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Intermittency and conditionally-averaged concentration fluctuation statistics in plumes
~NT~R~~~~NC~ AND C~ND~T~~~ALL~-AVERAGED CONCENTRATION FLU~UAT~~N STATISTICS IN PLW~ES*
can give nomwo variina at the ground. Equation (5.16f was derived with d = 1. The recsnt paper by Wilson er al. (1985) also approaches the fluct~tion problem by means of a closed form model for co~nt~tion variance. Like the model of Netterv~l~e (1979a. b), the model they describe predicts that lateral and vertical profiles of con~ntratjon variance in a plume follow the Gaussian distribution, with an image sink term added IO simulate the effect of variance dissipation near the ground. Their equations are
recent work O~I the study of concentration fluctuation statistics. The subject is one of considerable interest to me, as my Doctoral research was devoted to the probllm of measuring and modeling the spatial dist~bution of ~n~nt~tion fl~tuations in plumes (N~tte~~~~~,1979af Sometime in early 1978, while plotting vertical profiles of concentration variance measuredin a wind tunnel plume, it occurredto me tbat the profile &apes would be reproduced by a Gaussian curve if the positive image source term of the mean field Gaussianmodel was replaced by an ~ni~lent negatjve image sink term. The fit of the Gaussian curve to variance data was remarkabty good, especially in the lateral dir~tjoo~ but at that time I had no explanation for why this was so, and was discouraged by objections that the model was nothing more than a fo~uitous curve fit with no basis in physics, that was suitable only for use by ‘engineers’. Of course, the fit was too good to be truly due to luck, and eventually I was able to show that the Gaussian variance profite results naturally from Csanady’s (1967,1973) variance balance equation for the special case o~negligibIe downwjnd variance pr~nction, and variance decay time scales equal to the travel time. Experimental data showed that these special case assumpt;ons were not ~nr~~ble. This work (i.e. Nettervitie, 1979af was later published as part of the Research Monog~ph Series produ~d by Syncrude Canada Ltd. (Netterville, 1979b), and its results were summarized by Netterville and Wilson (IQgO).The main theoreti~i contribution was the closed form Gaussian model For tbe spatial djstrib~tjon of co~cnt~tion variance in plumes, The paper by WiIson er rrI.(~Q$~~d~~
where
and
Fluctuation jntens~ty is ~~cuiated from
and
Here the equation burners corres~~d to those in Wilson ef nt. (19851. It is easily shown by simple substitution that their Equations (14)‘ (IS), (16) and (I I) combine to reproduce Equation (5.17) for concentration variance, with only minor
s = 7: = so(~~cx*(~)(~xp[~]
-d.exp[-IIE$!f]) and. for the flnct~tion
(5.17)
intc~sity,
where the equation numbers correspond to those appearing in Netterville (?979a, b). With the exception of the empirical parameters SOand d, ah symbols arc as used in the familiar Gaussian model for mean concentration. The empirical parameter so is the hypothetical variance ‘source intensity’ required for tht model to match observed variance profiles downwind, and the empirical ratio d ( d 1) of sink strength to source sI~ngtb in ~nation (5.17) is introduced so the model
*Wilson D. J., Robins A. G. and Fackrell 3, E. (1985) .Irmosphrric Enrironmennr I9, l&53- $064.
