- Email: [email protected]
On plume meandering in unstable stratification
ARTICLE IN PRESS
Atmospheric Environment 39 (2005) 2995–2999 www.elsevier.com/locate/atmosenv
Technical note
On plume meandering in unstable stratification Chun-Ho Liu, Dennis Y.C. Leung Department of Mechanical Engineering, The University of Hong Kong, Hong Kong Received 3 September 2004; received in revised form 21 January 2005; accepted 1 February 2005
Abstract Vertical plume meandering of gaseous pollutant is commonly experienced in the daytime atmospheric boundary layer (also know as convective boundary layer, CBL) that arose from the complicated interaction between buoyancygenerated turbulence and gravitational force. It leads to rapid pollutant mixing that cannot be accurately modeled by conventional Gaussian plume model. In the light of explaining the mechanism of plume rises and descents in CBLs, this study employs a direct numerical simulation (DNS) technique to compute the plume behaviors for pollutant emitted from line sources placed parallel to the spanwise direction in an unstably stratified turbulent open channel flow. The DNS results show that the plume meandering is due to the domination of uni-directional mean vertical pollutant fluxes above and below the mean plume height. r 2005 Elsevier Ltd. All rights reserved. Keywords: Convective boundary layer (CBL); Plume rise and descent; Direct numerical simulation (DNS); Gaussian plume model
1. Introduction The Gaussian plume model is the most commonly adopted practical approach to handle transport of passive and inert gaseous pollutants in the atmospheric boundary layer (ABL) due to its versatility in use. However, its accuracy is in doubt in convective boundary layers (CBLs) as the turbulence structure and the plume trajectories are complicated by the upward heat flux at ground level. The plume meandering, in the form of rises and descents, has been confirmed by a series of laboratory experiments (e.g. Willis and Deardorff, 1976, 1978, 1981). Several studies of second-order closure turbulence model (e.g. Liu and Leung, 2001a, b) and the more sophisticated large-eddy simulation (LES, e.g. Dosio et al., 2003) have further supported the findings in this area. As there has been no Corresponding author. Tel.: +852 2857 8615; fax:+852 2858 5415. E-mail address: [email protected] (C.-H. Liu).
agreed subgrid-scale parameterization for second-order closure turbulence model or LES yet, more detailed mathematical modeling is deemed for our better understanding on the mechanism of plume meandering in CBLs. By calculating explicitly all the significant energycarrying scales, direct numerical simulation (DNS) is an alternative to explain the detailed turbulent pollutant transport in that regard. This paper is a quick and brief description on our recent DNS findings explaining the behaviors and mechanism of plume meandering in unstably stratified flow.
2. Methodology Because of the highly refined (spatial and temporal) scales required in DNS, the turbulent flow in an engineering-scale open channel of height H is modeled instead of the ABL. The current DNS model employs the incompressible Navier–Stokes, continuity, energy conservation, and pollutant transport equations whose
1352-2310/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2005.02.024
ARTICLE IN PRESS 2996
C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
findings are then used to explain the plume meandering in CBLs. In the open channel, the flow field is assumed to be horizontally homogeneous, and is bounded by a non-slip wall and a shear-free boundary, respectively, at the bottom and the top. The flow is driven by a background pressure force in the streamwise direction x. A temperature difference is prescribed across the vertical height z of the channel with heating at the bottom so that the buoyancy is opposite to the gravity (unstable stratification). Pollutant line sources of strength C are placed parallel to the spanwise direction y at the channel inlet. The pollutant emission and boundary condition are assumed to be periodic in the spanwise direction. An open boundary is prescribed at the downstream channel outlet. Zero Neumann boundary conditions are assumed elsewhere. Based on the aforementioned flow configuration and a reference velocity scale U, the Reynolds number is 3,000, the Prandtl and Schmidt numbers are 0.72, and the Richardson number is about 0.2. The spatial domain, which is discretized into 448 64 96 ¼ 2; 752; 512 brick elements, is solved by the three-dimensional Galerkin finite element method (FEM). Equal-order trilinear interpolating polynomials are employed for the approximation of the velocity, pressure, temperature, and pollutant concentration in the spatial domain. In the temporal domain, the advection and diffusion terms are solved by the explicit third-order accurate Runge–Kutta and the implicit second-order accurate Crank-Nicolson schemes, respectively. The buoyancy term is solved explicitly by the first-order accurate Euler scheme. The resulting linear equation systems are solved by the Conjugate Gradient method with the Jacobian preconditioning. The calculation was performed on a 16-node Linux Intel PIII PC cluster. Detailed numerical methodology is discussed elsewhere (Liu et al., 2003). The DNS FEM code was validated by calculating turbulent plane Couette flow and scalar transport (Liu, 2003). It should be mentioned that one of the major weaknesses of the current DNS calculation is that it is unable to represent all the processes in the ABL in true scale (for example, the full spectrum of turbulence motions and the eddies of size extending across the ABL thickness). Nevertheless, it is expected that the use of DNS to calculate explicitly all the significant energycarrying scales of fluid turbulence and pollutant transport in reduced scale can enrich our understanding on the basic physics of turbulent plume behaviors in CBLs and help to explain the commonly observed pollutant plume behaviors in the ABL.
