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A simple and fast model to compute concentration moments in a convective boundary layer
Atmospheric Environment 36 (2002) 4717–4724
A simple and fast model to compute concentration moments in a convective boundary layer Massimo Cassiani, Umberto Giostra* Environmental Science Faculty, University of Urbino, Campus Scientifico Sogesta, 61029 Urbino Italy Received 10 September 2001; received in revised form 5 July 2002; accepted 15 July 2002
Abstract Recently, a modified meandering plume model for concentration fluctuations in a convective boundary layer has been formulated (Atmos. Environ. 34 (2000) 3599). This model is based on a hybrid Eulerian–Lagrangian approach and it accounts for the skewed and inhomogeneous turbulence characteristics of the convective flow. Using the same hypotheses, but eliminating the need for Lagrangian particle model, we propose a generalised approach, that only requires the knowledge of mean concentration field. The proposed model is independent from the method used to obtain the mean concentration field. The evaluation of the concentration fluctuation field needs a computational time of only few seconds on a standard PC. Therefore, the model is suitable for practical applications. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Air pollution modelling; Mean concentration field; Meandering plume model; Higher order concentration moment modelling
1. Introduction Dispersion in non-homogeneous turbulence is a stimulating topic. A convective boundary layer (CBL) can be seen as a prototype of non-homogeneous turbulence. Whereas modelling the mean concentration in a CBL is an almost settled matter, concentration fluctuations modelling is still an open argument, especially for models devoted to practical applications. In the past years, a number of models aimed at predicting vertical profiles of mean concentration in a CBL has been proposed, and a short list can be found in Cassiani and Giostra (2002; hereinafter CG). However, there are many circumstances in which concentration fluctuations have to be modelled. That is the case, for example, when evaluation of the probability of exceeding specified concentration limits is needed. In such a situation, to our knowledge, only a few models are available: (i) large eddy simulation (LES) techniques *Corresponding author. Tel./fax: +39-0722-304265. E-mail address: [email protected] (U. Giostra).
(Henn and Sykes, 1992); (ii) Weil’s (1994) hybrid Lagrangian approach; and (iii) the meandering plume model proposed by Luhar, Hibberd, and Borgas (2000, hereinafter LHB). LES techniques are definitely a reliable approach but they are computationally very expensive, and are thus not effective for most purposes. The Weil (1994) and LHB models use the concept of meandering (Gifford, 1959), that is the simulation of the trajectories of the centre-of-mass of instantaneous plumes coupled with relative dispersion. Weil’s (1994) model shows some inconsistencies. First of all, since the model uses particle trajectories to simulate the plume meandering, it results in a double count of the relative dispersion. Secondly, the model is based on the further assumption of neglecting the inplume relative fluctuations. The LHB model has been shown to be efficient and more physically consistent. The model includes in-plume fluctuations using a gamma probability density function (pdf). The meander trajectories are evaluated from the particle trajectories, in order to preserve the total
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diffusion. Moreover, the LHB model uses a skewed relative dispersion, allowing a correct reproduction of the third moment of the total dispersion. However, the hybrid Eulerian–Lagrangian approach, on which the LHB model is based, makes the model itself a bit laborious and computationally expensive and useful only coupled with a Lagrangian stochastic model. In this paper, starting from the LHB procedure, we suggest a generalised method for the evaluation of the second and higher order concentration moments. The improved model does not require any evaluation through Lagrangian stochastic model, but it works using a mean concentration field derived from measurements or numerically evaluated. In this paper, we use the mean concentration field calculated as proposed in CG.
crosswind direction and using the expression derived from Taylor statistical theory: 2 s2y ¼ 2s2v TLv ðt=TLv 1 þ et=TLv Þ;
ð1Þ
where the Lagrangian time scale is TLv ¼ 2s2v =C0 e: According to LHB, the dimensionless velocity variance s2v =w2 ¼ 0:2; the Kolmogorov constant C0 ¼ 3 and the dimensionless dissipation ezi =w3 is chosen constant and equal to 0.4. Based on the crosswind Gaussian homogeneous assumption, the contribution to the higher concentration moments due to the crosswind meander can be calculated analytically, as shown in LHB model, summarised in the following section. 2.2. The LHB model
2. A model for the concentration moments First of all, we give some definitions and assumptions; subsequently, we recall the major aspects of LHB model; and, finally, we propose our generalisation. 2.1. Some assumptions and definitions
With the previous assumptions and following LHB, the probability density function pc ðc; t; y; zÞ of the concentration c in a fixed frame of reference (i.e., relative to the source) is written as Z zi Z N pc ðc; y; z; tÞ ¼ pcr ðc; t; y; z; ym ; zm Þ 0
N
