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Representation of internal plume structure in Gifford's meandering plume model
Atmospheric Environment 34 (2000) 2539}2545
Representation of internal plume structure in Gi!ord's meandering plume model Andrew Michael Reynolds* Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received 5 March 1999; received in revised form 5 November 1999; accepted 11 November 1999
Abstract Gi!ord's (1959. Advances in Geophysics 6, 117}138) meandering plume model is extended to account for internal plume structure. The applicability of the model is thereby extended to include the near "eld of large sources and the far "eld. Agreement with measured root-mean-square #uctuating concentrations of scalars dispersing from elevated compact-area and line sources into surface layers with neutral stability is shown to be as good or better than that obtained with Gi!ord's model. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Turbulent dispersion; Concentrations; Fluctuations; Meandering plume
1. Introduction The need to predict the #uctuations in concentration of scalars dispersing in turbulent #ows arises in numerous situations. It has for example relevance to: predicting the probability of exposure to a speci"ed level of concentration; the determination of reaction rates in turbulent chemical reactors; the prediction of malodour nuisance; predicting the ignition of potentially #ammable gas clouds; predicting the received dosage of dispersing toxic gases; and the simulation of &temperature noise' downwind of heat sources in turbulent #ows. The earliest and one of the most useful models for the prediction of concentration #uctuations is Gi!ord's (1959) meandering plume model. This model describes the concentrations caused by the meandering of a small &instantaneous' plume or &pu! ' by large eddies in a turbulent #ow. There are, however, several restrictive assumptions in Gi!ord's model, the most serious of which is to neglect concentration #uctuations within pu!s. This limits the applicability of the model to short distances downwind of small sources, where it is the meandering of
* Tel.: #44-1525-860000; fax: #44-1525-860156. E-mail address: [email protected] (A.M. Reynolds)
pu!s rather than the concentration #uctuations within pu!s that is the dominant source of concentration #uctuations. In this situation, Gi!ord's model is very useful and found to be in good agreement with measured near"eld concentration #uctuations in surface layers with neutral stability (Fackrell and Robins, 1982a) and the atmospheric boundary layer (Kimura et al., 1981). An extension of the model to include buoyant plume rise has also been found to be in reasonable agreement with measured concentration #uctuations in laboratory-scale convective boundary layers (Weil, 1994). In the far "eld, the model predicts wrongly that the intensity of concentration #uctuations vanishes whilst, for short distances downwind of large sources, root-mean-square (rms) #uctuating concentrations are signi"cantly underpredicted; e.g. Fackrell and Robins (1982a) and Sykes et al. (1986) found that for the largest source they considered, rms #uctuating concentrations were underpredicted by about a factor of 2. In the next section an extension of Gi!ord's model is presented which allows for a representation of internal plume structure (in-plume concentration #uctuations). The applicability of Gi!ord's model is thereby extended to include the near "eld of large sources and the far "eld in general. The approach is akin to that of Yee et al. (1994), but here plume meander is calculated rather than being a model input, making the model is applicable to inhomogeneous turbulence.
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2. Gi4ord's model and its extension 2.1. Theory The quantities of key importance in the formulation of Gi!ord's model and its extension are the probability density function (pdf ) for concentration at any point (x, y, z), p(c, x, y, z), the pdf for in-plume concentration, p (c, y , z , x), and the pdf for the location of a pu! centroid p (y , z , x). Here x is the downwind position relative to the source, and y and z represent position in the crosswind plane relative to the position of the pu! centroid at y and z . These three pdf's are linked by the convolution relation
p(c, x, y, z)" p (c, y , z , x)p (y , z , x) dy dz .
(1)
The corresponding moments of concentration are given by
1cL(x, y, z)2" cLp(c, x, y, z) dc " 1cL (y , z , x)2p (y , z , x) dy dz ,
(2)
where 1cL (y , z , x)2 are the moments of in-plume con centration, which are prescribed by
1cL (y , z , x)2" cLp (c, y , z , x) dc.
(3)
In Gi!ord's model, in-plume concentration #uctuations are ignored and the in-plume concentration is assumed to be exactly equal to the ensemble expectation 1c (y , z , x)2, so that p (y , z , x)"d(c!1c(y , z , x)2), (4) where d is a Dirac delta function. Thus, according to Gi!ord's model the moments of in-plume concentration are given by langle 1cL(y , z , x2"1c(y , z , x)2L with the consequence that, contrary to experimental observations (Fackrell and Robins, 1982b), the intensity i "p /1c 2 of in-plume concentrations vanishes. A natural extension of Gi!ord's model, which allows for a representation of internal plume structure, is to use a more realistic pdf for in-plume concentrations in place of the delta-function in Eq. (4), or more simply, but equivalently when predicting rms #uctuating concentrations, is to use a more realistic intensity of in-plume concentration, i , in place of i "0. This is an attractive possibility because the intensities in-plume concentration are much less variable than the intensities of total concentration c(x, y, z) and hence more readily characterized and represented by simple model distributions. For example, in the near "eld of su$ciently small sources, zeroes in concentration arise predominately from inter-
mittency (i.e. meandering of the whole plume by turbulent eddies comparable to or larger than the plume size) rather than by the internal #uctuations in concentration. Consequently, in this regime, in-plume concentration statistics are expected to be well approximated by conditional concentration statistics, which have the zero-concentration intervals censored. Also the measured intensities of conditional concentrations are found to be only weakly dependent upon source size and remain approximately constant along y and z pro"les through plumes from elevated sources (Wilson et al., 1985a,b; Sawford, 1985). The tendency of conditional intensity to establish a universal structure is also demonstrated by concentration #uctuation data from the laboratory-scale convective boundary-layer experiments of Deardor! and Willis (1984), who studied plume dispersion from buoyant and non-buoyant sources. In the far "eld, where concentration #uctuations are dominated by internal #uctuations in concentration rather than by plume meander, experimental measurements of concentration intensities are scarce. However, the experimental dispersion data of Fackrell and Robins (1982b) for a surface layer with neutral stability, suggests that in the far-"eld and independent of source size, i +0.45. A non-zero value of i implies that 1c (x, y, z)2" (i#1)1c (x, y, z)2 rather than 1c (x, y, z)2" 1c (x, y, z)2 as in by Gi!ord's model. Here Gi!ord's model is extended to account for i O0 by writing