3. Results and discussions In this paper, statistical means are denoted by parenthesis /S that represent the spatial (in the
spanwise direction) and temporal averaged values. The double primes 00 denotes the deviation from the statistical mean c00 ¼ c hci. The mean plume height z¯ ðxÞ and the dispersion coefficient sz ðxÞ are calculated in R H R H accordance with z¯ ðxÞ ¼ 0 cðx; zÞ z dz= 0 cðx; zÞ dz R H R H and s2z ðxÞ ¼ 0 cðx; zÞ ½z z¯ ðxÞ 2 dz= 0 cðx; zÞ dz, respectively, where c is the pollutant concentration. A rapid plume rise is observed for the pollutant emitted at the lower part of the channel at zs =H ¼ 0:25 (Fig. 1a) where zs is the pollutant emission height. The pollutant travels near the wall-level in the near field ðx=Hp4Þ that leads to high pollutant concentration there. Afterward, the plume ascends (at 4ox=Ho5Þ that drifts the mean plume height from the lower part to the center core of the channel. This plume rise also causes a rapid increase in pollutant concentration at the upper part of the channel. The plume is carried aloft with a maximum mean plume height z¯ =H ¼ 0:62 at x=H ¼ 8:5 before converges to the channel center thereafter. Engineering practice often assumes Gaussian-shape plume model in the crosswind direction that calculates the spatial distribution of pollutant. Using the same dispersion coefficient determined by the current DNS output and assuming Gaussian-shape pollutant dispersion, the pollutant distributions on the vertical plane for zs =H ¼ 0:25 shows neither plume rise nor descent (Fig. 1b). As a result, the plume generally travels horizontally near the wall-level along the streamwise direction and does not switch the ground-level maximum pollutant concentration to the top throughout the channel. A rapid plume descent is observed for the scalar emitted at the upper part of the channel at zs =H ¼ 0:75 (Fig. 1c). The plume rises slightly in the near field before x=Hp4 then descends rapidly from the upper part to the center core and lower parts of the channel at 4ox=Ho5. The descending plume eventually develops a local maximum of pollutant concentration traveling at the wall level in the far field ðx=HX5Þ. The descending mean plume height passes through the channel center at x=H ¼ 5 followed by an overshoot of minimum mean plume height z¯ =H ¼ 0:4 at x=H ¼ 8:5. The Gaussian-shape pollutant dispersion for an emission height zs =H ¼ 0:75 is illustrated in Fig. 1d. Unlike the plume behavior determined by the current DNS model, no significant plume meandering is calculated by Gaussian plume model and the plume generally travels toward the downstream outlet around its emission height. It is worth mentioning that the aforementioned pollutant plume meandering has been confirmed by a series of convective water tank experiments (Willis and Deardorff 1976, 1978, and 1981), the empirical secondorder closure model of the ABL (Liu and Leung 2001a, b), and the LES of CBLs (Dosio et al. 2003). In
ARTICLE IN PRESS C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
2997
/C 1.0E-05 3.7E-04 7.4E-04 1.1E-03 1.5E-03 1.8E-03 2.2E-03 2.5E-03 2.9E-03 3.3E-03 3.6E-03 4.0E-03
z/H
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x/H Fig. 1. Comparison of the spatial contours of the dimensionless mean pollutant concentration hci=C calculated by the current DNS model and the conventional Gaussian plume model. Dimensionless mean pollutant concentration for emission heights zs =H ¼ 0:25: (a) DNS model and (b) Gaussian model; and zs =H ¼ 0:75: (c) DNS model and (d) Gaussian model. Also shown are the mean plume height z¯ =H: dashed line and the mean plume coverage ¼ 2sz =H: white bars.