digerences in nomc~c~ture, e.g. q = Qsif2,a is the same as rl, his the height of the varianceson~e in ~uatjon (5.17kand h, is the (empirically redefined)source height in Equation f 15)of Wilson et al. (1985). Similarly, Equation (5.16) for concentration ~uctuation intensity is recovered in every fundamental detail by combining their Equations (18) and (19). The empirical refin~ent of setting oc +zz1 and h, f h does not extend the model physics beyond that already cantained in (5.16) and (5.17). I realize it has been 5 or 6 years since the authors first learned of the single source image sink Gaussian variance model and its derivation in Netterville (1979a, bj. and decided
f831
Discussions
1832
to use it as the springboard from which to launch the empirical refinements described in Wilson et al. (1982a, b). 1 can only assume that, while the authors now appear to have forgotten about my 1979 mode!,a subconcious recollection of it has helped guide the current resimplification of their 1982 model. By abandoning their earlier dual-source empiricism, they have, in effect, dropped back onto the springboard. and have unintentionally- recovered the original Gaussian variance model in al! its essential details. This may help explain why my work, referred to by Wilson ef a/. (f982a, b), is no longer referenced by Wilson ez al. (1985). 1 am pleased that my Gaussian variance mode! is playing a useful role in the work of others (e.g. Hanna, 1984% b; Bara, 1985; Wilson and Bara 1985). as it justifies thehard work, long hours and stressful circumstances that surrounded its development. Research and Deoelopmenr, Syncrude Canada Lrd., Edmonton, Alberta, Canada
DENNETT D. J. NET~ERVILLE
REFERENCES
Bara B. (1985) Water channel simulation of concentration fluctuatjons from a ground level source. M.Sc. thesis, Mechanical Engineering ~pa~ment, University of Alberta, Canada. Csanady G. T. (1967) Concentration fluctuations in turbulent diffusion. J. armos. Sci. 24, 21-28. Csanady G. T. (1973) Turbulenr Diflusion in rhe Environment, Chapter 7. Geophysics and Astrophysics Monograph Series, D. Reide!, Dordrecht. Hanna S. R. (1984a) Concentration fluctuations in a smoke plume. Atmospheric Enuironmenl 18, 1091-I 106. Hanna S. R. (1984b) The exponential PDFand concentration fluctuations in smoke plumes. Boundary-Layer Net. 29, 361-375. Netterville D. D. J. (1979a) Concentration fluctuations in plumes. Ph.D. thesis, Mechanical Engineering Department, University of Alberta, Canada. Netterville D. D. J. (1979b) Concenrrurion F~ucfuutjons in Plumes. Environmental Research Monograph 1979-4, Syncrude Canada Ltd., P.O. Box 5790, Station L, Edmonton, Alberta, Canada T6C 4G3. Nctterville D. D. J. and Wilson D. J. (1980) Probability estimates of high concentrations in plumes. Proc. 5rh Infernationa/ Clean Air Congress, IUAPPA, October 1980, Buenos Aires, Argentina. Wilson D. J. and Bara B. (1985) Concentration profiles of conditionally averaged mean concentration in plumes, Proc. 7th Symposium on Turbulence ond DifJitsion, AMS, November 1985, Boulder, Colorado. Wilson D. J., Fackre!! J. E. and Robins A. G. (1982a). Concentration notations in an ekvated plume: a Di~~ion~~i~tion approximation. .&nospheric Enuironmenr f&2581-2589. Wilson D. J., Robins A. G. and Fackre!! J. E. (1982b) Predicting the spatial distribution of concentration fluctuations from a ground-level source. Atmospheric Environment 16,497-504. Wilson D. J., Robins A. G. and Fackrell J. E. (1985) lntermittency and conditionally averaged concentration fluctuation statistics in plumes. Atmospheric Enoironment 19, 1053-1064.