1c(x, y, z)2" 1c (y , z , x)2p (y , z , x) dy dz , 1c(x, y, z)2" (i #1)1c (y , z , x)2p (y , z , x) dy dz . (5) It should be noted that Bara et al. (1992) used i and Gi!ord's model in a similar way but did not address internal concentration #uctuations per se. Gi!ord's model is semi-quantitative in the sense that it predicts concentration #uctuations in terms of the distribution of pu! centroids and the relative dispersion or spreading of a pu!, which must be modelled or parameterized to obtain fully quantitative predictions. The extension of Gi!ord's model proposed here also requires parameterization of the intensity of in-plume concentrations. Gi!ord assumed that p (y , z ;x) is Gaussian, which is appropriate for homogeneous turbulence. Here, for inhomogeneous turbulence, p (y , z ;x) is calculated from the trajectories of pu! centroids simulated using a Lagrangian stochastic model. A detailed description of this approach and the Lagrangian stochastic model can be found in Reynolds (1999). Reynolds (1999) showed that model predictions for the mean concentrations of scalars dispersing from elevated line and compact area into a surface layers with neutral stability are in good agreement with the experimental dispersion data of
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Fackrell and Robins (1982b) and Raupach and Legg (1983). Also well predicted are rms #uctuating concentrations in the near "eld for the line source and the smaller compact-area sources. Less well predicted, presumably because of the neglect of internal concentration #uctuations, are rms #uctuating concentrations in the far "eld and in the near "eld of the larger compact-area sources. The model for the spreading of a pu! and the parameterization of the intensity of in-plume concentrations are described in Sections 2.1 and 2.2. 2.2. In-plume concentrations In Gi!ord's model, the spatial distribution of concentration, c , in a pu! is assumed to be a non-random Gaussian distribution about the pu! centroid. That is Q c (x, y , z ; y , z )" 2n;p p W X (y !y ) (z !z ) ! , exp ! (6) 2p 2p W X where Q is the source strength, p and p are the rms W X sizes of the pu! in the y and z directions, and ; is the mean #uid velocity at the pu! centroid. The distinction between p and p allows for departure from axisymW X metry due to the anisotropy of turbulence in boundarylayers. Batchelor (1950) distinguished &near-"eld', &intermediate range' and &far-"eld' approximate descriptions of the rms growth in pu! size. For travel times t(p / 1v(0)2 , Batchelor (1950) showed that p (t)"p# 1v(0)2t, where p is the source size and v (0) is the initial velocity of pu! particles relative to the velocity of the centre of mass of the pu!. For the &intermediate range', Batchelor (1950) showed, for an instantaneous point source in homogeneous isotropic turbulence, that the rms pu! size at time t is given by p "(ae t) where a is a constant of O(0.1) and e is the mean rate of dissipation of turbulent kinetic energy. This result, which is strictly appropriate only when the size of the pu! lies in the inertial subrange, was subsequently generalized by Sykes (1988) to anisotopic turbulence and to sources of size p . Sykes proposed that p "(p #ap K\ t) where K is the background scale of the turbulence. Here, for inhomogeneous turbulence, it is assumed that Sykes proposal can be applied locally, so that p (t#*t)" (p (t) #ap K\ *t) where e and K are evaluated at the pu! centroid and p (0)"p . That is, at time t the pu! of size p (t) acts as a source from which the pu! at time t#Dt develops. Over the time-interval *t, the turbulence is taken to be homogeneous. In the &far "eld', where particles in the pu! are essentially moving independently, Batchelor (1950) found that p(t)"2p Jt, where p is the rms single-particle dispersion. However, the coe$cient 2 in the above (p(t)"2p ) should be
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omitted as Thomson (1990), shows from his two-particle model that p (t)&p Jt. It is doubtful whether the &far "eld' is attained in the wind-tunnel experiments of Fackrell and Robins (1982a) and, Raupach and Legg (1983), with which model predictions will be compared. This is simply because for both these experiments, the size of pu!s is predicted to remain smaller than or comparable in size to the &mixing length'. Fackrell and Robins (1982a) also found, in their numerical simulations of their experiments (Fackrell and Robins, 1982b) that, the far "eld regime in which the instantaneous pu! size grows as t was not attained. Fackrell and Robins (1982a) adopted the more complicated parameterization of pu! size proposed by Hay and Pasquill (1959) and Smith and Hay (1961). Consequently, in the present study only the &near "eld' and &intermediate range' forms for pu! growth are used. That is, the &near "eld' form of Batchelor (1950) is utilized for times t( p /1v(0)2 , and then the 'intermediate range' form of Sykes (1988) is used for all subsequent times. The determination of 1v(0)2 is non-trivial because only those turbulent eddies which are smaller than the source should be represented, so that for sources smaller than the largest turbulent eddies, 1v(0)2 is less than the local #uid velocity variance. Here 1v(0)2 is evaluated follow ing the approach of de Haan and Rotach (1998). The pu!s described by Eq. (6) e!ectively extend to arbitrarily large values of "z" whereas in reality the presence of the ground creates a barrier to the pu!. When predicting mean concentrations, the presence of the ground can be represented, in part, by &folding' pu!s at some height z (de"ned below), so that there is a contri bution to the predicted mean concentration 1c(x, y, z)2 at x, y, z from 1c (x, y , 2z !z )2. The extension of this approach to the prediction of rms #uctuating concentrations is, however, problematic because near the ground i is suppressed in a manner which is not readily para meterized. Near the ground, the high mean shear and turbulence intensity rapidly distort plume "laments, stretching them to increase surface area available for molecular di!usion e!ects to dissipate concentration #uctuations. Because the dissipation of concentration near the surface is neglected in the model, it is anticipated that the rms #uctuating concentrations will tend to be overpredicted close to the ground. This tendency of the model to overpredict rms #uctuating concentrations will be further worsened by &pu!-folding' because there will be a non-physical contribution to the total rms #uctuating concentration from unsuppresssed #uctuating concentrations close to the ground. However, pu!-folding is necessary to ensure mass conservation. A related problem, is that measurements of the statistical properties of the #ow, required here as model inputs, were not made below some height z , and so it is not possible to simulate pu! centroid trajectories below this height. In view of this and because the #ow close to the ground e!ectively
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creates a barrier to pu! centroid motions, in the numerical simulations, pu! centroids were re#ected at height z . This is, if during a timestep, a pu! centroid was transported to a location z (z , then it was re#ected to a location 2z !z and the signs of the #uctuating com ponents of pu! centroid velocity were reversed. For Gaussian turbulence, as in the cases studied later in Section 3, this re#ection scheme does not lead to a violation of the well-mixed condition (Wilson and Flesch, 1993).