particular, the mean plume heights and dispersion coefficients calculated by the LES and second-order closure models are close to those determined from the water tank experiments. In fact, the current DNS results are inline with the findings from various approaches discussed above that signifies the accuracy of the DNS FEM model and its representative of the physical processes in the realistic ABL. In the following discussion, the plume meandering mechanism will be explained using the current DNS results from a mathematical point of view. Turbulent transport (or mixing) of pollutant in the vertical direction is represented mathematically by the mean vertical pollutant flux hc00 w00 i (w is the vertical velocity) which also accounts for the correlation between vertical velocity and pollutant concentration. Because of plume development and widening, the mean vertical pollutant fluxes are generally positive and negative, respectively, above and below the mean plume height. This configuration signifies that upward flow ðw00 40Þ results in an increase in pollutant concentration ðc00 40Þ while downward flow ðw00 o0Þ results in a decrease in pollutant concentration ðc00 o0Þ, for region above the mean plume height. On the other hand, for region below the mean plume height, upward flow results in a decrease in pollutant concentration while downward flow results in an increase in pollutant concentration.
As there is no vertical mean flow ðhwi ¼ 0Þ, pollutant is transported in the vertical direction by diffusion in which most are by turbulence while an insignificant amount is by molecular scales. Looking into the structure of mean vertical pollutant flux helps elucidate our understanding on the mechanism of vertical plume meandering. Generally, in the near field ðx=Hp2Þ, the pollutant emitted from zs =H ¼ 0:25 (Fig. 2a) and 0:75 (Fig. 2b) shows positive and negative mean vertical pollutant fluxes above and below the mean plume height, respectively. This finding is inline with our aforementioned hypothesis of plume development. In the far field ðx=H42Þ, the pollutant emitted from the lower and upper parts of the channel exhibits different structures of mean vertical pollutant flux. For the pollutant emitted at zs =H ¼ 0:25, the positive mean vertical pollutant flux ascends from the emission height to the center core of the channel at 4ox=Ho5 (Fig. 2a). It covers the mean plume height and almost half of the vertical height of the channel. The domination of positive mean vertical pollutant flux signifies that the pollutant transport is in a single direction (upward) regardless of the position. It thus explains the mechanism of the rapid plume rise in CBLs. A similar rationale can be used to explain the rapid plume descent for the pollutant emitted at zs =H ¼ 0:75. After descending from the emission height and a slight rebound at 4ox=Ho5, a broad negative mean vertical pollutant flux is developed at the center core of the channel
ARTICLE IN PRESS C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
2998
/(CU) -2.0E-04 -1.6E-04 -1.2E-04 -8.4E-05 -4.5E-05 -6.0E-06 3.3E-05 7.2E-05
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x/H Fig. 2. Dimensionless mean vertical pollutant flux hc00 w00 i=ðCUÞ calculated by the current DNS model for emission heights: (a) zs =H ¼ 0:25 and (b) zs =H ¼ 0:75. Also shown are the mean plume height z¯ =H: dashed line and the mean plume coverage ¼ 2sz =H: white bars.
