12 1866 S Ghosal anjbkeller Sawfor B L an Hunt J C R 1986 Effects of turbulence structure, molecular iffusion an source size on scalar fluctuations in homogenous turbulence J. Flui Mech Stapountzis H, Sawfor L, Hunt J C R an Britter R E 1986 Structure of the temperature fiel ownwin of a line source in gri turbulence J. Flui Mech Taylor G I 1915 Ey motion in the atmosphere Phil. Trans. R. Soc. A Diffusion by continuous movements Proc. Lon. Math. Soc Thompson D J 199 A stochastic moel for the motion of particle pairs in isotropic high-reynols-number turbulence an its application to the problem of concentration variance J. Flui Mech