pym ðym ; tÞ pzm ðzm ; tÞ dym dzm ; In the meandering plume approach, the ensemble dispersion of a plume is viewed as the sum of a number of instantaneous plumes. The motion of the centroid of each instantaneous plume is considered in a fixed coordinate system relative to the source, whereas the concentration distribution within the instantaneous plume is calculated relatively to the plume centroid. Following Gifford (1959), we assume the contributions due to meandering and to relative diffusion are statistically independent. Moreover, we assume that: (i) the plume meander in the crosswind direction is independent from that in the vertical direction; (ii) the relative dispersion is homogeneous in the vertical irrespective of the non-homogeneous turbulence; (iii) the tracer is passive and it is emitted from a continuous point source; (iv) the turbulence field is stationary and horizontally homogeneous; (v) the wind shear can be neglected. Therefore, the streamwise diffusion can be ignored in comparison with the mean advection. These last restrictions allow the use of Taylor frozen turbulence hypothesis, i.e., x ¼ Ut; where x is the downwind distance, U is the mean wind and t is the plume travel time. Recalling the assumption of independence between vertical and lateral diffusion, in order to predict the pdf of the spatial distribution of the instantaneous centroid, we need the crosswind cy and the vertically cz integrated concentration fields, instead of the full concentration field. The Gaussian crosswind spread sy can evaluated assuming a homogeneous Gaussian turbulence in the
where pcr is the pdf of the relative concentration field and pym and pzm are the pdfs of the position of the instantaneous plume centroid, ym and zm ; respectively. The concentration moments of order n are given by Z zi Z N Z N cn ðt; y; zÞ ¼ cn ðt; y; z; ym ; zm Þ 0
N
0
pcr ðc; t; y; z; ym ; zm Þ dc pym ðym ; tÞpzm ðzm ; tÞ dym dzm :
ð2Þ
Following Yee et al. (1994) and Yee and Wilson (2000), pcr can be represented by the gamma distribution, l1 ll c lc exp ; ð3Þ pcr ðc; t; y; z; ym ; zm Þ ¼ c%r GðlÞ c%r c%r 2 ; icr ð¼ scr ðt; y; zÞ=c%ðt; y; zÞÞ is the relative where l ¼ 1=icr concentration fluctuation intensity, GðlÞ is the gamma function, and c%r is the mean concentration relative to the instantaneous plume centroid. As in LHB, substituting Eq. (3) in Eq. (2) and integrating, it is possible to obtain: Z Z 1 Gðn þ lÞ zi N n cn ðt; y; zÞ ¼ n c%r ðt; y; z; ym ; zm Þ l GðlÞ N 0 ð4Þ pym ðym ; tÞpzm ðzm ; tÞ dym dzm :
It is possible to write c%r as a product between two term pyr and c%zr where " # 1 ðy ym Þ2 exp : ð5Þ pyr ¼ 2s2yr ð2pÞ1=2 syr
M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
c%zr ¼
2 N X X j¼1 K¼N
(
"
in Eq. (7) is evaluated at each z point as the mean of the concentration (raised to the power n) derived from 20 000 centroids. We suggest a different way to evaluate the integral in Eq. (7). In our opinion, our approach is more general and less laborious than the LHB one.
Qaj ð2pÞ1=2 sj u%
# ðz zm þ 2kzi z%j Þ2 exp 2s2j " # ðz zm þ 2kzi z%j Þ2 : þ exp 2s2j
ð6Þ
Here Q is the emission rate and N is the number of reflection. The other parameters are as explained in LHB and are here reported in Appendix A. Here, we just remark that the relative mean concentration c%zr is a sum of two Gaussians and is skewed, that sj are functions of szr (see Appendix B) and that the relative skewness Sr must be computed to calculate all the other parameters. We will show later how to compute the skewness Sr : The assumption of independence between relative expansion and meander leads to s2ym ¼ s2y s2yr : Moreover, we have supposed the total crosswind expansion to be a Gaussian with no boundary and variance from Eq. (1), so that pym can be easily made explicit. Therefore, the contribution to concentration moments from the lateral meander can be carried out analytically to yield (see LHB), cn ðt; y; zÞ ¼
syr 1 Gðn þ lÞ 1 pffiffiffiffiffiffi ln GðlÞ ð 2psyr Þn ðns2ym þ s2yr Þ1=2 " # ny2 exp 2ðns2ym þ s2yr Þ Z zi c%nzr ðt; z; zm Þpzm ðzm ; tÞ dzm :
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2.3. LHB model revisited Let us emphasise what in our opinion are the ‘weak’ aspects of the LHB approach: (i) the approach can be used only coupled with a Lagrangian stochastic model for the particle trajectories; (ii) the method is computationally very intense. On the contrary, our approach simply requires the knowledge of mean concentration field; moreover, its computational need is very modest. Once the vertical profile of the mean concentration is known, we can obviously compute its centroid z%; its variance s2z and its third centred moment z03 ¼ ðz z%Þ3 ; here the prime denotes a displacement from the mean value. Now, we note that the definition of the crosswindintegrated concentration, Z N cy ðz; tÞ ¼ cðt; % y; zÞ dy N
ð7Þ
0
The relative expansions szr and syr can be parameterised (see Appendix B), consequently in order to obtain cn ðt; y; zÞ we have just to evaluate the integral in Eq. (7). Let us summarise the procedure followed in LHB to evaluate this integral. LHB derive the trajectory of the instantaneous plume centre of mass from the particle trajectory generated by a single particle Lagrangian stochastic model using the linear transformation, 2 s ðtÞ s2zrr ðtÞ zm ðtÞ ¼ z ½zL ðtÞ zL ðtÞ þ zL ðtÞ; ð8Þ s2z ðtÞ where zm ðtÞ and zL ðtÞ are the instantaneous plume centroid and particle positions, s2z ðtÞ is the vertical mean plume variance, szrr ðtÞ is the relative expansion including boundary reflections (see Appendix B) and zL ðtÞ is the mean height of the mean plume. To calculate the value of zL ðtÞ and s2z ðtÞ; LHB use a pre-run of the Lagrangian model with 4000 particles. Then, they release 20 000 particles and derive at each time step 20 000 instantaneous centroids. Hence, each centroid is expanded using Eq. (6). Finally, the integral
corresponds to the definition of a marginal density function. We recall that we are neglecting the along wind dispersion, so that, apart from the normalization factor, the crosswind-integrated concentration corresponds to the probability density for the position z at time t; namely, pz ðz; tÞ: In particular, it results that, pz ðz; tÞ ¼ R zi 0
cy ðz; tÞ : cy ðz; tÞ dz
From this definition it is straightforward that, if we consider the crosswind integrated mean concentration field normalised over the well mixed state in the dimensionless domain z=zi ; then its value in each point is exactly the value of the pdf pz for this particular point z in the dimensionless domain. This is an obvious consideration, but it will be useful for subsequent discussion. Sampling the normalised crosswind integrated mean concentration field in N points, we know exactly the discrete value of pz dz in a point: that is, pz Dz; where Dz ¼ 1=N: Therefore z% ¼
N X
zi piz Dz;
ð9Þ
i¼1
where the superscript i is referred to the ith sampling point. Analogously, relationships can be obtained for the high order moments.