Fig. 3. Comparison of the dimensionless mean plume height z=H calculated by the current DNS model and the conventional Gaussian plume model. Emission height zs =H ¼ 0:25: DNS model, solid line; Gaussian model, dotted line. Emission height zs =H ¼ 0:75: DNS model, dashed line; Gaussian model, dashed-dotted line.
covering the mean plume height (Fig. 2b). Hence, the mean pollutant transport is solely downward and leads to the rapid plume descent. The mean plume heights for emission heights zs =H ¼ 0:25 and 0:75 calculated by the current DNS FEM model and the conventional Gaussian plume models are contrasted in Fig. 3 for comparison. For the pollutant emitted at zs =H ¼ 0:25, the maximum mean plume height calculated by the current DNS FEM model is z¯ =H ¼ 0:6 while the value calculated by Gaussian plume model is around 0.3. Besides, the current DNS results show that the pollutant emitted at zs =H ¼ 0:75 descends down to z¯ =H ¼ 0:4. However, the corresponding maximum plume descent calculated by Gaussian plume model is z¯ =H ¼ 0:65. Hence, Gaussian plume model is over-simplified for handling the meandering of mean plume height in unstable stratification accurately. In particular, it is unable to determine the overshoots of pollutant plume ascent and descent. It is thus recommended that the conventional Gaussian plume models should be applied with caution to explain pollutant plume behaviors in CBLs.
4. Conclusion The plume transport phenomenon in an unstably stratified open channel flow is studied using DNS
technique. In contrasts to the conventional Gaussian plume model calculation, plume meandering is found in the flow which is due to the domination of a singledirection mean vertical pollutant flux beyond the near field. The plume rise with pollutant emitted at zs =H ¼ 0:25 is due to the broad positive mean vertical pollutant flux covering the mean plume height. Similarly, the plume descent with pollutant emitted at zs =H ¼ 0:75 is due to the broad negative mean vertical pollutant flux covering the mean plume height. The present work is targeted on the plume meandering in unstable stratification from a mathematical point of view. Future works will be focused on DNS of pollutant plume behaviors in unstable stratification at higher Reynolds numbers as well as LES of CBLs of realistic ABL scales so as to verify the current DNS findings.
Acknowledgements The authors would like to thank the Computer Centre (CC) of the University of Hong Kong (HKU) for permitting us to access to its High Performance Computing Clusters systems. In particular, we are grateful to Mr. Wing-Keung Kwan and Mr. Chat-Ming Woo of HKU CC for their kind assistance during the development of the DNS FEM code.
References Dosio, A., de Arellano, J.V.-G., Holtslag, A.A.M., Builtjes, P.J.H., 2003. Dispersion of a passive tracer in buoyancyand shear-driven boundary layers. Journal of Applied Meteorology 42, 1116–1130. Liu, C.-H., 2003. Turbulent plane Couette flow and scalar transport at low Reynolds number. Journal of Heat Transfer, Transaction of the American Society of Mechanical Engineers 125, 988–998.
ARTICLE IN PRESS C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999 Liu, C.-H., Leung, D.Y.C., 2001a. Study of turbulence and pollutant dispersion in a neutrally and unstably stratified atmosphere using a second-order closure boundary layer model. International Journal of Environment and Pollution 16, 16–27. Liu, C.-H., Leung, D.Y.C., 2001b. Turbulence and dispersion studies using a three-dimensional second-order closure Eulerian model. Journal of Applied Meteorology 40, 92–113. Liu, C.-H., Leung, D.Y.C., Woo, C.-M., 2003. Development of a scalable finite element solution to the Navier-Stokes equations. Computational Mechanics 32, 185–198.
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Willis, G.E., Deardorff, J.W., 1976. A laboratory model of diffusion into the convective planetary 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 modeled 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 convectively mixed layer. Atmospheric Environment 15, 109–117.