wind data for total intensity. Eq. (7) based as it is on wind-tunnel data at the height of an elevated source, is not appropriate in the far "eld nor is it strictly applicable near the ground. No attempt is made here to parameterize near-ground e!ect. For the intermediate-range, we assume, consistent with experimental observations (Fackrell and Robins, 1982b), that i "0.45.
2.3. Intensity of in-plume concentrations
3.1. Compact-area source in a surface layer with neutral stability
The intensity of in-plume concentrations will change with downwind distance with the evolving in-plume rms #uctuating concentration. In the near "eld, the intensity of in-plume concentrations are well approximated, for reasons stated above, by the intensities of conditional concentrations. Using the experimentally observed balance between advection and dissipation, and a simple model for the dissipation timescale, Wilson et al. (1982,1985a,b) suggested that the intensity of conditional concentrations at the source height is given 0.34j i " , j#0.0144
(7)
where j"((p p )/H is the along-wind distance in W X plume coordinates. The virtual origin at j "0.0144 accounts for the e!ect of source size on the contribution of meandering to the total concentration variance and the empirical constant, 0.34, was found by "tting along-
3. Model comparisons with experimental data
The most comprehensive experimental study of concentration #uctuations is that of Fackrell and Robins (1982b). They measured mean concentrations and rms #uctuating concentrations of tracers dispersing from ground-level and elevated compact-area sources, with a variety of sizes, into a surface layer with neutral stability, in a wind tunnel. Here attention is focused on the elevated sources. The scale of the turbulence, K, is taken to be the &mixing length', which near to the surface is iz. In accord which previous studies (Batchelor, 1950; Sykes, 1988) which indicate that a&O(0.1), with both Gi!ord's model and its extension, optimal agreement with the experimental dispersion data of Fackrell and Robins (1982b) was obtained when a"0.12. This is the value of a used to obtained the model predictions discussed below. Fackrell and Robins (1982a) did not report on properties of the air #ow below z"0.05H, where H is the height of the
Fig. 1. Comparison of predicted and measured (symbols; Fackrell and Robins, 1982a,b) relative intensities of concentration #uctuations for elevated compact-area sources of sizes p /H "0.013 (x), 0.039 (£), 0.066 (䊐), 0.11 (䉫), and 0.154 (*) at height H "0.19H. H is the boundary-layer height. Predictions are shown for Gi!ord's model (dashed-lines) and its extension (solid-lines) using the near "eld (left) and the intermediate forms (right) of the modelled in-plume concentration intensities.
A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
boundary layer. Consequently in the numerical simulations, pu! centroids were re#ected at z "0.05H. Fig. 1 shows a comparison of predicted and measured maximum rms #uctuating concentration, p , at x-loca tions downwind of the source location divided by the maximum mean concentration, C , at the same x locations; the maxima are not always at the same location in the transverse plane. Comparisons are shown for di!erent sizes of elevated source. Gi!ord's model is seen to capture of the major features of the experimental data. Particularly well represented is the tendency for larger values of rms #uctuating concentrations to occur at earlier locations with smaller source sizes and the dependency of the intensity of #uctuating concentrations, p /C , on source size. These dependencies on source
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size, which extend over a considerable fetch, arise because the source size is typically so small that it takes time for the relative dispersion (even with an x growth) to make a di!erence. In contrast, mean concentrations are largely independent of the pu! width and so are largely independent of source size. The experimental data show a somewhat smaller variation with source size than do the predictions obtained by Gi!ord's model. As remarked by Fackrell and Robins (1982a), this suggests that internal structure of the plume is not irrelevant for the larger sources. Further support for this view comes from comparisons of predictions obtained using the extension of Gi!ord's model with the experimental data. These predictions, which take a partial account of the internal plume structure are in better agreement with experiment
Fig. 2. Comparison of predicted and measured (symbols; Raupach and Legg, 1983) normalized mean temperatures and normalized rms #uctuating temperatures at locations x/H "2.5 (solid-line, x), 7.5 (dashed-line, 䊐) and 15.0 (dotted-line, £) of an elevated line-source at height H "0.11H, where H is the boundary-layer height. Model predictions are shown for Gi!ord's model (left) and its extension (right) with the intermediate-range1 form for the modelled in-plume concentration intensity. Predictions for mean concentrations are the same for both models. Mean temperatures and temperature variances were rendered non-dimensional using h "Q /oc H u where Q is the H electrical heating power supplied to the source, o and c are the density and speci"c heat of the air at constant pressure, and u ";(H ).
