1 Scalar Transport and Diffusion

Transcription

1 ME 543 Scalar Transport and Diffusion February 8 Scalar Transport and Diffusion As we will see, scalar transport is most naturally addressed in Lagrangian form, although it is often used in Eulerian form. We will consider both, starting with scalar transport in Eulerian form.. Eulerian approach Consider the advection diffusion equation for a scalar Θ, i.e., t Θ + U j Θ = κ Θ. () where κ is a molecular diffusivity, taken to be a constant here. Θ can be, for example, the concentration of a chemical reactant (in which case we may need a reaction rate term on the right-hand side of the equation), the temperature, salinity, the concentration of a plume from a stack, etc. With the Reynolds decomposition Θ = Θ + θ, as previously discussed the equation for Θ is: t Θ + U j Θ = x j ( κ ) Θ u j θ, () where the term u j θ represents the turbulent flux of Θ, and is analogous to the Reynolds stress. In a similar manner to working with the turbulence kinetic energy equation, some significant insights into the scalar transport problem can be obtained by developing the equation for θ /. To do this, first the equation for θ can be found by using the Reynolds decomposition for both Θ and U i in Equation, and subtracting Equation from it, giving: θ t + U j θ + u j θ Θ + u j = κ x θ + Multiplying this by θ and averaging then gives the following equation for θ /. ( t + U j ) θ }{{} time rate-of-change following mean flow using θ θ x = ( θ θ ) j Note the following. = u j θ Θ }{{} production, i.e., exchange with Θ / θ ( θ ) = x j ( θ + { κ ( ) θ } θ u j }{{} molecular and turbulent transport ) θ θ. u j θ. (3) ( θ κ ) θ }{{} molecular dissipation rate Equation 6 is analogous to the turbulence kinetic energy equation, with θ analogous to u i. χ = κ θ θ, the scalar dissipation rate, will be seen to be important, similar to ɛ. It is important in combustion, being closely related to the mixing rate of the chemical species at the molecular scale. (4),

2 It is important in geophysics, for example, being related to the transfer of available potential energy into background potential energy. As with the turbulence kinetic energy equation, it is useful to estimate the various terms in Equation 6. The following estimates will be used, with θ as an estimate for both θ / and Θ. θ θ. Θ θ. ( ). l u t ( ) U. l u u j θ θ U. This implies that the correlation coefficient, u j θ [ θ u i ] /, is of order. u j θ θ U. To estimate χ, the Taylor microscale for θ is introduced, that is: Therefore the scaling for χ is: ( θ ) κ θ λ. θ ( θ ) = θ λ. (5) θ When these estimates are used, the terms in the equation for θ have the following estimates. ( t + U j ) θ }{{} Uθ /l u = u j θ Θ + { κ ( ) θ θ } u j }{{}}{{}}{{} Uθ /l u κθ /l u Uθ /l u κ/ul u ( θ κ ) θ } {{ } κθ /λ θ (κ/ul u )(l u /λ θ ) (6) In the first line below each term in the equation is the estimate, while the second line is the estimate divided by Uθ /l u. Note the following. ( κ ) ν The molecular diffusion term is of the order: =, which is again small for ν Ul u P r Re l large Re l, provided the Prandtl number P r = ν/κ is of order or greater. (Note that ν/κ could also be referred to as the Schmidt number, depending on what the scalar Θ is.) Again we require that the dissipation rate χ is the same order as the other terms of order, so that κ ( lu ), or Ul u λ θ ( λθ l u ) = O [ P r / Re / l ]. (7),

3 Therefore, for P r of order or greater, this ratio is small for large Re l, just as for the Taylor microscale for the turbulence kinetic energy. Furthermore, using the scaling for (λ/l u ) obtained from the turbulence kinetic energy equation, then ( λθ ) = O(P r / ). (8) λ For Θ as temperature in air, P r.7, so λ θ λ. For temperature in water, P r 7, so λ is somewhat larger than λ θ. For salt in water, P r 7, so λ θ λ, which is important to consider in making measurements, since then θ would be expected to have much smaller length scales than U. In this case the appropriate dissipation scale corresponding to the Kolmogorov scale η, called the Batchelor scale, is: ( ν η B = ηp r / 3 = ɛ ) /4( κ ν ) / ( κ ν ) /4 =. (9) ɛ Generally, for high Prandtl (Schmidt) number fluids, e.g., salt or dye in water, it is difficult to make measurements down to the scale η B. Such fluids are difficult to numerically simulate as well, because of the range of length (and time) scales required. For P r = ν/κ, e.g., metals, the earth s core, then U would have much smaller length scales that θ. When P r, then the useful length scale corresponding to the Kolmogorov scale is the Oboukov-Corrsin scale, defined by: ( ν η θ = ηp r 3/4 3 ) /4 ( κ ) 3/4 ( κ 3 ) /4 = =. () ɛ ν ɛ Very roughly, the spectra for u i and θ look somewhat like what is sketched in Figure for various values of the Prandtl number. Figure : Sketch of the spectra of the velocity E u (k) and the scalar E θ (k), the latter for cases when P r and P r... Chemically-reacting flows When considering a chemically-reacting flow, there will be equations for the concentrations of the reactants of the form of Equation, but with a reaction rate term on the right-hand-side of the 3