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Now, if we make the linear transformation 2 s ðtÞ s2zrr ðtÞ þ z%ðtÞ; zim ðtÞ ¼ ½zi ðtÞ z%ðtÞ z s2z ðtÞ
Therefore, we can calculate the relative skewness Sr ð10Þ
then the previous value piz Dz is exactly the value of pizm Dzm for an instantaneous plume centroid located at zim : We note that Dzm is smaller than Dz by the factor ðs2z ðtÞ s2zrr ðtÞÞ=s2z ðtÞ; so that the value pzm is in correspondence greater than pz : This is a ‘compression’ of the probability density field that reduces the variance conserving at the same time the skewness, the kurtosis and all the scaled moments. The resulting form of the probability density function is pzm ¼ 8 < 0 : pz Dz Dzm
ðin the compressed concentration fieldÞ:
N X ðzim z%m Þ2 pizm Dzm ;
ð11Þ
and N X ðzim z%m Þ3 pizm Dzm ; i¼1
z%m ¼ z%:
ð13Þ
Using the Eq. (6) for the mean concentration in a point z produced by a centroid in zm ; the integral in Eq. (7) is therefore simply discretised as N X
c%nzr ðt; z; zim Þpzm ðzim ; tÞDzm :
ð14Þ
i¼1
This method can be, of course, adopted irrespectively of the method used to obtain the mean concentration field. In such a sense, the method is a generalisation of the LHB approach, whereas from a physical point of view the method is equivalent to the LHB one.
3. Results
i¼1
z0m 3 ¼
z03 z03 m : s3zrr
ðout of the compressed concentration fieldÞ;
Once the probability density function is computed, we can calculate the value of s2zm ¼
Sr ¼
ð12Þ
As we have described in Section 2.3, the proposed method needs the mean concentration field in order to evaluate the higher order concentration moments. Here we use the mean concentration field obtained by the simple CG model. In the CG model, vertical profiles of the second and third vertical velocity moments as in LHB have been chosen. In order to be confident of the CG model performances, some basic results (used by the generalisation in Section 2.3) will be shortly discussed. Further details and validations on the CG model simulations can be found in CG. The classical dimensionless
Fig. 1. Contours of the dimensionless-integrated concentration predicted by the simple mean concentration model (d, e, f) and measured by WD (a, b, c) for the source heights ðzs =zi Þ ¼ 0:067 zi ; (left) 0:24 zi (centre) and 0:49 zi (right).
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Fig. 2. Variation of the normalised mean plume height with T: The lines are the predictions by the simple model the open symbols the tank data of WD.
crosswind integrated concentration fields measured by Willis and Deardorff (1976, 1978, 1981; hereinafter WD) are compared in Fig. 1 to the CG model evaluations. An overall agreement comes out from Fig. 1. The normalised mean plume height z%=zi as a function of the dimensionless travel time is reported in Fig. 2. Again, a comparison between the WD experiments and the CG model results is shown. The model results show an excellent agreement for the source at zs =zi ¼ 0:067; whereas there is some over prediction for the source at zs =zi ¼ 0:49 and, especially, at zs =zi ¼ 0:24: At large times, all the curves converge to 0:5 zi ; since the well-mixed condition is approached. Fig. 3 shows normalised total spread in the vertical direction sz =zi : A comparison between the proposed simple model, the Nieuwstadt (1992) LES simulation, and the classical WD experiments is shown. Following LHB, s0 is chosen equal to 0:02 zi in order to match the experiments. In Fig. 4, the normalised vertical spreads of the plume meander szm =zi is compared to the LES results (Nieuwstadt, 1992). There is a general over prediction, partially due to the over prediction of the total spread and partially due to the under prediction in the szrr (not shown here, see Fig. 2a in LHB for a comparison).
Fig. 3. Variation of the normalised total vertical spread. The lines are the predictions by the simple model. The open symbols the tank data of WD. The solid symbols are the LES prediction of Nieuwstadt (1992).
Fig. 4. Variation of the normalised vertical meander spread. The lines are the predictions by the simple model. The solid symbols are the LES prediction of Nieuwstadt (1992).
discussion about the choice of this parameter, see LHB. Following LHB, the concentration fluctuation intensity computed by the model Eq. (7) using Eq. (14)
3.1. Concentration fluctuation intensity ic ðt; y; zÞ ¼ 2 In order to determine the parameter lð¼ 1=icr Þ in Eq. (3), the in-plume fluctuation in the relative coordinate system ðicr Þ must be known. For a complete
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sc ðt; y; zÞ c%ðt; y; zÞ
is compared with the data of Deardorff and Willis (1984), focusing on data from their non-buoyant source
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Fig. 5. Near surface concentration fluctuation intensity ðic Þ as a function of T: zs =zi ¼ 0:22: The open symbols are the measurements of Deardorff and Willis (1984) averaged over the lateral location jyjosy =2 (triangles) and sy =2ojyjosy (circles). The crosses refer to the result from the LHB model for jyjosy =2 and icr ¼ 1: The lines are the results from our model for jyjosy =2 and icr ¼ 1 (continuous line) and icr as in Fig. 6 (dashed line).
experiment. We also choose s0 ¼ 0:003zi and zs ¼ 0:22 zi ; as in LHB. The results shown in Fig. 5 refer to near-surface ðz ¼ 0:08 zi Þ predicted (by our model and by LHB model) and measured fluctuation intensities. The measurements denoted by open triangles represent an average over jyjosy =2; and those denoted by open circles represent average over sy =2ojyjosy ; where sy is the total lateral spread. Crosses refer to the simulation by the LHB model with l ¼ 1: Recalling that 2 l ¼ 1=icr ; the continuous and dashed lines refer to our model predictions using respectively icr ¼ 1 and icr as deduced from a simple analytical fit of the curve proposed by LHB (see Fig. 6). Finally, in Fig. 7 a comparison between the concentration fluctuation intensity ic fields predicted by the LHB model and by our model (with the icr value as in Fig. 6) is shown. A substantial equivalence between the results obtained from the two models emerges. Obviously, different parameterisations for the relative expansion szrr ; as well as for the vertical velocity variance (s2w ) and vertical velocity third moment (w3 ) used in the mean concentration model (CG) can be used in order to improve the agreement with data and LES. However, here we preferred to use the same parameters proposed in LHB. Nevertheless, the LHB model and our generalisation exhibit a very good agreement in repro-
Fig. 6. Variation of the relative in-plume concentration fluctuation intensity ðicr Þ with T computed by LHB (crosses) and fitted by an analytic curve (continuous line).