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than those obtained with Gi!ord's model and show a variation with source size comparable to that found in experiment. The greatest improvement over Gi!ord's model is obtained in the near "eld for the largest source size. For example, at x/H"2 Gi!ord's model underpredicts the intensity of concentrations arising from the largest source by about a factor of about two whilst the new model is in close agreement with experiment. As expected in the near "eld, di!erences between predictions obtained using the two models decrease with decreasing source size, i.e. with decreasing contribution to concentration #uctuations from in-plume concentration relative to plume meandering. Not surprisingly, Fig. 1 shows that with the near-"eld form of i , the close correspondence between the exten sion of Gi!ord's model and experiment breaks down at downwind distances x/H+2 (travel times t+¹ , # the Eulerian timescale), the location at which the tracer "rst reaches the ground in signi"cant quantities. For 2(x/H(7, Gi!ord's model and its extension with i "0.45 are both in reasonable agreement with experi ment. Asymptotically, however, Gi!ord's model predicts wrongly that the intensity of concentration #uctuations i"0 whilst its extension, by construction, predicts that i"i "0.45. 3.2. Line source in a surface layer with neutral stability In the wind-tunnel experiments of Raupach and Legg (1983), heat was released as a passive tracer from an elevated line source stretched across a surface layer with neutral stability. The scale of the turbulence is taken to be K"iz. Optimal model agreement with the experimental data is obtained when a"0.4, which is the value of a used to obtain the predictions described below. This estimate for a is somewhat greater than the estimate of a"0.12 obtained for the case of a compact-area source and simply re#ects the fact that there are di!erences between the total dispersion for area and line sources. In the numerical simulations, pu! centroids were re#ected at a height d #z , where d is the displacement height and z is the roughness length. Shown in Fig. 2 are comparisons of predicted and measured values of mean temperatures, h, and rms #uctuating temperatures, p . Predictions are shown for F Gi!ord's model and its extension with the &intermediate range' form for i . The &near "eld' form for i has not been utilized because it is only applicable for point sources. It is seen that p are better predicted by the extension F of Gi!ord's model, which accounts for internal plume structure. For example, at the source height, H , and at locations x/H "7.5 and 15 downwind of the source, Gi!ord's model underpredicts p by approximately 50%, F whilst the extended model underpredicts p by only 20%. F This suggests that the modelled internal plume structure
Fig. 3. Comparison of predicted and measured (symbols; Raupach and Legg, 1983) relative intensities of temperature #uctuations for elevated line source at height H "0.11H where H is the boundary-layer height. Predictions are shown for Gi!ord's model (dashed-line) and its extension (solid-line) using i "0.45.
accounts for some of the underprediction, by Gi!ord's model, of p . Close to the source (x/H "2.5), however, F the extended model overpredicts p . This is most probF ably a consequence of applying the &intermediate range' form of i in the &near "eld'. As anticipated, both Gi!ord's model and its extension are seen to overpredict p close to the ground. A related F de"ciency of both models is the predicted minima and secondary maxima in p /h , which are not seen in
experiment (see Fig. 3). This is a consequence of plume interactions with the ground, which "rst become signi"cant at x/H +6. That is, in the models there are unrealistically large contributions to the total rms #uctuating temperature from the folded portions of pu!s those #uctuations have not been suppressed. When &pu!folding' is not invoked there is no minima or secondary maxima in the predicted values of p /h , and the
extended model is found to be in close agreement with the experiment.
4. Summary Gi!ord's meandering plume has been extended to account for internal plume structure. The approach combines a good parameterization of in-plume concentration statistics with an appropriate model for plume meander. The greatest strength of the model is its ability to simulate most of the processes that contribute to concentration #uctuations while retaining a simple form. Its major weakness is the limitation to the prediction of the "rst two moments of concentration rather intermittency
A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
factors and peak concentrations, and there is a clear need for a better, more generally applicable, parameterization of i . Acknowledgements I acknowledge with gratitude the many useful comments and suggestions made by two anonymous referees.
References Bara, B.M., Wilson, D.J., Zelt, B.W., 1992. Concentration #uctuations pro"les from a water channel simulation of a groundlevel release. Atmospheric Environment 26A, 1053}1062. Batchelor, G.B., 1950. The application of the similarity theory of turbulence to atmospheric di!usion. Quarterly Journal of Royal Meteorological Society 76, 133}146. Deardor!, J.W., Willis, G.E., 1984. Ground-level concentration #uctuations from a buoyant and non-buoyant source within a laboratory convective mixed layer. Atmospheric Environment 18, 1297}1309. Fackrell, J.E., Robins, A.G., 1982a. The e!ects of source size on concentration #uctuations in plumes. Boundary-Layer Meteorology 22, 335}350. Fackrell, J.E., Robins, A.G., 1982b. Concentration #uctuations and #uxes in plumes from point sources in a turbulent boundary layer. Journal of Fluid Mechanics 117, 1}26. Gi!ord, F.A., 1959. Statistical properties of a #uctuation plume dispersion model. Advances in Geophysics 6, 117}138. de Haan, P., Rotach, M.W., 1998. A novel approach to atmospheric dispersion modelling: the pu!-particle model. Quarterly Journal of Royal Meteorological Society 124, 2771}2792. Hay, J.S., Pasquill, F., 1959. Di!usion from a continuous source in relation to the spectrum and scale of turbulence. Advances in Geophysics 6, 345}365. Kimura, F., Yoshikawa, T., Sato, J., 1981. Frequency distribution of the concentration on the center of mean plume axis in the surface layer. Papers in Meteorology and Geophysics 32, 149}154.
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Raupach, M.R., Legg, B.J., 1983. Turbulent dispersion from an elevated line source: measurements of wind-concentration moments and budgets. Journal of Fluid Mechanics 136, 111}137. Reynolds, A.M., 1999. On the application of a Lagrangian particle-pu! model to elevated sources in surface layers with neutral stability. Journal of Applied Meteorology, in press. Sawford, B.L., 1985. Concentration statistics for surface plumes in the atmospheric boundary layer. Seventh Symposium on Turbulence and Di!usion, Boulder, November 12}15, AMS, Boston, MA. Sykes, R.I., Lewellen, W.S., Parker, S.F., 1986. A Gaussian plume model of atmospheric dispersion based on secondorder closure. Journal of Climate Applied Meteorology 25, 322}331. Sykes, R.I., 1988. Concentration #uctuations in dispersing plumes. In: Venkatrum, Wyngaard (Eds.), Lecture Notes on Air Pollution Modelling. American Meteorological Society. Smith, F.B., Hay, J.S., 1961. The expansion of clusters of particles in the atmosphere. Quarterly Journal of the Royal Meteorological Society 87, 89}91. Thomson, D.J., 1990. A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. Journal of Fluid Mechanics 210, 113}153. Weil, J.C., 1994. A hybrid Lagrangian dispersion model for elevated sources in the convective boundary layer. Atmospheric Environment 28, 3433}3448. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1982. Concentration #uctuations in an elevated plume: a di!usion-dissipation approximation. Atmospheric Environment 16, 2581}2589. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1985a. Intermittency and conditionally-averaged concentration #uctuations statistics in plumes. Atmospheric Environment 19, 1053}1063. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1985b. Intermittency and conditionally-averaged concentration #uctuation statistics in plumes. Atmospheric Environment 19, 1053}1064. Wilson, J.D., Flesch, T.K., 1993. Flow boundaries in random#ight dispersion models: enforcing the well-mixed condition. Journal of Applied Meteorology 32, 1695}1707. Yee, E., Chan, R., Kosteniuk, P.R., Chandler, G.M., Biltoft, C.A., Bowers, J.F., 1994. Incorporation of internal #uctuations in a meandering plume model of concentration #uctuations. Boundary-Layer Meteorology 67, 11}39.