4 form: w(x, t) = AY m F Y n O e E/(R T ), () where A is a constant, Y F and Y O are the concentrations of the fuel and oxidizer, respectively, E is called the overall activation energy, R is a gas constant, T is the temperature, and m and n are overall reaction orders. The term e E/(R T ) is often called the Arrhenius term, and creates significant difficulty in modeling. For reactions that do not generate a significant amount of heat, especially for liquids, then the Arrhenius term can be considered to be a constant. Then, for example, with Θ replaced by Y F, so that Equation can be taken to be the equation for the concentration of the fuel, when averaged a term of the form A YF my O n result on the right-hand-side. This can result in some signficant modeling difficulties. In the case of combustion, however, where there is significant heat release, then the Arrhenius term cannot be neglected, and averaging the equation results in a term on the right-hand-side of the form A YF m YO n e E/(R T ). The temperature T is now one of the unknowns, and it is impossible to propose useful models for this term in terms of the other quantities being computed, such as Y F, Y O, Y F Y O, T. (Here Y F and Y O are defined using the Reynolds decomposition of Y F anf Y O.) To avoid these averaging issues, two very different approaches are often used. One is the probability density function (PDF) approach, where the PDF of Y F and Y O is considered. In this case there is no closure problem for the reaction-rate term, but other new closure problems enter and the mathematical problem becomes much more complex; in particular the dimensionality of the PDF equation becomes very large. To handle the latter issue, Monte-Carlo numerical methods are often used. The second approach, often called flamelet modeling, approximates the reaction zone as a very narrow region, so that the equations simplify locally. The reactant concentrations can then be written in terms of a mixture fraction, and the reaction terms can be computed directly and tabulated.. Lagrangian approach Consider the problem of diffusion from a point source in a turbulent flow, as shown in Figure. This might be considered as a Green s function problem, since the equation for Θ is linear, and the solution to this problem can be used to construct more general solutions. The turbulence problem considered here then is often the following: Problem: given the point source of Θ (e.g., temperature) in a turbulent flow, find the average value of the scalar field, Θ, given information about the turbulent velocity field, such as U i, u i (x, t)u j (x + r, t + τ), etc. The equation for Θ is given by Equation. But scaling arguments give: κ Θ u j θ κθ /l u Uθ κ Ul u κ Θ u j θ ν κ Ul u ν Re l Re l P r,, that is, P r and the effect of molecular diffusion is very small for P r O() and Re l large. Therefore to solve for Θ we can neglect molecular effects (but molecular effects must be included to solve for θ, 4

5 Figure : Sketch of the problem of the point source of heat (temperature) in a turbulent flow. Note the difference between the length scale defining the mean temperature Θ and the width of the plume as discussed in the previous section). This suggests addressing the following problem: which when averaged gives: ( t + U j ) Θ =, () t Θ + U j Θ = u j θ, (3) which is just Equation with the molecular term neglected. Equation is often written as: D Dt Θ =, (4) where D ( ) is the substantial (or material) derivative, i.e., the derivative following a fluid particle. Dt Equation 4 implies the following: the time rate-of-chafe of Θ (e.g., temperature) following a fluid point is zero, or Θ is constant following a fluid point (particle). This suggests using Lagrangian variables, where the problem should be simpler since Lagrangian variables track the fluid particles. Recall the following notation: Θ(x, t) is called an Eulerian dependent variable (the unknown), while (x, t) are the Eulerian independent variables (coordinates), since Θ is considered a function of space and time. 5