Fig. 7. Total concentration fluctuation intensity ðic Þ at y ¼ 0 predicted by the LHB model (up) with icr as in Fig. 6 (crosses) and by our model (down) with icr as in Fig. 6 (continuous line).
ducing the second moment concentration field in convective condition, irrespective of the different formulations.
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4. Conclusion
s1 ¼ szr ½a2 =fa1 ð1 þ f 2 Þg1=2 ;
Following the skewed meandering plume approach proposed by LHB, we introduce a general method to compute the higher order concentration moments, given a mean concentration field. The requested mean concentration field can be obtained either from models or from experiments. The proposed model, as well as the LHB model, makes the limiting assumption of a homogeneous relative expansion. The LHB model, based on a hybrid Eulerian–Lagrangian approach, needs the further knowledge of the trajectories of the centroids of each instantaneous plume to compute the high order concentration fields. Our generalisation does not require any trajectories, thus relaxing the need for Lagrangian modelling. In such a way, the skewed meandering plume approach is referred to a typical meandering plume model (i.e. to a model based on fixed profiles of mean concentration), although the solution is not yet analytical. This method seems to be simpler and more efficient than the LHB one. The comparison of the proposed model results with experimental data set and with numerical (LES) results, shows good overall agreement. The proposed procedure for the vertical co-ordinate is general and can be easily adapted for the crosswind meander in case it is not possible to find an easy analytical solution. The proposed technique can be improved in order to consider a two-dimensional mean concentration field that includes correlation between lateral and vertical dispersion. On the contrary, to include a full nonhomogeneous relative expansion, the Lagrangian point of view used in the LHB approach must be preferred, although it is a waste of computer time if the homogeneous relative expansion is used. We emphasise the generality of our technique and the low demand on computational time, giving higher order concentration moments evaluated in few seconds on a standard PC.
s2 ¼ szr ½a1 =fa2 ð1 þ f 2 Þg1=2 ;
Acknowledgements Authors are grateful to B.L. Sawford for his helpful comments and to P. Franzese for fruitful discussions. They greatly contributed to the improvement of this work. This work was partially supported by the Italian National Research Council (CNR) in the frame of Arctic Project.
Appendix A z1 ¼ Sr f s1 =jSr j; z2 ¼ Sr f s2 =jSr j;
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a1 ¼ ½1 fr=ð4 þ rÞg1=2 =2; a2 ¼ 1 a1 ; r ¼ ½ð1 þ f 2 Þ3 Sr2 =½ð3 þ f 2 Þ2 f 2 ; f ¼ c4 jSr j1=3 ; c4 ¼ 2=3:
Appendix B Following LHB, the growth of the relative variance is divided in two regimes, 1 1 s2r ¼ s20 þ Ce ðes0 Þ2=3 t2 ðC0 þ 4Þet3 3 3 iftZ 5tots 1 ¼ s# 20 þ C# e ðes0 Þ2=3 t2 þ ar et3 3 if ts ot5tL ; ts ¼ ct ðs20 =eÞ1=3 ; 3 C# e ¼ Ce ct ðC0 þ 4 þ 3ar Þ; 2 s# 20 ¼ s20 =3½3 þ c2t ðCe C# e Þ c3t ðC0 þ 4 þ 3ar Þ; where tZ is the Kolmogorov time scale, TL is the Lagrangian integral time scale , Ce ¼ 9:8; ar ¼ 0:3; C0 ¼ 3; ct ¼ 1=3: Supposing that for large travel time the expansion follows the Taylor statistical theory, the expansion sr is modified for time greater than ts by s2zr ðtÞ ¼ s2r t¼ts s2r ðtÞ s2r t¼ts þ ; 2 4=5 g5=4 f1 þ ½ðs2r ðtÞ s2r jt¼ts Þ=C0 etTLW s2yr ðtÞ ¼ s2r t¼ts s2r ðtÞ s2r t¼ts þ ; 2 4=5 g5=4 f1 þ ½ðs2r ðtÞ s2r jt ¼ ts Þ=C0 etTLV with TLv ¼ 2s2v =C0 e and TLW ¼ 2w2av =C0 e where w2av ¼ 0:36 w2 : However, in the vertical direction a further interpolate expansion is considered including boundary reflections s2zrr ðtÞ ¼
s2r ðtÞ f1 þ ½ðs2r ðtÞ s2r jt¼ts Þ=z2eq 4=5 g5=4
;
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where z2eq ¼ 0:083 z2i is the equilibrium value of the total vertical mean square spread including boundary reflections.
References Cassiani, M., Giostra, U., 2002. A semi-analytical model for mean concentration in a convective boundary layer. Atmospheric Environment 36, 4707–4715. Deardorff, J.W., Willis, G.E., 1984. Ground level concentration fluctuations from a buoyant and a non-buoyant source within a laboratory convective mixed layer. Atmospheric Environment 18, 1297–1309. Gifford, F.A., 1959. Statistical properties of a fluctuating plume dispersion model. Advances in Geophysics 6, 117–137. Henn, D.S., Sykes, R.I., 1992. Large-Eddy simulation of dispersion in the convective boundary layer. Atmospheric Environment 26A, 3145–3159. Luhar, A.K., Hibberd, M.F., Borgas, M.S., 2000. A skewed meandering plume model for concentration statistics in the convective boundary layer. Atmospheric Environment 34, 3599–3616. Nieuwstadt, F.T.M., 1992. A large-eddy simulation of the line source in a convective atmospheric boundary layer—I.