Representation of internal plume structure in Gi!ord's meandering plume model Andrew Michael Reynolds* Silsoe Research Institute, Wrest Park, Silsoe, Bedford MK45 4HS, UK Received 5 March 1999; received in revised form 5 November 1999; accepted 11 November 1999
Abstract Gi!ord's (1959. Advances in Geophysics 6, 117}138) meandering plume model is extended to account for internal plume structure. The applicability of the model is thereby extended to include the near "eld of large sources and the far "eld. Agreement with measured root-mean-square #uctuating concentrations of scalars dispersing from elevated compact-area and line sources into surface layers with neutral stability is shown to be as good or better than that obtained with Gi!ord's model. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Turbulent dispersion; Concentrations; Fluctuations; Meandering plume
1. Introduction The need to predict the #uctuations in concentration of scalars dispersing in turbulent #ows arises in numerous situations. It has for example relevance to: predicting the probability of exposure to a speci"ed level of concentration; the determination of reaction rates in turbulent chemical reactors; the prediction of malodour nuisance; predicting the ignition of potentially #ammable gas clouds; predicting the received dosage of dispersing toxic gases; and the simulation of &temperature noise' downwind of heat sources in turbulent #ows. The earliest and one of the most useful models for the prediction of concentration #uctuations is Gi!ord's (1959) meandering plume model. This model describes the concentrations caused by the meandering of a small &instantaneous' plume or &pu! ' by large eddies in a turbulent #ow. There are, however, several restrictive assumptions in Gi!ord's model, the most serious of which is to neglect concentration #uctuations within pu!s. This limits the applicability of the model to short distances downwind of small sources, where it is the meandering of
* Tel.: #44-1525-860000; fax: #44-1525-860156. E-mail address: [email protected] (A.M. Reynolds)
pu!s rather than the concentration #uctuations within pu!s that is the dominant source of concentration #uctuations. In this situation, Gi!ord's model is very useful and found to be in good agreement with measured near"eld concentration #uctuations in surface layers with neutral stability (Fackrell and Robins, 1982a) and the atmospheric boundary layer (Kimura et al., 1981). An extension of the model to include buoyant plume rise has also been found to be in reasonable agreement with measured concentration #uctuations in laboratory-scale convective boundary layers (Weil, 1994). In the far "eld, the model predicts wrongly that the intensity of concentration #uctuations vanishes whilst, for short distances downwind of large sources, root-mean-square (rms) #uctuating concentrations are signi"cantly underpredicted; e.g. Fackrell and Robins (1982a) and Sykes et al. (1986) found that for the largest source they considered, rms #uctuating concentrations were underpredicted by about a factor of 2. In the next section an extension of Gi!ord's model is presented which allows for a representation of internal plume structure (in-plume concentration #uctuations). The applicability of Gi!ord's model is thereby extended to include the near "eld of large sources and the far "eld in general. The approach is akin to that of Yee et al. (1994), but here plume meander is calculated rather than being a model input, making the model is applicable to inhomogeneous turbulence.
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2. Gi4ord's model and its extension 2.1. Theory The quantities of key importance in the formulation of Gi!ord's model and its extension are the probability density function (pdf ) for concentration at any point (x, y, z), p(c, x, y, z), the pdf for in-plume concentration, p (c, y , z , x), and the pdf for the location of a pu! centroid p (y , z , x). Here x is the downwind position relative to the source, and y and z represent position in the crosswind plane relative to the position of the pu! centroid at y and z . These three pdf's are linked by the convolution relation
p(c, x, y, z)" p (c, y , z , x)p (y , z , x) dy dz .
(1)
The corresponding moments of concentration are given by
1cL(x, y, z)2" cLp(c, x, y, z) dc " 1cL (y , z , x)2p (y , z , x) dy dz ,
(2)
where 1cL (y , z , x)2 are the moments of in-plume con centration, which are prescribed by
1cL (y , z , x)2" cLp (c, y , z , x) dc.