6 Another approach is to label the fluid particles, and consider Θ (e.g., temperature) for each fluid particle as a function of time. If the particles were countable, then, following particle mechanics, we could write: Θ + (t, n) n =,, 3,... as the temperature at time t for the n th particle. Note that Θ + (t; n) is not the same function as Θ(x, t). In fact there is an uncountable continuum of fluid particles in 3 spatial dimensions in a flow field. Choose Y to be a three-dimensional vector as a label for a fluid particle. Assume that Y spans the field of fluid particles, i.e., every fluid particle could be defined by its own value of Y. Then, say, call the position of the fluid particle labelled Y at time t to be X + (t, Y). In this case now X + (t, Y), the position at time t of the particle labelled Y is a Lagrangian dependent variable, and Y and t are now the independent variables. The question is then how to define Y. One possible method to label the fluid particles is using the particles position in space at a fixed time, say at t =. Each fluid particle has a unique position and a unique label. We write: Y = X + (, Y). (5) Therefore X + (t, Y) now means the position at time t of the fluid particle which started at Y at time t =. The particle velocity is given by: U + (t, Y) = t X+ (t, Y) = t X+ (t, Y), (6) Y i.e., the time derivative following the fluid particle labelled Y. In distinction with Θ(x, t), X + (t, Y) is called the Lagrangian dependent variable, and (t, Y) are the Lagrangian independent variables (coordinates). The Eulerian and Lagrangian dependent variables are related through the Lagrangian particle displacement X + in the following way: Θ + (t, Y) = Θ[X + (t, Y), t], (7) i.e., the fluid temperature at the location X + (t, Y) is just the temperature of the fluid particle which was at Y at time t =. Again note that Θ and Θ + are different functions. In a similar manner, the Lagrangian and Eulerian velocities are related by: and similarly for other dependent variables. U + (t, Y) = U[X + (t, Y), t], (8) Example. Stagnation point flow (see Figure 3). Consider the following two-dimensional, steadystate velocity field: U (x, x ) = αx, U (x, x ) = +αx, with α = constant >. Note that U = and Ω =. This flow is a solution for an inviscid, irrotational flow approaching a surface; the boundary layer has been neglected. The Lagrangian particle trajectories are given by, using Equation 6: t X+ (t, Y, Y ) = U + (t, Y) = U (X + (t, Y), t) = αx + (t, Y, Y ), and 6

7 Figure 3: Sketch of the flow towards a stagnation point, located at the origin (, ). t X (t, Y, Y ) = U (t, Y) = U (X + (t, Y), t) = +αx (t, Y, Y ). Note the distinction between the Lagrangian, U + (t, Y), and Eulerian, U(x, t), velocities. Integrating these two equations in time gives, using the initial condition X + (, Y) = Y: X + (t, Y) = Y e αt, and X (t, Y) = Y e +αt, with Y = (Y, Y ). These equations give the trajectory, (X +, X ), as a function of time of the fluid particle which started at Y = (Y, Y ) at t =. Note that X + X = Y Y = constant, or X + = constant/x+, i.e., the trajectories are hyperbolas. Suppose that the Eulerian temperature field for this stagnation point flow was measured to be: T (x, t) = T ( x ) n, with n > and x <, then the temperature of the particle starting at (Y, Y ) at t = would be: T + (t, Y) = T [X + (t, Y), t] = T [ X + (t, Y)] n = T [ Y e αt ] n, where T (x, t) and T + (t, Y) are the Eulerian and Lagrangian temperature fields, respectively. Returning to Equation 4, in Lagrangian form it can be written as t Θ+ (t, Y) =, (9) where Θ + (t, Y) is the Lagrangian field corresponding to the Eulerian field Θ(x, t). Equation 9 says that the time derivative of Θ + with Y held fixed, i.e., following the fluid particle labelled Y, is, or that Θ + is constant following this fluid particle. Equation 9 can be easily integrated in time to give: Θ + (t, Y) = Θ + (, Y) = F (Y), where F (Y) is the initial condition for Θ + (t, Y). The problem of scalar dispersion is clearly simpler and more natural in the Lagrangian frame. The difficulty, however, is that most information, e.g., regarding the velocity field and the interpretation of the flow, is given in an Eulerian reference frame. The problem of relating the Eulerian and Lagrangian velocity statistics will be discussed later. 7