Dispersion Characteristics. Atmospheric Environment 26A, 285–495. Weil, J.C., 1994. A hybrid Lagrangian dispersion model for elevated sources in the convective boundary layer. Atmospheric Environment 28, 3433–3448. Willis, G.E., Deardorff, J.W., 1976. A laboratory model of diffusion into the convective planetary boundary layer. Quarterly Journal of the Royal Meteorological Society 102, 427–445. Willis, G.E., Deardorff, J.W., 1978. A laboratory study of dispersion from an elevated source within a modelled convective planetary boundary layer. Atmospheric Environment 12, 1305–1311. Willis, G.E., Deardorff, J.W., 1981. A laboratory study of dispersion from a source in the middle of the convective mixed layer. Atmospheric Environment 15, 109–117. Yee, E., Wilson, D.J., 2000. A comparison of the detailed structure in dispersing tracer plumes measured in gridgenerated turbulence with a meandering plume model incorporating internal fluctuations. Boundary Layer Meteorology 94, 235–296. Yee, E., Chan, R., Kosteniuk, P.R., Chandler, G.M., Biltoft, C.A., Bowers, J.F., 1994. Incorporation of internal fluctuations in a meandering plume model of concentration fluctuations. Boundary Layer Meteorology 67, 11–39.
A simple and fast model to compute concentration moments in a convective boundary layer Massimo Cassiani, Umberto Giostra* Environmental Science Faculty, University of Urbino, Campus Scientifico Sogesta, 61029 Urbino Italy Received 10 September 2001; received in revised form 5 July 2002; accepted 15 July 2002
Abstract Recently, a modified meandering plume model for concentration fluctuations in a convective boundary layer has been formulated (Atmos. Environ. 34 (2000) 3599). This model is based on a hybrid Eulerian–Lagrangian approach and it accounts for the skewed and inhomogeneous turbulence characteristics of the convective flow. Using the same hypotheses, but eliminating the need for Lagrangian particle model, we propose a generalised approach, that only requires the knowledge of mean concentration field. The proposed model is independent from the method used to obtain the mean concentration field. The evaluation of the concentration fluctuation field needs a computational time of only few seconds on a standard PC. Therefore, the model is suitable for practical applications. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Air pollution modelling; Mean concentration field; Meandering plume model; Higher order concentration moment modelling
1. Introduction Dispersion in non-homogeneous turbulence is a stimulating topic. A convective boundary layer (CBL) can be seen as a prototype of non-homogeneous turbulence. Whereas modelling the mean concentration in a CBL is an almost settled matter, concentration fluctuations modelling is still an open argument, especially for models devoted to practical applications. In the past years, a number of models aimed at predicting vertical profiles of mean concentration in a CBL has been proposed, and a short list can be found in Cassiani and Giostra (2002; hereinafter CG). However, there are many circumstances in which concentration fluctuations have to be modelled. That is the case, for example, when evaluation of the probability of exceeding specified concentration limits is needed. In such a situation, to our knowledge, only a few models are available: (i) large eddy simulation (LES) techniques *Corresponding author. Tel./fax: +39-0722-304265. E-mail address: [email protected] (U. Giostra).
(Henn and Sykes, 1992); (ii) Weil’s (1994) hybrid Lagrangian approach; and (iii) the meandering plume model proposed by Luhar, Hibberd, and Borgas (2000, hereinafter LHB). LES techniques are definitely a reliable approach but they are computationally very expensive, and are thus not effective for most purposes. The Weil (1994) and LHB models use the concept of meandering (Gifford, 1959), that is the simulation of the trajectories of the centre-of-mass of instantaneous plumes coupled with relative dispersion. Weil’s (1994) model shows some inconsistencies. First of all, since the model uses particle trajectories to simulate the plume meandering, it results in a double count of the relative dispersion. Secondly, the model is based on the further assumption of neglecting the inplume relative fluctuations. The LHB model has been shown to be efficient and more physically consistent. The model includes in-plume fluctuations using a gamma probability density function (pdf). The meander trajectories are evaluated from the particle trajectories, in order to preserve the total
1352-2310/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 2 - 2 3 1 0 ( 0 2 ) 0 0 5 6 4 - 2
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diffusion. Moreover, the LHB model uses a skewed relative dispersion, allowing a correct reproduction of the third moment of the total dispersion. However, the hybrid Eulerian–Lagrangian approach, on which the LHB model is based, makes the model itself a bit laborious and computationally expensive and useful only coupled with a Lagrangian stochastic model. In this paper, starting from the LHB procedure, we suggest a generalised method for the evaluation of the second and higher order concentration moments. The improved model does not require any evaluation through Lagrangian stochastic model, but it works using a mean concentration field derived from measurements or numerically evaluated. In this paper, we use the mean concentration field calculated as proposed in CG.