(3)
In Gi!ord's model, in-plume concentration #uctuations are ignored and the in-plume concentration is assumed to be exactly equal to the ensemble expectation 1c (y , z , x)2, so that p (y , z , x)"d(c!1c(y , z , x)2), (4) where d is a Dirac delta function. Thus, according to Gi!ord's model the moments of in-plume concentration are given by langle 1cL(y , z , x2"1c(y , z , x)2L with the consequence that, contrary to experimental observations (Fackrell and Robins, 1982b), the intensity i "p /1c 2 of in-plume concentrations vanishes. A natural extension of Gi!ord's model, which allows for a representation of internal plume structure, is to use a more realistic pdf for in-plume concentrations in place of the delta-function in Eq. (4), or more simply, but equivalently when predicting rms #uctuating concentrations, is to use a more realistic intensity of in-plume concentration, i , in place of i "0. This is an attractive possibility because the intensities in-plume concentration are much less variable than the intensities of total concentration c(x, y, z) and hence more readily characterized and represented by simple model distributions. For example, in the near "eld of su$ciently small sources, zeroes in concentration arise predominately from inter-
mittency (i.e. meandering of the whole plume by turbulent eddies comparable to or larger than the plume size) rather than by the internal #uctuations in concentration. Consequently, in this regime, in-plume concentration statistics are expected to be well approximated by conditional concentration statistics, which have the zero-concentration intervals censored. Also the measured intensities of conditional concentrations are found to be only weakly dependent upon source size and remain approximately constant along y and z pro"les through plumes from elevated sources (Wilson et al., 1985a,b; Sawford, 1985). The tendency of conditional intensity to establish a universal structure is also demonstrated by concentration #uctuation data from the laboratory-scale convective boundary-layer experiments of Deardor! and Willis (1984), who studied plume dispersion from buoyant and non-buoyant sources. In the far "eld, where concentration #uctuations are dominated by internal #uctuations in concentration rather than by plume meander, experimental measurements of concentration intensities are scarce. However, the experimental dispersion data of Fackrell and Robins (1982b) for a surface layer with neutral stability, suggests that in the far-"eld and independent of source size, i +0.45. A non-zero value of i implies that 1c (x, y, z)2" (i#1)1c (x, y, z)2 rather than 1c (x, y, z)2" 1c (x, y, z)2 as in by Gi!ord's model. Here Gi!ord's model is extended to account for i O0 by writing
1c(x, y, z)2" 1c (y , z , x)2p (y , z , x) dy dz , 1c(x, y, z)2" (i #1)1c (y , z , x)2p (y , z , x) dy dz . (5) It should be noted that Bara et al. (1992) used i and Gi!ord's model in a similar way but did not address internal concentration #uctuations per se. Gi!ord's model is semi-quantitative in the sense that it predicts concentration #uctuations in terms of the distribution of pu! centroids and the relative dispersion or spreading of a pu!, which must be modelled or parameterized to obtain fully quantitative predictions. The extension of Gi!ord's model proposed here also requires parameterization of the intensity of in-plume concentrations. Gi!ord assumed that p (y , z ;x) is Gaussian, which is appropriate for homogeneous turbulence. Here, for inhomogeneous turbulence, p (y , z ;x) is calculated from the trajectories of pu! centroids simulated using a Lagrangian stochastic model. A detailed description of this approach and the Lagrangian stochastic model can be found in Reynolds (1999). Reynolds (1999) showed that model predictions for the mean concentrations of scalars dispersing from elevated line and compact area into a surface layers with neutral stability are in good agreement with the experimental dispersion data of
A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
Fackrell and Robins (1982b) and Raupach and Legg (1983). Also well predicted are rms #uctuating concentrations in the near "eld for the line source and the smaller compact-area sources. Less well predicted, presumably because of the neglect of internal concentration #uctuations, are rms #uctuating concentrations in the far "eld and in the near "eld of the larger compact-area sources. The model for the spreading of a pu! and the parameterization of the intensity of in-plume concentrations are described in Sections 2.1 and 2.2. 2.2. In-plume concentrations In Gi!ord's model, the spatial distribution of concentration, c , in a pu! is assumed to be a non-random Gaussian distribution about the pu! centroid. That is Q c (x, y , z ; y , z )" 2n;p p W X (y !y ) (z !z ) ! , exp ! (6) 2p 2p W X where Q is the source strength, p and p are the rms W X sizes of the pu! in the y and z directions, and ; is the mean #uid velocity at the pu! centroid. The distinction between p and p allows for departure from axisymW X metry due to the anisotropy of turbulence in boundarylayers. Batchelor (1950) distinguished &near-"eld', &intermediate range' and &far-"eld' approximate descriptions of the rms growth in pu! size. For travel times t(p / 1v(0)2 , Batchelor (1950) showed that p (t)"p# 1v(0)2t, where p is the source size and v (0) is the initial velocity of pu! particles relative to the velocity of the centre of mass of the pu!. For the &intermediate range', Batchelor (1950) showed, for an instantaneous point source in homogeneous isotropic turbulence, that the rms pu! size at time t is given by p "(ae t) where a is a constant of O(0.1) and e is the mean rate of dissipation of turbulent kinetic energy. This result, which is strictly appropriate only when the size of the pu! lies in the inertial subrange, was subsequently generalized by Sykes (1988) to anisotopic turbulence and to sources of size p . Sykes proposed that p "(p #ap K\ t) where K is the background scale of the turbulence. Here, for inhomogeneous turbulence, it is assumed that Sykes proposal can be applied locally, so that p (t#*t)" (p (t) #ap K\ *t) where e and K are evaluated at the pu! centroid and p (0)"p . That is, at time t the pu! of size p (t) acts as a source from which the pu! at time t#Dt develops. Over the time-interval *t, the turbulence is taken to be homogeneous. In the &far "eld', where particles in the pu! are essentially moving independently, Batchelor (1950) found that p(t)"2p Jt, where p is the rms single-particle dispersion. However, the coe$cient 2 in the above (p(t)"2p ) should be
2541
omitted as Thomson (1990), shows from his two-particle model that p (t)&p Jt. It is doubtful whether the &far "eld' is attained in the wind-tunnel experiments of Fackrell and Robins (1982a) and, Raupach and Legg (1983), with which model predictions will be compared. This is simply because for both these experiments, the size of pu!s is predicted to remain smaller than or comparable in size to the &mixing length'. Fackrell and Robins (1982a) also found, in their numerical simulations of their experiments (Fackrell and Robins, 1982b) that, the far "eld regime in which the instantaneous pu! size grows as t was not attained. Fackrell and Robins (1982a) adopted the more complicated parameterization of pu! size proposed by Hay and Pasquill (1959) and Smith and Hay (1961). Consequently, in the present study only the &near "eld' and &intermediate range' forms for pu! growth are used. That is, the &near "eld' form of Batchelor (1950) is utilized for times t( p /1v(0)2 , and then the 'intermediate range' form of Sykes (1988) is used for all subsequent times. The determination of 1v(0)2 is non-trivial because only those turbulent eddies which are smaller than the source should be represented, so that for sources smaller than the largest turbulent eddies, 1v(0)2 is less than the local #uid velocity variance. Here 1v(0)2 is evaluated follow ing the approach of de Haan and Rotach (1998). The pu!s described by Eq. (6) e!ectively extend to arbitrarily large values of "z" whereas in reality the presence of the ground creates a barrier to the pu!. When predicting mean concentrations, the presence of the ground can be represented, in part, by &folding' pu!s at some height z (de"ned below), so that there is a contri bution to the predicted mean concentration 1c(x, y, z)2 at x, y, z from 1c (x, y , 2z !z )2. The extension of this approach to the prediction of rms #uctuating concentrations is, however, problematic because near the ground i is suppressed in a manner which is not readily para meterized. Near the ground, the high mean shear and turbulence intensity rapidly distort plume "laments, stretching them to increase surface area available for molecular di!usion e!ects to dissipate concentration #uctuations. Because the dissipation of concentration near the surface is neglected in the model, it is anticipated that the rms #uctuating concentrations will tend to be overpredicted close to the ground. This tendency of the model to overpredict rms #uctuating concentrations will be further worsened by &pu!-folding' because there will be a non-physical contribution to the total rms #uctuating concentration from unsuppresssed #uctuating concentrations close to the ground. However, pu!-folding is necessary to ensure mass conservation. A related problem, is that measurements of the statistical properties of the #ow, required here as model inputs, were not made below some height z , and so it is not possible to simulate pu! centroid trajectories below this height. In view of this and because the #ow close to the ground e!ectively
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creates a barrier to pu! centroid motions, in the numerical simulations, pu! centroids were re#ected at height z . This is, if during a timestep, a pu! centroid was transported to a location z (z , then it was re#ected to a location 2z !z and the signs of the #uctuating com ponents of pu! centroid velocity were reversed. For Gaussian turbulence, as in the cases studied later in Section 3, this re#ection scheme does not lead to a violation of the well-mixed condition (Wilson and Flesch, 1993).
