hemial, Biologial and Radiologial Haard Assessment: A New Model of a Plume in a omplex Urban Environment Skvortsov, A.T., P.D. Dawson, M.D. Roberts and R.M. Gailis HPP Division, Defene Siene and Tehnology Organisation, PO Box 4, Melbourne, VI, Email: alex.skvortsov@dsto.defene.gov.au Keywords: pollutant, plume, urban anopy, mean onentration profile, model, omparison EXTENDED ABSTRAT There has been a growing interest in reent years in the modelling of haards arising from the atmospheri dispersion of hemial, biologial and radiologial (BR) agents in the environment, and the threat that they pose to the population and military fores. This is a partiularly hallenging problem in an urban setting. Dispersion of BR agents in an atmospheri boundary layer (ABL) over a heterogeneous (urban) anopy is a omplex proess to be desribed by advaned methods of fluid dynamis, turbulene theory, diffusion and statistis. Using omprehensive modelling is omputationally intensive and too time onsuming when applied to operational problems when reliable outome has to be produed within a limited time frame. Plume haraterisation requires the development of simplified analytial models of turbulent dispersion based on physial assumptions and first priniples physis onsiderations. These models must still be simple enough to be easily treated numerially in an operationally viable way. Suh models an also provide a theoretial foundation for baktraking problems, i.e. finding a BR soure in a omplex anopy under various meteorologial onditions. The purpose of this paper is to summarise the reent researh onduted by DSTO (HPPD) in the development of suh models. There undoubtedly exists a vast amount of literature dediated to the simplified models of traer dispersion (for instane see an extensive literature review and referenes in our reent publiations: Gailis et al 6, Gailis et al 7). The main goal of our study was the extension of the elebrated flutuating plume model of traer dispersion to two ases: namely, a simple sheared boundary layer and a large array of regular obstales (model of urban anopy, Fig ). We tried to inorporate these ases based on the physial models of the assoiated advetiondiffusion proess in turbulent flow, rather than based on ad-ho empirial relationships. We present a new mathematial model of RB plume dispersion in an urban environment. The model uses parameters that expliitly take into aount turbulent flow lose to the ground and the urban anopy parameters enabling an analyti alulation of the plume onentration profiles. Model preditions are ompared with some reent experimental data, showing a lose math. The model developed an be used as an analytial tool for prediting BR plume behaviour in omplex urban environments, or as a prototype and performane hek for a new generation of dispersion models. This will lead to a set of improved tools for planning and support of military operations in BR threat environments. Figure : An example of one of the obstale arrays 645
. INTRODUTION Sine air flow in the ABL is a key driver for BRN pollutant advetion and dispersion, the development of a high fidelity model of this flow is a ruial step in the modelling of the whole turbulent dispersion proess. The flutuations are presented as a omposition of two distint omponents (large-sale plume entroid meandering and fine-sale internal plume flutuations) allowing reasonable analytial preditions. () d U av* where U() is the horiontal veloity, is distane from the ground, V * is the frition veloity, a and m are onstants (m is a main parameter from the theory), is the roughness height and d is the soalled displaement height. Both d and should be onsidered as fitting parameters of the ABL flow over the anopy. m, The profile desribed by Eq (), being algebraially simpler than the elebrated log-law profile, has been used merely as a onvenient engineering approximation, but reently it has attrated muh attention sine it has been shown that it an be justified based on a self-similarity property of the ABL flow (see Barenblatt et al ). For the boundary layer over a flat smooth surfae it has been rigorously shown that Figure : Shemati representation of the experimental setup, depiting the sheet of laser light interseting the dispersing fluoresent dye in a surfae layer flow with anopy objets. Figure : The alulated wind profile in sparse (solid line) and dense (dashed line) anopies based on the proposed model. The dashed line orresponds to the lower value of the parameter ε (dense anopy).. A NEW MODELLING FRAMEWORK.. Mean Flow in a omplex anopy The flow model in surfae layer within and above the anopy should orretly desribe the average (i.e. non-flutuating) veloity field. It has been known for a long time (dating bak to Prandtl, see Monin and Yaglom 978) that the ABL mean veloity profile an be fairly approximated by a power-law funtion (see Fig ): where Re is the Reynolds number of the flow. Observed values of m in the atmosphere range from nearly in very unstable onditions, representing perfet mixing and a uniform veloity profile, to nearly in very stable onditions, approahing the ouette linear profile of laminar motion over a plane surfae. For neutral onditions m /7 (Monin and Yaglom 978). The value of m also depends on surfae roughness: roughness promotes mixing near the surfae, whih redues the veloity gradient at small and thus leads to larger variation in m. Based on the so-alled distributed drag approah it has been reently shown (see Harman et al 7) that the entire influene of the anopy on the ABL flow () an be desribed by only one parameter that desribes the ratio of the anopy surfae area to the total area. For an array of idential ylinders (similar to Fig ) this parameter is approximately equal to () m H ε πr ln Re λ p λ p where all parameters in this formula are determined by the anopy morphology (H is the,, 646