crosswind direction and using the expression derived from Taylor statistical theory: 2 s2y ¼ 2s2v TLv ðt=TLv 1 þ et=TLv Þ;
ð1Þ
where the Lagrangian time scale is TLv ¼ 2s2v =C0 e: According to LHB, the dimensionless velocity variance s2v =w2 ¼ 0:2; the Kolmogorov constant C0 ¼ 3 and the dimensionless dissipation ezi =w3 is chosen constant and equal to 0.4. Based on the crosswind Gaussian homogeneous assumption, the contribution to the higher concentration moments due to the crosswind meander can be calculated analytically, as shown in LHB model, summarised in the following section. 2.2. The LHB model
2. A model for the concentration moments First of all, we give some definitions and assumptions; subsequently, we recall the major aspects of LHB model; and, finally, we propose our generalisation. 2.1. Some assumptions and definitions
With the previous assumptions and following LHB, the probability density function pc ðc; t; y; zÞ of the concentration c in a fixed frame of reference (i.e., relative to the source) is written as Z zi Z N pc ðc; y; z; tÞ ¼ pcr ðc; t; y; z; ym ; zm Þ 0
N
pym ðym ; tÞ pzm ðzm ; tÞ dym dzm ; In the meandering plume approach, the ensemble dispersion of a plume is viewed as the sum of a number of instantaneous plumes. The motion of the centroid of each instantaneous plume is considered in a fixed coordinate system relative to the source, whereas the concentration distribution within the instantaneous plume is calculated relatively to the plume centroid. Following Gifford (1959), we assume the contributions due to meandering and to relative diffusion are statistically independent. Moreover, we assume that: (i) the plume meander in the crosswind direction is independent from that in the vertical direction; (ii) the relative dispersion is homogeneous in the vertical irrespective of the non-homogeneous turbulence; (iii) the tracer is passive and it is emitted from a continuous point source; (iv) the turbulence field is stationary and horizontally homogeneous; (v) the wind shear can be neglected. Therefore, the streamwise diffusion can be ignored in comparison with the mean advection. These last restrictions allow the use of Taylor frozen turbulence hypothesis, i.e., x ¼ Ut; where x is the downwind distance, U is the mean wind and t is the plume travel time. Recalling the assumption of independence between vertical and lateral diffusion, in order to predict the pdf of the spatial distribution of the instantaneous centroid, we need the crosswind cy and the vertically cz integrated concentration fields, instead of the full concentration field. The Gaussian crosswind spread sy can evaluated assuming a homogeneous Gaussian turbulence in the
where pcr is the pdf of the relative concentration field and pym and pzm are the pdfs of the position of the instantaneous plume centroid, ym and zm ; respectively. The concentration moments of order n are given by Z zi Z N Z N cn ðt; y; zÞ ¼ cn ðt; y; z; ym ; zm Þ 0
N
0
pcr ðc; t; y; z; ym ; zm Þ dc pym ðym ; tÞpzm ðzm ; tÞ dym dzm :
ð2Þ
Following Yee et al. (1994) and Yee and Wilson (2000), pcr can be represented by the gamma distribution, l1 ll c lc exp ; ð3Þ pcr ðc; t; y; z; ym ; zm Þ ¼ c%r GðlÞ c%r c%r 2 ; icr ð¼ scr ðt; y; zÞ=c%ðt; y; zÞÞ is the relative where l ¼ 1=icr concentration fluctuation intensity, GðlÞ is the gamma function, and c%r is the mean concentration relative to the instantaneous plume centroid. As in LHB, substituting Eq. (3) in Eq. (2) and integrating, it is possible to obtain: Z Z 1 Gðn þ lÞ zi N n cn ðt; y; zÞ ¼ n c%r ðt; y; z; ym ; zm Þ l GðlÞ N 0 ð4Þ pym ðym ; tÞpzm ðzm ; tÞ dym dzm :
It is possible to write c%r as a product between two term pyr and c%zr where " # 1 ðy ym Þ2 exp : ð5Þ pyr ¼ 2s2yr ð2pÞ1=2 syr
M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
c%zr ¼
2 N X X j¼1 K¼N
(
"
in Eq. (7) is evaluated at each z point as the mean of the concentration (raised to the power n) derived from 20 000 centroids. We suggest a different way to evaluate the integral in Eq. (7). In our opinion, our approach is more general and less laborious than the LHB one.
Qaj ð2pÞ1=2 sj u%
# ðz zm þ 2kzi z%j Þ2 exp 2s2j " # ðz zm þ 2kzi z%j Þ2 : þ exp 2s2j
ð6Þ
Here Q is the emission rate and N is the number of reflection. The other parameters are as explained in LHB and are here reported in Appendix A. Here, we just remark that the relative mean concentration c%zr is a sum of two Gaussians and is skewed, that sj are functions of szr (see Appendix B) and that the relative skewness Sr must be computed to calculate all the other parameters. We will show later how to compute the skewness Sr : The assumption of independence between relative expansion and meander leads to s2ym ¼ s2y s2yr : Moreover, we have supposed the total crosswind expansion to be a Gaussian with no boundary and variance from Eq. (1), so that pym can be easily made explicit. Therefore, the contribution to concentration moments from the lateral meander can be carried out analytically to yield (see LHB), cn ðt; y; zÞ ¼
syr 1 Gðn þ lÞ 1 pffiffiffiffiffiffi ln GðlÞ ð 2psyr Þn ðns2ym þ s2yr Þ1=2 " # ny2 exp 2ðns2ym þ s2yr Þ Z zi c%nzr ðt; z; zm Þpzm ðzm ; tÞ dzm :
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2.3. LHB model revisited Let us emphasise what in our opinion are the ‘weak’ aspects of the LHB approach: (i) the approach can be used only coupled with a Lagrangian stochastic model for the particle trajectories; (ii) the method is computationally very intense. On the contrary, our approach simply requires the knowledge of mean concentration field; moreover, its computational need is very modest. Once the vertical profile of the mean concentration is known, we can obviously compute its centroid z%; its variance s2z and its third centred moment z03 ¼ ðz z%Þ3 ; here the prime denotes a displacement from the mean value. Now, we note that the definition of the crosswindintegrated concentration, Z N cy ðz; tÞ ¼ cðt; % y; zÞ dy N
ð7Þ
0
The relative expansions szr and syr can be parameterised (see Appendix B), consequently in order to obtain cn ðt; y; zÞ we have just to evaluate the integral in Eq. (7). Let us summarise the procedure followed in LHB to evaluate this integral. LHB derive the trajectory of the instantaneous plume centre of mass from the particle trajectory generated by a single particle Lagrangian stochastic model using the linear transformation, 2 s ðtÞ s2zrr ðtÞ zm ðtÞ ¼ z ½zL ðtÞ zL ðtÞ þ zL ðtÞ; ð8Þ s2z ðtÞ where zm ðtÞ and zL ðtÞ are the instantaneous plume centroid and particle positions, s2z ðtÞ is the vertical mean plume variance, szrr ðtÞ is the relative expansion including boundary reflections (see Appendix B) and zL ðtÞ is the mean height of the mean plume. To calculate the value of zL ðtÞ and s2z ðtÞ; LHB use a pre-run of the Lagrangian model with 4000 particles. Then, they release 20 000 particles and derive at each time step 20 000 instantaneous centroids. Hence, each centroid is expanded using Eq. (6). Finally, the integral
corresponds to the definition of a marginal density function. We recall that we are neglecting the along wind dispersion, so that, apart from the normalization factor, the crosswind-integrated concentration corresponds to the probability density for the position z at time t; namely, pz ðz; tÞ: In particular, it results that, pz ðz; tÞ ¼ R zi 0
cy ðz; tÞ : cy ðz; tÞ dz
From this definition it is straightforward that, if we consider the crosswind integrated mean concentration field normalised over the well mixed state in the dimensionless domain z=zi ; then its value in each point is exactly the value of the pdf pz for this particular point z in the dimensionless domain. This is an obvious consideration, but it will be useful for subsequent discussion. Sampling the normalised crosswind integrated mean concentration field in N points, we know exactly the discrete value of pz dz in a point: that is, pz Dz; where Dz ¼ 1=N: Therefore z% ¼
N X
zi piz Dz;
ð9Þ
i¼1
where the superscript i is referred to the ith sampling point. Analogously, relationships can be obtained for the high order moments.