wind data for total intensity. Eq. (7) based as it is on wind-tunnel data at the height of an elevated source, is not appropriate in the far "eld nor is it strictly applicable near the ground. No attempt is made here to parameterize near-ground e!ect. For the intermediate-range, we assume, consistent with experimental observations (Fackrell and Robins, 1982b), that i "0.45.
2.3. Intensity of in-plume concentrations
3.1. Compact-area source in a surface layer with neutral stability
The intensity of in-plume concentrations will change with downwind distance with the evolving in-plume rms #uctuating concentration. In the near "eld, the intensity of in-plume concentrations are well approximated, for reasons stated above, by the intensities of conditional concentrations. Using the experimentally observed balance between advection and dissipation, and a simple model for the dissipation timescale, Wilson et al. (1982,1985a,b) suggested that the intensity of conditional concentrations at the source height is given 0.34j i " , j#0.0144
(7)
where j"((p p )/H is the along-wind distance in W X plume coordinates. The virtual origin at j "0.0144 accounts for the e!ect of source size on the contribution of meandering to the total concentration variance and the empirical constant, 0.34, was found by "tting along-
3. Model comparisons with experimental data
The most comprehensive experimental study of concentration #uctuations is that of Fackrell and Robins (1982b). They measured mean concentrations and rms #uctuating concentrations of tracers dispersing from ground-level and elevated compact-area sources, with a variety of sizes, into a surface layer with neutral stability, in a wind tunnel. Here attention is focused on the elevated sources. The scale of the turbulence, K, is taken to be the &mixing length', which near to the surface is iz. In accord which previous studies (Batchelor, 1950; Sykes, 1988) which indicate that a&O(0.1), with both Gi!ord's model and its extension, optimal agreement with the experimental dispersion data of Fackrell and Robins (1982b) was obtained when a"0.12. This is the value of a used to obtained the model predictions discussed below. Fackrell and Robins (1982a) did not report on properties of the air #ow below z"0.05H, where H is the height of the
Fig. 1. Comparison of predicted and measured (symbols; Fackrell and Robins, 1982a,b) relative intensities of concentration #uctuations for elevated compact-area sources of sizes p /H "0.013 (x), 0.039 (£), 0.066 (䊐), 0.11 (䉫), and 0.154 (*) at height H "0.19H. H is the boundary-layer height. Predictions are shown for Gi!ord's model (dashed-lines) and its extension (solid-lines) using the near "eld (left) and the intermediate forms (right) of the modelled in-plume concentration intensities.
A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
boundary layer. Consequently in the numerical simulations, pu! centroids were re#ected at z "0.05H. Fig. 1 shows a comparison of predicted and measured maximum rms #uctuating concentration, p , at x-loca tions downwind of the source location divided by the maximum mean concentration, C , at the same x locations; the maxima are not always at the same location in the transverse plane. Comparisons are shown for di!erent sizes of elevated source. Gi!ord's model is seen to capture of the major features of the experimental data. Particularly well represented is the tendency for larger values of rms #uctuating concentrations to occur at earlier locations with smaller source sizes and the dependency of the intensity of #uctuating concentrations, p /C , on source size. These dependencies on source
2543
size, which extend over a considerable fetch, arise because the source size is typically so small that it takes time for the relative dispersion (even with an x growth) to make a di!erence. In contrast, mean concentrations are largely independent of the pu! width and so are largely independent of source size. The experimental data show a somewhat smaller variation with source size than do the predictions obtained by Gi!ord's model. As remarked by Fackrell and Robins (1982a), this suggests that internal structure of the plume is not irrelevant for the larger sources. Further support for this view comes from comparisons of predictions obtained using the extension of Gi!ord's model with the experimental data. These predictions, which take a partial account of the internal plume structure are in better agreement with experiment
Fig. 2. Comparison of predicted and measured (symbols; Raupach and Legg, 1983) normalized mean temperatures and normalized rms #uctuating temperatures at locations x/H "2.5 (solid-line, x), 7.5 (dashed-line, 䊐) and 15.0 (dotted-line, £) of an elevated line-source at height H "0.11H, where H is the boundary-layer height. Model predictions are shown for Gi!ord's model (left) and its extension (right) with the intermediate-range1 form for the modelled in-plume concentration intensity. Predictions for mean concentrations are the same for both models. Mean temperatures and temperature variances were rendered non-dimensional using h "Q /oc H u where Q is the H electrical heating power supplied to the source, o and c are the density and speci"c heat of the air at constant pressure, and u ";(H ).