8 Figure 4: The trajectory of a fluid particle X + starting at Y at time t =. To address the problem of dispersion of a scalar from a Lagrangian perspective, consider again the Green s function problem of dispersion from a point source (see Figure 4). Instead of a steady release of heat, consider an ensemble of realizations where in each realization a fluid particle is released at t = from the point Y with a temperature T p into a turbulent flow. Assume that there is no molecular diffusion, and that the ambient temperature is T =. The average temperature in a volume dx centered at the point x will be proportional to the average number of times that the particle position X + (t, Y) falls into dx, i.e., T (x, t) T p Prob {x x X+ (t, Y) < x + } x = T p f X (x; t Y)dx, where f X (x; t Y) is the probability density of X + (t, Y), the fluid particle displacement, taking on the value of x. Therefore the Lagrangian probability density f X (x; t Y) becomes a key quantity in dealing with scalar transport. Of special interest are the first two moments of f X, X + (t, Y) and X + (t, Y) X + (t, Y)..3 Lagrangian Dispersion from a Point Source As discussed above and in the text (page 497), when molecular effects are neglected, turbulent dispersion is more naturally viewed from a Lagrangian framework. And when it is, the principal quantity of interest is the probability density function (pdf) of fluid particle displacement. Using the notation in the text, with X + (t, Y) being the position at time t of the fluid particle which started at Y at time t =, then its pdf is written as f X (x; t Y). The first two moments of this distribution were obtained by G.I. Taylor in a famous paper entitled Diffusion by Continuous Movements. (See also the text, page 498 and the following.) He used the term continuous movements to distinguish the motion from the discontinuous motion in Einstein s theory of Brownian motion. Assume a turbulent flow that is statistically homogeneous, isotropic, and stationary (an idealization) in a Lagrangian frame. (The isotropic assumption will be dropped in a later discussion of shear dispersion.) It can be shown from examining the mean momentum equation that these Taylor, G.I. Diffusion by continuous movements, Proc. Lond. Math. Soc., :96-, 9. Einstein, A. On the movement of small particles suspended in a stationary liquid demanded by the molecularkinetic theory of heat, Annalen der Physik, 3(8):549-56, 95 8

9 assumptions imply that the mean velocity U(x, t) is a constant. Assume the coordinate system is moving with the mean flow, so that U =. It can also be shown that the Lagrangian mean velocity U + = as well. With U + (t, Y) the velocity of the fluid particle that started at Y at t =, then, noting that ( ) denotes the time derivative holding Y fixed, i.e., the time derivative following a specific fluid t particle, t X+ (t, Y) = U + (t, Y), () and integrating in time from to t, X + (t, Y) = Y + The mean value of X + (t, Y) is therefore X + (t, Y) = Y + U + (t, Y) dt. U + (t, Y) dt = Y, so X + (t, Y) = Y, i.e., the particle just wanders around its original location. We now consider the second moment of the displacement about the mean. Define Z + (t, Y) = X + (t, Y) Y, i.e., the displacement about the mean position, and consider a component Z + (t, Y). Because of the assumption of isotropy, Z = Z = Z 3, and Z + Z + = Z+ Z+ 3 = =, so that we only need to consider this one component. With Z Z+ 3 then, using Equations () and (), Therefore, averaging yields Z + (t, Y) = X+ (t, Y) Y = Z + (t, Y)U + (t, Y) = Z+ (t, Y) t Z+ (t, Y) = t = U + (t, Y)U + (t, Y) dt. U + (t, Y) dt, () Z+ t t Z (t, Y) = U + (t, Y)U + (t, Y) dt. (t, Y) We expect Z (t, Y) to be independent of Y since the flow is statistically homogeneous. It is useful to define the Lagrangian velocity autocorrelation coefficient by R L (τ) = U + (t, Y)U + (t + τ, Y), (t, Y) U + where U + (t, Y) is independent of Y and t because of homogeneity and stationarity. Therefore, d t dt Z (t, Y) = U (t, Y) R L (t t ) dt. 9

10 But, substituting τ = t t, and changing the variable of integration from t to τ, Integrating this in time gives R L (t t ) dt = Z+ Now, integrating by parts, so finally { t R L (τ) d( τ) = R L (τ) dτ, so d t dt Z (t, Y) = U (t, Y) R L (τ) dτ. () { (t, Y) = U (t, Y) R L (τ) dτ }dt = t Z+ = R L (τ) dτ } R L (τ) dτ dt. t (t τ)r L (τ) dτ, t R L (t ) dt (t, Y) = U (t, Y) (t τ)r L (τ) dτ. (3) This is Taylor s main result, and expresses the second moment of displacement in terms of the second moment of the Lagrangian velocity and its autocorrelation function. Therefore, given the mean square Lagrangian velocity and its autocorrelation function, the first two moments of the Lagrangian displacement pdf f X (x; t Y) can be determined. This result has two important limits. Take T L to be the integral time scale for U +, i.e., The limits are the following. T L = R L (τ) dτ. i. Short times: t T L. Then, since R L (τ) under this condition, and since (t τ) dτ = (tτ τ ) Z + (t, Y) U + (t, Y) t t = t t = t, then for t T L. (4) This is referred to as the ballistic limit, since the fluid particle velocity does not change during this time period. ii. Long times: t T L. Under this condition (t τ)r L (τ) dτ = t R L (τ) dτ tt L I tt L, τr L (τ) dτ