M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
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Now, if we make the linear transformation 2 s ðtÞ s2zrr ðtÞ þ z%ðtÞ; zim ðtÞ ¼ ½zi ðtÞ z%ðtÞ z s2z ðtÞ
Therefore, we can calculate the relative skewness Sr ð10Þ
then the previous value piz Dz is exactly the value of pizm Dzm for an instantaneous plume centroid located at zim : We note that Dzm is smaller than Dz by the factor ðs2z ðtÞ s2zrr ðtÞÞ=s2z ðtÞ; so that the value pzm is in correspondence greater than pz : This is a ‘compression’ of the probability density field that reduces the variance conserving at the same time the skewness, the kurtosis and all the scaled moments. The resulting form of the probability density function is pzm ¼ 8 < 0 : pz Dz Dzm
ðin the compressed concentration fieldÞ:
N X ðzim z%m Þ2 pizm Dzm ;
ð11Þ
and N X ðzim z%m Þ3 pizm Dzm ; i¼1
z%m ¼ z%:
ð13Þ
Using the Eq. (6) for the mean concentration in a point z produced by a centroid in zm ; the integral in Eq. (7) is therefore simply discretised as N X
c%nzr ðt; z; zim Þpzm ðzim ; tÞDzm :
ð14Þ
i¼1
This method can be, of course, adopted irrespectively of the method used to obtain the mean concentration field. In such a sense, the method is a generalisation of the LHB approach, whereas from a physical point of view the method is equivalent to the LHB one.
3. Results
i¼1
z0m 3 ¼
z03 z03 m : s3zrr
ðout of the compressed concentration fieldÞ;
Once the probability density function is computed, we can calculate the value of s2zm ¼
Sr ¼
ð12Þ
As we have described in Section 2.3, the proposed method needs the mean concentration field in order to evaluate the higher order concentration moments. Here we use the mean concentration field obtained by the simple CG model. In the CG model, vertical profiles of the second and third vertical velocity moments as in LHB have been chosen. In order to be confident of the CG model performances, some basic results (used by the generalisation in Section 2.3) will be shortly discussed. Further details and validations on the CG model simulations can be found in CG. The classical dimensionless
Fig. 1. Contours of the dimensionless-integrated concentration predicted by the simple mean concentration model (d, e, f) and measured by WD (a, b, c) for the source heights ðzs =zi Þ ¼ 0:067 zi ; (left) 0:24 zi (centre) and 0:49 zi (right).
M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
Fig. 2. Variation of the normalised mean plume height with T: The lines are the predictions by the simple model the open symbols the tank data of WD.
crosswind integrated concentration fields measured by Willis and Deardorff (1976, 1978, 1981; hereinafter WD) are compared in Fig. 1 to the CG model evaluations. An overall agreement comes out from Fig. 1. The normalised mean plume height z%=zi as a function of the dimensionless travel time is reported in Fig. 2. Again, a comparison between the WD experiments and the CG model results is shown. The model results show an excellent agreement for the source at zs =zi ¼ 0:067; whereas there is some over prediction for the source at zs =zi ¼ 0:49 and, especially, at zs =zi ¼ 0:24: At large times, all the curves converge to 0:5 zi ; since the well-mixed condition is approached. Fig. 3 shows normalised total spread in the vertical direction sz =zi : A comparison between the proposed simple model, the Nieuwstadt (1992) LES simulation, and the classical WD experiments is shown. Following LHB, s0 is chosen equal to 0:02 zi in order to match the experiments. In Fig. 4, the normalised vertical spreads of the plume meander szm =zi is compared to the LES results (Nieuwstadt, 1992). There is a general over prediction, partially due to the over prediction of the total spread and partially due to the under prediction in the szrr (not shown here, see Fig. 2a in LHB for a comparison).
Fig. 3. Variation of the normalised total vertical spread. The lines are the predictions by the simple model. The open symbols the tank data of WD. The solid symbols are the LES prediction of Nieuwstadt (1992).
Fig. 4. Variation of the normalised vertical meander spread. The lines are the predictions by the simple model. The solid symbols are the LES prediction of Nieuwstadt (1992).
discussion about the choice of this parameter, see LHB. Following LHB, the concentration fluctuation intensity computed by the model Eq. (7) using Eq. (14)
3.1. Concentration fluctuation intensity ic ðt; y; zÞ ¼ 2 In order to determine the parameter lð¼ 1=icr Þ in Eq. (3), the in-plume fluctuation in the relative coordinate system ðicr Þ must be known. For a complete
4721
sc ðt; y; zÞ c%ðt; y; zÞ
is compared with the data of Deardorff and Willis (1984), focusing on data from their non-buoyant source
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M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
Fig. 5. Near surface concentration fluctuation intensity ðic Þ as a function of T: zs =zi ¼ 0:22: The open symbols are the measurements of Deardorff and Willis (1984) averaged over the lateral location jyjosy =2 (triangles) and sy =2ojyjosy (circles). The crosses refer to the result from the LHB model for jyjosy =2 and icr ¼ 1: The lines are the results from our model for jyjosy =2 and icr ¼ 1 (continuous line) and icr as in Fig. 6 (dashed line).