) anopy hight, r is the radius of the ylinders and λ p is the paking density of anopy elements). The limiting values of ε orrespond to sparse (ε >> ) and dense (ε << ) anopies. We have developed a onsistent theoretial framework that allows us to derive a modified veloity profile U() () for a given value of ε, i.e. for a given anopy. Our framework is based on smooth mathing of the two solutions of momentum balane (below and above the anopy) near the anopy top. We have derived general funtions for d(ε), (ε) and U() within the anopy. good agreement between our model and the measured veloity profiles... Mean BRN onentration Profiles For the derived veloity profile in and above the anopy we omputed the mean onentration field from the advetion-diffusion equation. For the power-law profile the mean onentration an be modelled by the well-known strethed exponential solution (see Monin and Yaglom 978) y ( y, x ) (, x ), Urban array - U mean (mm s 5 5 5 5 Point A Point Point K Point L where x is downstream distane, y is the distane from the plume entroid in the lateral diretion, y is the lateral onentration profile (Gaussian funtion of y) and () B exp ( ( σ ) α ) -5 4 6 8 4 6 Height (mm) Figure 4: Measured veloity profiles for a simulated urban anopy at different positions relative to anopy objets. We found that for a large ε funtion d(ε) H, (ε) as a power law (i.e. rather slowly) and d() () if H. It should be emphasied that in the proposed framework, the entire morphologial variety of anopies manifests itself only in different values of parameter ε. Examples of alulated veloity profiles are presented in Fig. In Fig 4 we show our experimental data from a water hannel experiment (Gailis et al 7). The urban anopy was modelled by an array of ubi obstales that were paked in regular or random patterns (see Fig and Fig ). The veloity measurements were onduted in various positions within a anopy ell (inluding wake areas). The solid line in Fig 4 is our model predition, whih represents an average veloity profile for the whole ell. This is to be ompared to the individual point measurements of veloity within eah ell, whih vary signifiantly from point to point. The point orresponds to the position straight behind the obstale (wake area) with a lear visible reverse flow (negative veloity). Our simplified models attempt to apture the averaged behaviour. For a variety of obstale array onfigurations, we observed a reasonably is a strethed exponential profile. The funtions B(x) and σ(x) are determined by soure intensity and downstream position, and the parameter α + m is determined only by the veloity profile parameter m. The profile () is valid above the anopy top (i.e. > d) and should be mathed with the pollutant onentration modelled within the anopy. Two models of the onentration profile within the anopy were validated. The first model was a lipped profile, when we simply assumed a onstant value of onentration for < d. The justifiation for suh a model is the strong proess of turbulent mixing that ours within the anopy that should smooth out all onentration gradients. The data fit to the lipped model for different downstream positions is presented in Fig 5. The left olumn is the vertial onentration profile and the right olumn is the lateral struture of the plume with a Gaussian fit. The onentration is normalised to the known pollutant at the soure rate. This removes soure strength as a free parameter and enfores onservation of pollutant material at eah downstream position. 647
onentration 4.5 X6.5 mm.5 X7. mm.5.5 4 5 6 7 8.4..8.6.4. X9.5 mm X7.5 mm 4 5 6 7 8..5..5..5 X58. mm X85.5 mm 4 5 6 7 8 Height above ground (mm).5.5.5.5-8 -6-4 - 4 6 8.8.6.4. -8-6 -4-4 6 8...8.6.4...8.6-8 -6-4 - 4 6 8 ross-stream position (mm) Figure 5: onentration data fit in the lipped model (, for the left olumn and y for right olumn). Horiontal axis is the distane from the ground (left olumn) and from the entre of the hannel (right olumn). The seond evaluated onentration model was based on allowing the variation of α with height to provide the best data fit i.e. α α(). The rationale behind this framework was the known limiting values of α: α + m for > d, and α near << d (stagnation areas near ground in a deep anopy). As a reasonble approximation we proposed where is a funtion of the veloity profile parameter m and anopy parameter ε (sine d d(ε)). onentration 4.5 X6.5 mm.5 X7. mm.5.5 4 5 6 7 8.4. X9.5 mm.8 X7.5 mm.6.4. 