11 4.5 Mean square particle displacement versus time <U > t 3 <Z >.5.5 <Z > <U > T L t t Figure 5: Mean-square particle displacement versus time. using the definition of the integral scale, and assuming that the following integral exists: Therefore Z + I = τr L (τ) dτ. (t, Y) U + (t, Y) TL t for t T L. (5) This long time behavior is similar to that for Brownian motion, where the velocity field is a discontinuous function of time, and the integral time scale approaches zero (so that time t is always large compared to it). The general result, Equation (3), and the two limiting cases, Equations (4) and (5), are plotted in the Figure 5. In addition to helping to determine the properties of the pdf f X, these results can be used to estimate the turbulent diffusivity for a passive scalar, say Θ. As discussed above (Equation 3) and in the text (section.4, page 494), the equation satisfied by the average of Θ for high Reynolds numbers (neglecting molecular diffusion) is, assuming the Reynolds decomposition Θ = Θ + θ, Θ + ( U ) Θ = u θ. (6) t For this case of homogeneous, isotropic turbulence, as discussed above, U =. Furthermore, we use an eddy-diffusivity model for u θ, i.e., u θ = K(t) Θ, (7) where K is the turbulent diffusivity and, because of statistical homogeneity, it is independent of x. We consider the mathematical problem of a point source, in an unbounded domain, which is initially at x =, i.e., Θ(x, ) = Θ δ(x), where δ(x) is the Dirac delta function and the point

12 source amplitude is Θ. (Note that this can be considered a Green s function problem so that, since the equation is linear in Θ, a problem with more general initial conditions can in theory be obtained from the solution to this problem.) The mathematical solution to this problem is Θ(x, t) = Θ δ(x + (t, ) x), i.e., without molecular diffusion, the perturbation in Θ moves with the fluid particle which started at x = at t =. Averaging this gives, setting the amplitude Θ to be, Θ(x, t) = δ(x + (t, ) x) = δ(y x)f X (y; t ) dy = f X (x; t ), where f X (x; t ) is the probability density of displacement of the fluid particle that started at x = at t =. Therefore Θ(x, t) = f X (x; t ). (8) With Equations (6), (7), and (8), then f X satisfies t f X(x; t ) = K(t) f X (x; t ), (9) i.e., the diffusion equation, where is the three-dimensional Laplacian operator. But Multiplying Equation (34) by x and integrating over all x gives x t f X(x; t ) dx = K(t) x f X (x; t ) dx. x x x t f X(x; t ) dx = d dt x x f X (x; t ) dx = d dt X (t, ). Also, using integration by parts twice, K(t) x f X (x; t ) dx = K(t). Therefore, the turbulent diffusivity K satisfies x d dt X (t, ) = K(t). Using Equation (), the expression above for the mean square displacement in terms of Lagrangian velocity statistics, then, since X + = Z+ for this case with Y =, K(t) = d dt X+ (t, ) = U (t, ) R L (τ) dτ. This gives the turbulent diffusivity solely in terms of the Lagrangian mean square velocity and its autocorrelation function. See Figure 6 for a plot of this result. Again, there are two asymptotic limits, which are the following. i. K(t) = d dt X (t, ) U (t, ) t for t TL, and ii. K(t) U + (t, ) TL for t T L.

13 .4 Turbulent Diffusivity versus t/t. K/<U >T t/t Figure 6: K/( U + TL ) versus t/t L. This latter expression is sometimes the starting point for turbulence diffusivity models. However, there is usually still a need to relate U (t, ) and TL to the statistical properties of the Eulerian velocity field, a task sometimes referred to as the Euler-Lagrange problem. Usually, turbulent diffusivities are approximated as K T = u l, where u and l are suitably chosen velocities and lengths. This Lagrangian analysis suggests that u = U (t, Y) /, and l = U (t, Y) / T L. Note that it has been established, at least for homogeneous flows, that u (x, t) = U (t, Y), i.e., the Eulerian and Lagrangian mean square velocities are equal (Tennekes and Lumley 3 ). Therefore the Lagrangian analysis suggests that u should be the Eulerian root-mean-square velocity. It also suggests that the length scale l should be related to the Lagrangian velocity integral time scale..4 Shear dispersion To understand how shearing of the mean velocity (i.e., a mean velocity gradient) affects fluid particle dispersion, it is useful to consider the hypothetical, simplified case of a homogeneous, stationary shear flow, with the Eulerian velocity field given by U(x, t) = U(x, t) + u(x, t) = d dx U x e + u(x, t), (3) 3 Tennekes, H., and J.L. Lumley. A First Course in Turbulence, MIT Press, 97, p. 7 3