AUTHOR’S REPLY Nttttilk’~ (1979) eddy di~usi~ty mode! for variance is an CXtCnSiOnof the original work of Csanady (f973). Nettervi]fe showed the assumptions rquired for the pro&s to reduce to
Gaussian form, and suggested the use of an image sink of fluctuation variance to account for the effect of the ground surface. His contribution of the image sink concept was adopted in our model and has been acknowledged in our previous papers ( I982a, b). However, Netterville is mistaken in his claim that our adoption of Gaussian profiles for variance is based on his mode!. In our case, the Gaussian profiles used for an elevated source are simply a convenienL commonly used empiricism. Their usedoes not imply our support for Nette~i!!e’sclaim of a physical basis related to the d~si~tion time scak, and the use of an eddy diffusivity for variance. The measured profiles in Fackrelt and Robins (1982) and in Ramsdtll and Hinds (1971)show t!rat a Gaussian is only a rough approximation to the spatial distribution of variance. The theoretical basis claimed by Netterville for Gaussian profiles seems rather weak, in light of the flattened and even bimoda! distributions actually observed. Fortunately, considering the other uncertainties in the process, a Gaussian is often adequate to represent the genera! shape of these profiles. In addition, our representation of the variance distribution differs from Csanady (1973) and Netterville (1979) in two important respects. We account for the along-wind dissipation of variance by deriving a variable effective source strength for variance, and we account for wind shear by displa~ng this effective source upward, above the actual point of release, by 0.7~~. Also, because we simply assume that the mean and variance profiles have thesamt functional form, the vertical profiles of both mean and variance for a ground leve! source are represented as exponentials of z’,~ rather than the zL ofa Gaussian. With these differences it seemsreasonable to refer the reader to our previous papers, where the original contributions from Csanady and Netterville are discussed. Finally, it is worth noting that the purpose of Wilson et u/. was to present a model for the conditionally averaged nonzero concentrations and for the intermittency; topics which are not dealt with in any way by the theories of Csanady (1973) or Netterville (1979). Dept. of ~echunjeul Engineering, nie Uni~ersify of Alberta, Edmonton, Afberfa, Cunudo T6G 2G8
D~viv
J. WKSOFI
REFERENCES Csanady G. T. ( 1973) Turbulenr DijDusion in the Environment, Chapter 7. Geophysics and Astrophysics Monograph Series, D. Reidel, Dordrecht. Fackrell J. E. and Robins A. G. (1982) J. Fluid Mech. 117, l-26. Nctterville D. D. J. (1979) Con~ntration ~uctuations in plumes. Ph.D. thesis,Mechanical ~ngin~ring Department, University of Aibcrta, Canada. Ramsde!! J. V. and Hinds W. T. (1971) Concentration fluctuations in plumes from a ground level continuous point source. Atmospheric Enuironmenr 5. 48M95. Wilson D. J., Fackrcl! J. E. and Robins A. G. (1982a) Concentration fluctuations in an elevated plume: a diflusiondissipation approximation. Afmospheric Entlironmenu l&2581-2589. Wilson D. J., Robins A. G. and Fackrel! J. E. (1982b) Predicting the spatial distribution of concentration fluctuations from a ground level source. Atmospheric Enuironmenf f&497-504. Wilson D. J., Robins A. G. and Fackrelt J. E. 119851 lnte~itte~y and conditionally averaged concentration fluctuation statistics in plumes. Armospher~~ En~~ironmenr 19, 1053-1064.
can give nomwo variina at the ground. Equation (5.16f was derived with d = 1. The recsnt paper by Wilson er al. (1985) also approaches the fluct~tion problem by means of a closed form model for co~nt~tion variance. Like the model of Netterv~l~e (1979a. b), the model they describe predicts that lateral and vertical profiles of con~ntratjon variance in a plume follow the Gaussian distribution, with an image sink term added IO simulate the effect of variance dissipation near the ground. Their equations are
recent work O~I the study of concentration fluctuation statistics. The subject is one of considerable interest to me, as my Doctoral research was devoted to the probllm of measuring and modeling the spatial dist~bution of ~n~nt~tion fl~tuations in plumes (N~tte~~~~~,1979af Sometime in early 1978, while plotting vertical profiles of concentration variance measuredin a wind tunnel plume, it occurredto me tbat the profile &apes would be reproduced by a Gaussian curve if the positive image source term of the mean field Gaussianmodel was replaced by an ~ni~lent negatjve image sink term. The fit of the Gaussian curve to variance data was remarkabty good, especially in the lateral dir~tjoo~ but at that time I had no explanation for why this was so, and was discouraged by objections that the model was nothing more than a fo~uitous curve fit with no basis in physics, that was suitable only for use by ‘engineers’. Of course, the fit was too good to be truly due to luck, and eventually I was able to show that the Gaussian variance profite results naturally from Csanady’s (1967,1973) variance balance equation for the special case o~negligibIe downwjnd variance pr~nction, and variance decay time scales equal to the travel time. Experimental data showed that these special case assumpt;ons were not ~nr~~ble. This work (i.e. Nettervitie, 1979af was later published as part of the Research Monog~ph Series produ~d by Syncrude Canada Ltd. (Netterville, 1979b), and its results were summarized by Netterville and Wilson (IQgO).The main theoreti~i contribution was the closed form Gaussian model For tbe spatial djstrib~tjon of co~cnt~tion variance in plumes, The paper by WiIson er rrI.(~Q$~~d~~
where
and
Fluctuation jntens~ty is ~~cuiated from
and
Here the equation burners corres~~d to those in Wilson ef nt. (19851. It is easily shown by simple substitution that their Equations (14)‘ (IS), (16) and (I I) combine to reproduce Equation (5.17) for concentration variance, with only minor
s = 7: = so(~~cx*(~)(~xp[~]
-d.exp[-IIE$!f]) and. for the flnct~tion
(5.17)
intc~sity,
where the equation numbers correspond to those appearing in Netterville (?979a, b). With the exception of the empirical parameters SOand d, ah symbols arc as used in the familiar Gaussian model for mean concentration. The empirical parameter so is the hypothetical variance ‘source intensity’ required for tht model to match observed variance profiles downwind, and the empirical ratio d ( d 1) of sink strength to source sI~ngtb in ~nation (5.17) is introduced so the model
*Wilson D. J., Robins A. G. and Fackrell 3, E. (1985) .Irmosphrric Enrironmennr I9, l&53- $064.