experiment. We also choose s0 ¼ 0:003zi and zs ¼ 0:22 zi ; as in LHB. The results shown in Fig. 5 refer to near-surface ðz ¼ 0:08 zi Þ predicted (by our model and by LHB model) and measured fluctuation intensities. The measurements denoted by open triangles represent an average over jyjosy =2; and those denoted by open circles represent average over sy =2ojyjosy ; where sy is the total lateral spread. Crosses refer to the simulation by the LHB model with l ¼ 1: Recalling that 2 l ¼ 1=icr ; the continuous and dashed lines refer to our model predictions using respectively icr ¼ 1 and icr as deduced from a simple analytical fit of the curve proposed by LHB (see Fig. 6). Finally, in Fig. 7 a comparison between the concentration fluctuation intensity ic fields predicted by the LHB model and by our model (with the icr value as in Fig. 6) is shown. A substantial equivalence between the results obtained from the two models emerges. Obviously, different parameterisations for the relative expansion szrr ; as well as for the vertical velocity variance (s2w ) and vertical velocity third moment (w3 ) used in the mean concentration model (CG) can be used in order to improve the agreement with data and LES. However, here we preferred to use the same parameters proposed in LHB. Nevertheless, the LHB model and our generalisation exhibit a very good agreement in repro-
Fig. 6. Variation of the relative in-plume concentration fluctuation intensity ðicr Þ with T computed by LHB (crosses) and fitted by an analytic curve (continuous line).
Fig. 7. Total concentration fluctuation intensity ðic Þ at y ¼ 0 predicted by the LHB model (up) with icr as in Fig. 6 (crosses) and by our model (down) with icr as in Fig. 6 (continuous line).
ducing the second moment concentration field in convective condition, irrespective of the different formulations.
M. Cassiani, U. Giostra / Atmospheric Environment 36 (2002) 4717–4724
4. Conclusion
s1 ¼ szr ½a2 =fa1 ð1 þ f 2 Þg1=2 ;
Following the skewed meandering plume approach proposed by LHB, we introduce a general method to compute the higher order concentration moments, given a mean concentration field. The requested mean concentration field can be obtained either from models or from experiments. The proposed model, as well as the LHB model, makes the limiting assumption of a homogeneous relative expansion. The LHB model, based on a hybrid Eulerian–Lagrangian approach, needs the further knowledge of the trajectories of the centroids of each instantaneous plume to compute the high order concentration fields. Our generalisation does not require any trajectories, thus relaxing the need for Lagrangian modelling. In such a way, the skewed meandering plume approach is referred to a typical meandering plume model (i.e. to a model based on fixed profiles of mean concentration), although the solution is not yet analytical. This method seems to be simpler and more efficient than the LHB one. The comparison of the proposed model results with experimental data set and with numerical (LES) results, shows good overall agreement. The proposed procedure for the vertical co-ordinate is general and can be easily adapted for the crosswind meander in case it is not possible to find an easy analytical solution. The proposed technique can be improved in order to consider a two-dimensional mean concentration field that includes correlation between lateral and vertical dispersion. On the contrary, to include a full nonhomogeneous relative expansion, the Lagrangian point of view used in the LHB approach must be preferred, although it is a waste of computer time if the homogeneous relative expansion is used. We emphasise the generality of our technique and the low demand on computational time, giving higher order concentration moments evaluated in few seconds on a standard PC.
s2 ¼ szr ½a1 =fa2 ð1 þ f 2 Þg1=2 ;
Acknowledgements Authors are grateful to B.L. Sawford for his helpful comments and to P. Franzese for fruitful discussions. They greatly contributed to the improvement of this work. This work was partially supported by the Italian National Research Council (CNR) in the frame of Arctic Project.
Appendix A z1 ¼ Sr f s1 =jSr j; z2 ¼ Sr f s2 =jSr j;
4723
a1 ¼ ½1 fr=ð4 þ rÞg1=2 =2; a2 ¼ 1 a1 ; r ¼ ½ð1 þ f 2 Þ3 Sr2 =½ð3 þ f 2 Þ2 f 2 ; f ¼ c4 jSr j1=3 ; c4 ¼ 2=3:
Appendix B Following LHB, the growth of the relative variance is divided in two regimes, 1 1 s2r ¼ s20 þ Ce ðes0 Þ2=3 t2 ðC0 þ 4Þet3 3 3 iftZ 5tots 1 ¼ s# 20 þ C# e ðes0 Þ2=3 t2 þ ar et3 3 if ts ot5tL ; ts ¼ ct ðs20 =eÞ1=3 ; 3 C# e ¼ Ce ct ðC0 þ 4 þ 3ar Þ; 2 s# 20 ¼ s20 =3½3 þ c2t ðCe C# e Þ c3t ðC0 þ 4 þ 3ar Þ; where tZ is the Kolmogorov time scale, TL is the Lagrangian integral time scale , Ce ¼ 9:8; ar ¼ 0:3; C0 ¼ 3; ct ¼ 1=3: Supposing that for large travel time the expansion follows the Taylor statistical theory, the expansion sr is modified for time greater than ts by s2zr ðtÞ ¼ s2r t¼ts s2r ðtÞ s2r t¼ts þ ; 2 4=5 g5=4 f1 þ ½ðs2r ðtÞ s2r jt¼ts Þ=C0 etTLW s2yr ðtÞ ¼ s2r t¼ts s2r ðtÞ s2r t¼ts þ ; 2 4=5 g5=4 f1 þ ½ðs2r ðtÞ s2r jt ¼ ts Þ=C0 etTLV with TLv ¼ 2s2v =C0 e and TLW ¼ 2w2av =C0 e where w2av ¼ 0:36 w2 : However, in the vertical direction a further interpolate expansion is considered including boundary reflections s2zrr ðtÞ ¼
s2r ðtÞ f1 þ ½ðs2r ðtÞ s2r jt¼ts Þ=z2eq 4=5 g5=4
;
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where z2eq ¼ 0:083 z2i is the equilibrium value of the total vertical mean square spread including boundary reflections.
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