14 d d where it is assumed that the mean velocity gradient U is a constant, say, U = λ. (A dx dx closely related flow, but one which is not statistically stationary, is discussed in Section of the text.) Here e is a unit vector in the x direction. Note that. the mean velocity component U is linear in x, and the coordinate system has been chosen such that U = at x = ;. by examining the averaged equations, it can be shown that this mean velocity profile admits statistical homogeneity of u(x, t) in all three directions (although this is not necessary); and 3. to maintain such a flow u(x, t) to be statistically stationary, a fictitious forcing is required. In the Lagrangian frame, since U + (t, Y) = U[X + (t, Y), t], one can write, using Equation (3), U + (t, Y) = λx (t, Y) e + U + (t, Y), where U + (t, Y) = u[x + (t, Y), t]. It is expected that U + (t, Y) is statistically homogeneous and stationary. Following the same procedures as for the previous case without shear in Section.3, using integration by parts, etc., and assuming reflective invariance with respect to the x x plane, i.e., u u 3 = u u 3 = U + U 3 + = U + U + 3 = X+ X+ 3 = X X+ 3 =, the following results are obtained for the nonzero components of the mean square displacement tensor X i + (t, Y)X+ j (t, Y), with Y = for simplicity. X + X + (t, Y)X X X + 3 { (t, Y) = λ U t (t, Y) 3 t3 R L (τ) dτ t τr L (τ) dτ + } τ 3 R L (τ) dτ 3 { + λ U + + (t, Y)U (t, Y) (t τ) [ R L (τ) + R L (τ) ] dτ } τ(t τ)r L (τ) dτ U (t, Y) (t τ)r L (τ) dτ (t, Y) = λ U + (t, Y) t(t τ)r L (τ) dτ + U + (t, Y)U + (t, Y) (t τ) [ R L (τ) + R L (τ) ] dτ (t, Y) = U (t, Y) (t τ)r L (τ) dτ (t, Y) = U 3 (t, Y) (t τ)r L33 (τ) dτ Here, the Lagrangian velocity autocorrelation tensor is defined by R Lij (τ) = U + i (t, Y)U j + (t + τ, Y) i (t, Y)U j +. (t, Y) U + 4

15 Note that the analytical results for X and X 3 are not affected by the shear, whereas those for X and the correlation X + X + are strongly affected. As in the previous case without shear, there are also important short and long time limits for these results. i. Short times: t T L, where now T L is an integral time scale representing all the components of the autocorrelation tensor R Lij. X (t, Y) U (t, Y) t X + (t, Y)X+ + + (t, Y) U (t, Y)U (t, Y) t X (t, Y) U (t, Y) t X (t, Y) U 3 (t, Y) t These are similar to the results for the unsheared case, as shear has very little effect for short times. ii. Long times: t T L. X (t, Y) 3 λ t 3 U (t, Y) TL X + (t, Y)X (t, Y) λ t U (t, Y) TL X X + 3 (t, Y) t U (t, Y) TL (t, Y) t U 3 (t, Y) TL33 Here T Lij is the integral time scale corresponding to R Lij. The particle position cross-correlation coefficient asymptotes as X + (t, Y)X (t, Y) 3 ( X X ) /.866, a fairly large value for a correlation coefficient. Note the much more rapid dispersion rate, especially in the x direction. combination of two effects working in combination.. vertical (x ) dispersion, represented by the factor T L U (t, Y), and. shearing by the mean horizontal (x ) velocity, represent by the factor λ. This is due to a As turbulence disperses the fluid particles in the x direction, they become increasingly more subject to the effects of shearing, which then tend to disperse them more strongly in the x direction. Some idea of the time development of the joint probability density of (X +, X ) (or, e.g., of the mean temperature field) can be obtained as follows. Using the Central Limit Theorem (see Section.5 below, and section 3.5 of the text), it can be argued that the joint probability density of the two-dimensional displacement vector (X +, X ) converges to a joint-gaussian distribution as t becomes large compared to T L. Therefore, this joint density f (x, x ) has the form: { f (x, x ) {4π X (t, Y) X (t, Y) ( ρ )} / exp ( ρ ( ) x x x X ρ (t, Y) [ X X + x )} (t, Y) ] / X. (t, Y) 5