digerences in nomc~c~ture, e.g. q = Qsif2,a is the same as rl, his the height of the varianceson~e in ~uatjon (5.17kand h, is the (empirically redefined)source height in Equation f 15)of Wilson et al. (1985). Similarly, Equation (5.16) for concentration ~uctuation intensity is recovered in every fundamental detail by combining their Equations (18) and (19). The empirical refin~ent of setting oc +zz1 and h, f h does not extend the model physics beyond that already cantained in (5.16) and (5.17). I realize it has been 5 or 6 years since the authors first learned of the single source image sink Gaussian variance model and its derivation in Netterville (1979a, bj. and decided
f831
Discussions
1832
to use it as the springboard from which to launch the empirical refinements described in Wilson et al. (1982a, b). 1 can only assume that, while the authors now appear to have forgotten about my 1979 mode!,a subconcious recollection of it has helped guide the current resimplification of their 1982 model. By abandoning their earlier dual-source empiricism, they have, in effect, dropped back onto the springboard. and have unintentionally- recovered the original Gaussian variance model in al! its essential details. This may help explain why my work, referred to by Wilson ef a/. (f982a, b), is no longer referenced by Wilson ez al. (1985). 1 am pleased that my Gaussian variance mode! is playing a useful role in the work of others (e.g. Hanna, 1984% b; Bara, 1985; Wilson and Bara 1985). as it justifies thehard work, long hours and stressful circumstances that surrounded its development. Research and Deoelopmenr, Syncrude Canada Lrd., Edmonton, Alberta, Canada
DENNETT D. J. NET~ERVILLE
REFERENCES
Bara B. (1985) Water channel simulation of concentration fluctuatjons from a ground level source. M.Sc. thesis, Mechanical Engineering ~pa~ment, University of Alberta, Canada. Csanady G. T. (1967) Concentration fluctuations in turbulent diffusion. J. armos. Sci. 24, 21-28. Csanady G. T. (1973) Turbulenr Diflusion in rhe Environment, Chapter 7. Geophysics and Astrophysics Monograph Series, D. Reide!, Dordrecht. Hanna S. R. (1984a) Concentration fluctuations in a smoke plume. Atmospheric Enuironmenl 18, 1091-I 106. Hanna S. R. (1984b) The exponential PDFand concentration fluctuations in smoke plumes. Boundary-Layer Net. 29, 361-375. Netterville D. D. J. (1979a) Concentration fluctuations in plumes. Ph.D. thesis, Mechanical Engineering Department, University of Alberta, Canada. Netterville D. D. J. (1979b) Concenrrurion F~ucfuutjons in Plumes. Environmental Research Monograph 1979-4, Syncrude Canada Ltd., P.O. Box 5790, Station L, Edmonton, Alberta, Canada T6C 4G3. Nctterville D. D. J. and Wilson D. J. (1980) Probability estimates of high concentrations in plumes. Proc. 5rh Infernationa/ Clean Air Congress, IUAPPA, October 1980, Buenos Aires, Argentina. Wilson D. J. and Bara B. (1985) Concentration profiles of conditionally averaged mean concentration in plumes, Proc. 7th Symposium on Turbulence ond DifJitsion, AMS, November 1985, Boulder, Colorado. Wilson D. J., Fackre!! J. E. and Robins A. G. (1982a). Concentration notations in an ekvated plume: a Di~~ion~~i~tion approximation. .&nospheric Enuironmenr f&2581-2589. Wilson D. J., Robins A. G. and Fackre!! J. E. (1982b) Predicting the spatial distribution of concentration fluctuations from a ground-level source. Atmospheric Environment 16,497-504. Wilson D. J., Robins A. G. and Fackrell J. E. (1985) lntermittency and conditionally averaged concentration fluctuation statistics in plumes. Atmospheric Enoironment 19, 1053-1064.