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A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
than those obtained with Gi!ord's model and show a variation with source size comparable to that found in experiment. The greatest improvement over Gi!ord's model is obtained in the near "eld for the largest source size. For example, at x/H"2 Gi!ord's model underpredicts the intensity of concentrations arising from the largest source by about a factor of about two whilst the new model is in close agreement with experiment. As expected in the near "eld, di!erences between predictions obtained using the two models decrease with decreasing source size, i.e. with decreasing contribution to concentration #uctuations from in-plume concentration relative to plume meandering. Not surprisingly, Fig. 1 shows that with the near-"eld form of i , the close correspondence between the exten sion of Gi!ord's model and experiment breaks down at downwind distances x/H+2 (travel times t+¹ , # the Eulerian timescale), the location at which the tracer "rst reaches the ground in signi"cant quantities. For 2(x/H(7, Gi!ord's model and its extension with i "0.45 are both in reasonable agreement with experi ment. Asymptotically, however, Gi!ord's model predicts wrongly that the intensity of concentration #uctuations i"0 whilst its extension, by construction, predicts that i"i "0.45. 3.2. Line source in a surface layer with neutral stability In the wind-tunnel experiments of Raupach and Legg (1983), heat was released as a passive tracer from an elevated line source stretched across a surface layer with neutral stability. The scale of the turbulence is taken to be K"iz. Optimal model agreement with the experimental data is obtained when a"0.4, which is the value of a used to obtain the predictions described below. This estimate for a is somewhat greater than the estimate of a"0.12 obtained for the case of a compact-area source and simply re#ects the fact that there are di!erences between the total dispersion for area and line sources. In the numerical simulations, pu! centroids were re#ected at a height d #z , where d is the displacement height and z is the roughness length. Shown in Fig. 2 are comparisons of predicted and measured values of mean temperatures, h, and rms #uctuating temperatures, p . Predictions are shown for F Gi!ord's model and its extension with the &intermediate range' form for i . The &near "eld' form for i has not been utilized because it is only applicable for point sources. It is seen that p are better predicted by the extension F of Gi!ord's model, which accounts for internal plume structure. For example, at the source height, H , and at locations x/H "7.5 and 15 downwind of the source, Gi!ord's model underpredicts p by approximately 50%, F whilst the extended model underpredicts p by only 20%. F This suggests that the modelled internal plume structure
Fig. 3. Comparison of predicted and measured (symbols; Raupach and Legg, 1983) relative intensities of temperature #uctuations for elevated line source at height H "0.11H where H is the boundary-layer height. Predictions are shown for Gi!ord's model (dashed-line) and its extension (solid-line) using i "0.45.
accounts for some of the underprediction, by Gi!ord's model, of p . Close to the source (x/H "2.5), however, F the extended model overpredicts p . This is most probF ably a consequence of applying the &intermediate range' form of i in the &near "eld'. As anticipated, both Gi!ord's model and its extension are seen to overpredict p close to the ground. A related F de"ciency of both models is the predicted minima and secondary maxima in p /h , which are not seen in
experiment (see Fig. 3). This is a consequence of plume interactions with the ground, which "rst become signi"cant at x/H +6. That is, in the models there are unrealistically large contributions to the total rms #uctuating temperature from the folded portions of pu!s those #uctuations have not been suppressed. When &pu!folding' is not invoked there is no minima or secondary maxima in the predicted values of p /h , and the
extended model is found to be in close agreement with the experiment.
4. Summary Gi!ord's meandering plume has been extended to account for internal plume structure. The approach combines a good parameterization of in-plume concentration statistics with an appropriate model for plume meander. The greatest strength of the model is its ability to simulate most of the processes that contribute to concentration #uctuations while retaining a simple form. Its major weakness is the limitation to the prediction of the "rst two moments of concentration rather intermittency
A.M. Reynolds / Atmospheric Environment 34 (2000) 2539}2545
factors and peak concentrations, and there is a clear need for a better, more generally applicable, parameterization of i . Acknowledgements I acknowledge with gratitude the many useful comments and suggestions made by two anonymous referees.
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Raupach, M.R., Legg, B.J., 1983. Turbulent dispersion from an elevated line source: measurements of wind-concentration moments and budgets. Journal of Fluid Mechanics 136, 111}137. Reynolds, A.M., 1999. On the application of a Lagrangian particle-pu! model to elevated sources in surface layers with neutral stability. Journal of Applied Meteorology, in press. Sawford, B.L., 1985. Concentration statistics for surface plumes in the atmospheric boundary layer. Seventh Symposium on Turbulence and Di!usion, Boulder, November 12}15, AMS, Boston, MA. Sykes, R.I., Lewellen, W.S., Parker, S.F., 1986. A Gaussian plume model of atmospheric dispersion based on secondorder closure. Journal of Climate Applied Meteorology 25, 322}331. Sykes, R.I., 1988. Concentration #uctuations in dispersing plumes. In: Venkatrum, Wyngaard (Eds.), Lecture Notes on Air Pollution Modelling. American Meteorological Society. Smith, F.B., Hay, J.S., 1961. The expansion of clusters of particles in the atmosphere. Quarterly Journal of the Royal Meteorological Society 87, 89}91. Thomson, D.J., 1990. A stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence, and its application to the problem of concentration variance. Journal of Fluid Mechanics 210, 113}153. Weil, J.C., 1994. A hybrid Lagrangian dispersion model for elevated sources in the convective boundary layer. Atmospheric Environment 28, 3433}3448. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1982. Concentration #uctuations in an elevated plume: a di!usion-dissipation approximation. Atmospheric Environment 16, 2581}2589. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1985a. Intermittency and conditionally-averaged concentration #uctuations statistics in plumes. Atmospheric Environment 19, 1053}1063. Wilson, D.J., Robins, A.G., Fackrell, J.E., 1985b. Intermittency and conditionally-averaged concentration #uctuation statistics in plumes. Atmospheric Environment 19, 1053}1064. Wilson, J.D., Flesch, T.K., 1993. Flow boundaries in random#ight dispersion models: enforcing the well-mixed condition. Journal of Applied Meteorology 32, 1695}1707. Yee, E., Chan, R., Kosteniuk, P.R., Chandler, G.M., Biltoft, C.A., Bowers, J.F., 1994. Incorporation of internal #uctuations in a meandering plume model of concentration #uctuations. Boundary-Layer Meteorology 67, 11}39.