16 Lines of constant probability density of displacement 8 Assumes t >> T L 6 4 λ t = 3 λ t = 6 X λ t = X Figure : Lines of constant probability density of displacement. Figure 7: Lines of constant probability density of displacement. Here X (t, Y) and X (t, Y) are given by their asumptotic forms in Equations () and (4) above, and ρ is the asymptotic form for the cross-correlation coefficient in Equation (6), that is, ρ X + (t, Y)X (t, Y) 3 ( X (t, Y) X (t, Y) ) /. The (x, x ) dependence of f is in the argument of the exponential. Therefore a line of constant f in the (x, x ) plane is given by X + x 4 x x ρ (t, Y) [ X X + x (t, Y) ] / X = constant. (t, Y) With the large time asymptotic values given by Equations (), (3), and (4) above, this equation simplifies to x (λt) 3 x x (λt) + x 3 3 (λt) = constant. In Figure 7 is presented plots of the line f = constant for three different values of λt, showing how the size of the joint probability density f (or, e.g., the mean temperature field from a point source) grows due to turbulent diffusion, and is stretched and tilted by the action of the mean shear. It is also useful to compute the turbulent diffusivity for this case with shear. Now, however, there are clearly direction preferences, which were not present in the isotropic case. To treat the directionality of the flow, a diffusivity tensor must be used; it is defined by u i θ = K ij (t) Θ. 6

17 Following the same procedures as in the previous case without shear, i.e., starting with the modeled equation for f X (x; t ), etc., it is found that K (t) = U + f X (x; t ) t (t, Y) + λx f X (x; t ) x = K ij x i f X (x; t ), R L (τ) dτ + λ U + + (t, Y)U (t, Y) τr L dτ K (t) + K (t) = U + + (t, Y)U (t, Y) [ RL (τ) + R L (τ) ] dτ + λ U (t, Y) τr L (τ) dτ K (t) = U + K 33 (t) = U + 3 (t, Y) R L (τ) dτ (t, Y) R L33 (τ) dτ Because of the reflective invariance with respect to the x x plane, then K 3 = K 3 = K 3 = K 3 =. The long time limits of these expressions are of most interest. For t T L, then K U + K (t) + K (t) U + Here I L = τr L (τ) dτ. (t, Y) TL + λ U + + (t, Y)U + (t, Y)U (t, Y) I L (t, Y) (T L + T L ) + λ U (t, Y) IL K (t) U (t, Y) TL K 33 (t) U 3 (t, Y) TL33 Because the off-diagonal terms K and K are clearly non-zero, and because the three diagonal terms, K, K, and K 33 are not expected to be equal, the tensorial form of the turbulent diffusivity is clearly required. Note that for most turbulence models involving a turbulent viscosity or turbulent diffusivity, e.g., k-ɛ models, the tensorial form for turbulent viscosity or diffusivity is not taken into account (see Chapter in the text.).5 Central Limit Theorem applied to turbulent dispersion Consider the sequence of random variables x, x,..., x N which are:. identically distributed, i.e., have the same probability densities, so that, for example, x = x = = x N m, say, and (x m) = (x m) = = (x N m) σ ; 7

18 . statistically independent so that, for example (x i m)(x j m) = if i j. (3) Now consider the sum of these random variables, say z N, i.e., N z N = x i. Of course z N is itself a random variable, with mean and standard deviation given by i z N = N x i = Nm m N, say, and i (z N m N ) = Nσ σ N using Equation (3) above. Denote the probability density of z N as f z (ζ). The Central Limit Theorem says that, as N becomes large, f z (ζ) approaches a Gaussian distribution, i.e., f z (ζ) σ N π e (ζ m N ) /σ N as N. (See a discussion of this in the text in section 3.5, and in problem I.3, page 79.) There are several important things to note about this result.. Only two moments, m N and σ N, are needed to define the entire distribution.. This can be easily extended to vectors, e.g., consider the sequence of vectors x, x,..., x N, and define the sum as N z N = x i. Now f z (ζ) is joint Gaussian. (See Equation 3.8, page 6 of the text for the joint probability density of two random variables which are joint-gaussian distributed.) 3. For example, in two dimensions, with the random variables (z, z ), the following 5 quantities are required to define the joint probability density: z = m, z = m, (z m ) = σ, (z m ) = σ, (z m )(z m ) = σ. Now the Central Limit Theorem is applied to turbulent diffusion. Consider the Lagrangian displacement function for statistically homogeneous, stationary, isotropic turbulence, as considered in previous notes: X + (t, Y) = Y + i U + (t, Y)dt, with X + (t, Y) = Y. For large times, i.e., t T L, where T L is the Lagrangian integral time scale, then it is convenient to write this equation as X + (t, Y) = Y + N i TL i T L (i ) U + (t, Y)dt } {{ } X i + U + (t, Y)dt. T L N 8