AUTHOR’S REPLY Nttttilk’~ (1979) eddy di~usi~ty mode! for variance is an CXtCnSiOnof the original work of Csanady (f973). Nettervi]fe showed the assumptions rquired for the pro&s to reduce to
Gaussian form, and suggested the use of an image sink of fluctuation variance to account for the effect of the ground surface. His contribution of the image sink concept was adopted in our model and has been acknowledged in our previous papers ( I982a, b). However, Netterville is mistaken in his claim that our adoption of Gaussian profiles for variance is based on his mode!. In our case, the Gaussian profiles used for an elevated source are simply a convenienL commonly used empiricism. Their usedoes not imply our support for Nette~i!!e’sclaim of a physical basis related to the d~si~tion time scak, and the use of an eddy diffusivity for variance. The measured profiles in Fackrelt and Robins (1982) and in Ramsdtll and Hinds (1971)show t!rat a Gaussian is only a rough approximation to the spatial distribution of variance. The theoretical basis claimed by Netterville for Gaussian profiles seems rather weak, in light of the flattened and even bimoda! distributions actually observed. Fortunately, considering the other uncertainties in the process, a Gaussian is often adequate to represent the genera! shape of these profiles. In addition, our representation of the variance distribution differs from Csanady (1973) and Netterville (1979) in two important respects. We account for the along-wind dissipation of variance by deriving a variable effective source strength for variance, and we account for wind shear by displa~ng this effective source upward, above the actual point of release, by 0.7~~. Also, because we simply assume that the mean and variance profiles have thesamt functional form, the vertical profiles of both mean and variance for a ground leve! source are represented as exponentials of z’,~ rather than the zL ofa Gaussian. With these differences it seemsreasonable to refer the reader to our previous papers, where the original contributions from Csanady and Netterville are discussed. Finally, it is worth noting that the purpose of Wilson et u/. was to present a model for the conditionally averaged nonzero concentrations and for the intermittency; topics which are not dealt with in any way by the theories of Csanady (1973) or Netterville (1979). Dept. of ~echunjeul Engineering, nie Uni~ersify of Alberta, Edmonton, Afberfa, Cunudo T6G 2G8
D~viv
J. WKSOFI
REFERENCES Csanady G. T. ( 1973) Turbulenr DijDusion in the Environment, Chapter 7. Geophysics and Astrophysics Monograph Series, D. Reidel, Dordrecht. Fackrell J. E. and Robins A. G. (1982) J. Fluid Mech. 117, l-26. Nctterville D. D. J. (1979) Con~ntration ~uctuations in plumes. Ph.D. thesis,Mechanical ~ngin~ring Department, University of Aibcrta, Canada. Ramsde!! J. V. and Hinds W. T. (1971) Concentration fluctuations in plumes from a ground level continuous point source. Atmospheric Enuironmenr 5. 48M95. Wilson D. J., Fackrcl! J. E. and Robins A. G. (1982a) Concentration fluctuations in an elevated plume: a diflusiondissipation approximation. Afmospheric Entlironmenu l&2581-2589. Wilson D. J., Robins A. G. and Fackrel! J. E. (1982b) Predicting the spatial distribution of concentration fluctuations from a ground level source. Atmospheric Enuironmenf f&497-504. Wilson D. J., Robins A. G. and Fackrelt J. E. 119851 lnte~itte~y and conditionally averaged concentration fluctuation statistics in plumes. Armospher~~ En~~ironmenr 19, 1053-1064.