4 5 6 7 8..5..5..5 X58. mm X85.5 mm 4 5 6 7 8 Height above ground (mm) α ( + m)( + φexp( / d)) φ ( m )( + m).5.5.5.5-8 -6-4 - 4 6 8.8.6.4....8.6.4...8.6-8 -6-4 - 4 6 8-8 -6-4 - 4 6 8 ross-stream position (mm) Figure 6: onentration data fit in variable α model. All notations are as in Fig 5. The data fit to the variable α model is presented in Fig 6 (left olumn). As in Fig 5, the right olumn is the lateral struture of the plume with a Gaussian fit. In general we observed that both models are in good agreement with the experimental data. The data fits are nearly indistinguishable downstream of the soure. loser to the soure the variable α model seems to be a better representation of the vertial plume struture. The better performane of the variable α model an be attributed to the more adequate desription of the proess of turbulent mixing in the anopy layer (i.e. mixing is hanging within the anopy with height). The lipped model orresponds to the onstant mixing in the anopy. For dense anopies with the stagnation flows near the ground hanges with height are not so importation (Harman et al 7) and both models produe very lose results... onentration Flutuations The approah outlined above models the development of a mean plume within a omplex environment. This is the time average behaviour of a real dispersing plume, or equivalently, the average pattern that would be seen if an idential release of material was performed many times. Model analysis of BR events also requires the development of onentration realisation models that give a statistially sound representation of possible instantaneous patterns of plume dispersion. This is important to enable the investigation of unertainty or risk in BR haard assessments, as well as to provide realisti syntheti environments for operational analysis studies. A model of plume onentration flutuations has been developed based on the so-alled flutuating plume approah, where overall flutuations are represented as a ombined effet of slowly osillating plume meander and fast in-plume flutuations. Thus, for the onditional Probability Density Funtion (PDF) of onentration in absolute frame f the following general representation was adopted (Gailis et al 7): f ( ; x) f ( ; x x ) f ( x; x ) dx r where f is the PDF of entroid meander, f r is the onentration PDF in relative frame (assoiated with plume entroid), x (t) is the position of entroid. 648
Development of realisti models for f r and f requires the appliation of rather ompliated statistial methods and are desribed in detail in our other publiations (see Gailis et al 6, Gailis et al 7, Skvortsov et al 7). We have proposed a model where the PDF for vertial plume meander an be desribed by a Gamma distribution while the PDF for horiontal meander is always lose to Gaussian. We have proposed the model for in-plume flutuations in a reent paper (Borgas et al 7) where we related the onentration flutuation intensity i < ( < with the flow parameters within and above the anopy. The important onlusion is that the statistial properties of the plume in the anopy an be parameterised with the flow parameters (m,ε), so a proposed two-parameter model of the veloity profile () also provides a onsistent framework for onentration flutuation modelling.. ONLUSIONS > > ) Simple Models for Flutuations, submitted to Boundary-Layer Meteorology. Monin, A.S., Yaglom, A.M., (978), Statistial Fluid Mehanis, MIT Press, ambridge Massahusetts. Gailis, R.M., Hill, A. (6), A Wind-Tunnel Simulation of Plume Dispersion within a Large Array of Obstales, Boundary-Layer Meteorology, Vol. 9, pp. 57-47. Gailis, R. M., Hill, A., Yee, E., Hilderman, T., (7). Extension of a Flutuating Plume Model of Traer Dispersion to a Sheared Boundary Layer, Boundary-Layer Meteorology, Vol., pp. 577-6. Harman, I.N., Finnigan, J.J, (7). A Simple Unified Theory For Flow in the anopy And Roughness Sublayer, Boundary- Layer Meteorology, Vol., pp. 96. Skvortsov, A.T., Gailis R.M., Hilderman T., (7), Statistial Properties of a Meandering Plume in a Sheared Boundary Layer, submitted to Boundary-Layer Meteorology. Physis based models of a plume in an urban anopy allow a simplified (but still adequate) analytial desription of pollution transport in a omplex environment, partiularly useful for BR appliations. The proposed theoretial framework has been validated against our experimental data (water hannel experiment) and provided a lose math. The proposed framework an help to validate and justify some more empirially based and heuristi assumptions of operational dispersion models urrently available to the ADF. Our modelling framework an thus be used as a valuable performane hek of suh models, or is in a form available for extension to an operational model prototype, able to be linked to larger modelling systems. With the development of enhaned BR defensive apabilities now seen as a priority within the ADF, the model desribed here is an important ontribution to this larger sale effort. 4. REFERENES Barenblatt, G. I., horin, A. J., and Prostokishin V.M.(), A model of a turbulent boundary layer with a non-ero pressure gradient, PNAS,, Vol.99, pp. 577-5776. Borgas, M.S., Gailis, R.M., Skvortsov, A.T., (7), Surfae Layer Mixing Properties: 649