19 From the definition of T L, the X i are approximately statistically independent. Furthermore, because the flow is statistically steady, the X i are identically distributed. Therefore it is expected that f X (x, t Y) σ (t) π e (x Y ) /σ (t) as t, where σ (t) = X + (t, Y) Y ) = Z (t, Y). There are a number of things to note about this result.. The problem has been solved if σ (t) is known, i.e., Z + = U (t τ)r L (τ)dτ, so that U + and RL are needed to obtain the solution.. This extends to two and three dimensions. For example, in two dimensions, Z, Z, and Z +! Z + are needed. 3. Generally the Gaussian function with σ a function of time is a solution to the advectiondiffusion equation, i.e., to t φ + U φ = K φ, motivating the use of a turbulent diffusivity. 4. For the present case, K(t) U + TL = u (x, t) T L as t. So a theory or model is needed to predict u and T L. 5. The same argument regarding the use of the Central Limit Theorem can be made for statistically homogeneous, stationary shear flows, and also the use of turbulent diffusivities. 6. This suggests an approach for treating turbulent diffusion in more general flows. There are great difficulties, however, with this suggestion, for example with the assumptions of homogeneity, of stationarity, of long time, etc. 7. Note that most information about turbulence is Eulerian. Also, the use of a turbulent diffusivity is Eulerian. The information needed, however, is Lagrangian. This leads to what is called the Euler-Lagrange problem..6 The Euler-Lagrange Problem The problem of relating the statistical properties of the Lagrangian displacement field X + (t, Y) to those of the Eulerian velocity field U(x, t) is called the Euler-Lagrange problem, and is one of the central problems in turbulent diffusion/dispersion theory. As shown in the previous section, the statistical properties of X + are directly related to the Lagrangian velocity time autocorrelation function R L (τ). Since much more information is available, from experiments and theory, regarding the statistics of the Eulerian velocity field, however, what is needed is the knowledge of how R L is related to the statistical properties of the Eulerian field. Some insight into this relationship can be obtained as follows. 9

20 For simplicity, assuming statistically isotropic flow, and working with the first component of the velocity (without loss of generality), the Lagrangian velocity time autocorrelation function R L (τ) can be written directly in terms of the Eulerian space-time autocorrelation as R L (τ) = U + (, Y)U + (τ, Y) = U (Y, )U [X + (τ, Y), τ], (3) since U + (τ, Y) = U [X + (τ, Y), τ], and, in particular, U + (, Y) = U (Y, ). But, using the properties of the δ-function, we can also express U [X + (t, Y), t)] as U [X + (t, Y), t] = U (x, t)δ[x + (t, Y) x] dx, x where the integral is over all space. Therefore, Equation (3) becomes R L (τ) = U (Y, )U (x, τ)δ[x + (τ, Y) x] dx. (33) x In terms of the joint probability density function of U (Y, ), U (x, τ) and X + (τ, Y), say p(υ, Y; υ, x, τ; y, Y, τ), this expression can be rewritten as R L = υυ δ(y x)p(υ, Y; υ, x, τ; y, Y, τ) dx dy dυ dυ υ υ y x = υυ p(υ, Y; υ, y, τ; y, Y, τ) dy dυ dυ. (34) But, from an identity from probability theory, υ υ y p XY Z (A, B, C) = p XY (A, B C)p Z (C), where p XY (A, B C) is probability density of X and Y, conditioned on the event Z = C. Using this in Equation (34), we can then write R L (τ) = υυ p(υ, Y, υ, y, τ X + (τ, Y) = y)f X (y; τ Y) dy dυ dυ υ υ y = U (a, )U (y, τ) X (y; τ Y) dy. (35) X (τ,y)=yf + y Here f X (y; τ Y) is the probability density of X + (t, Y). This is a general relationship between the Lagrangian velocity time autocorrelation and the conditionally-averaged Eulerian space-time autocorrelation, and gives insights into what information regarding the Eulerian field is needed to determine R L. From this expression it is possible to obtain short and long time results. In particular, for large τ, it is expected that U (x, τ) will not depend on the particular trajectory X + (τ, Y), so that U (a, )U (y, τ) U (a, )U (y, τ) = R E (y a, τ), X (τ,y)=y + where R E (x, τ) is the Eulerian space-time autocorrelation. This latter assumption is known as Corrsin s hypothesis, and results in, from Equation (35), R L (τ) = R E (y a, τ)f X (y; τ a) dy. (36) y This relationship has been used in several theoretical models, and is found to work rather well.