Atmospheric Environment 39 (2005) 2995–2999 www.elsevier.com/locate/atmosenv
Technical note
On plume meandering in unstable stratification Chun-Ho Liu, Dennis Y.C. Leung Department of Mechanical Engineering, The University of Hong Kong, Hong Kong Received 3 September 2004; received in revised form 21 January 2005; accepted 1 February 2005
Abstract Vertical plume meandering of gaseous pollutant is commonly experienced in the daytime atmospheric boundary layer (also know as convective boundary layer, CBL) that arose from the complicated interaction between buoyancygenerated turbulence and gravitational force. It leads to rapid pollutant mixing that cannot be accurately modeled by conventional Gaussian plume model. In the light of explaining the mechanism of plume rises and descents in CBLs, this study employs a direct numerical simulation (DNS) technique to compute the plume behaviors for pollutant emitted from line sources placed parallel to the spanwise direction in an unstably stratified turbulent open channel flow. The DNS results show that the plume meandering is due to the domination of uni-directional mean vertical pollutant fluxes above and below the mean plume height. r 2005 Elsevier Ltd. All rights reserved. Keywords: Convective boundary layer (CBL); Plume rise and descent; Direct numerical simulation (DNS); Gaussian plume model
1. Introduction The Gaussian plume model is the most commonly adopted practical approach to handle transport of passive and inert gaseous pollutants in the atmospheric boundary layer (ABL) due to its versatility in use. However, its accuracy is in doubt in convective boundary layers (CBLs) as the turbulence structure and the plume trajectories are complicated by the upward heat flux at ground level. The plume meandering, in the form of rises and descents, has been confirmed by a series of laboratory experiments (e.g. Willis and Deardorff, 1976, 1978, 1981). Several studies of second-order closure turbulence model (e.g. Liu and Leung, 2001a, b) and the more sophisticated large-eddy simulation (LES, e.g. Dosio et al., 2003) have further supported the findings in this area. As there has been no Corresponding author. Tel.: +852 2857 8615; fax:+852 2858 5415. E-mail address: [email protected] (C.-H. Liu).
agreed subgrid-scale parameterization for second-order closure turbulence model or LES yet, more detailed mathematical modeling is deemed for our better understanding on the mechanism of plume meandering in CBLs. By calculating explicitly all the significant energycarrying scales, direct numerical simulation (DNS) is an alternative to explain the detailed turbulent pollutant transport in that regard. This paper is a quick and brief description on our recent DNS findings explaining the behaviors and mechanism of plume meandering in unstably stratified flow.
2. Methodology Because of the highly refined (spatial and temporal) scales required in DNS, the turbulent flow in an engineering-scale open channel of height H is modeled instead of the ABL. The current DNS model employs the incompressible Navier–Stokes, continuity, energy conservation, and pollutant transport equations whose
1352-2310/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2005.02.024
ARTICLE IN PRESS 2996
C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
findings are then used to explain the plume meandering in CBLs. In the open channel, the flow field is assumed to be horizontally homogeneous, and is bounded by a non-slip wall and a shear-free boundary, respectively, at the bottom and the top. The flow is driven by a background pressure force in the streamwise direction x. A temperature difference is prescribed across the vertical height z of the channel with heating at the bottom so that the buoyancy is opposite to the gravity (unstable stratification). Pollutant line sources of strength C are placed parallel to the spanwise direction y at the channel inlet. The pollutant emission and boundary condition are assumed to be periodic in the spanwise direction. An open boundary is prescribed at the downstream channel outlet. Zero Neumann boundary conditions are assumed elsewhere. Based on the aforementioned flow configuration and a reference velocity scale U, the Reynolds number is 3,000, the Prandtl and Schmidt numbers are 0.72, and the Richardson number is about 0.2. The spatial domain, which is discretized into 448 64 96 ¼ 2; 752; 512 brick elements, is solved by the three-dimensional Galerkin finite element method (FEM). Equal-order trilinear interpolating polynomials are employed for the approximation of the velocity, pressure, temperature, and pollutant concentration in the spatial domain. In the temporal domain, the advection and diffusion terms are solved by the explicit third-order accurate Runge–Kutta and the implicit second-order accurate Crank-Nicolson schemes, respectively. The buoyancy term is solved explicitly by the first-order accurate Euler scheme. The resulting linear equation systems are solved by the Conjugate Gradient method with the Jacobian preconditioning. The calculation was performed on a 16-node Linux Intel PIII PC cluster. Detailed numerical methodology is discussed elsewhere (Liu et al., 2003). The DNS FEM code was validated by calculating turbulent plane Couette flow and scalar transport (Liu, 2003). It should be mentioned that one of the major weaknesses of the current DNS calculation is that it is unable to represent all the processes in the ABL in true scale (for example, the full spectrum of turbulence motions and the eddies of size extending across the ABL thickness). Nevertheless, it is expected that the use of DNS to calculate explicitly all the significant energycarrying scales of fluid turbulence and pollutant transport in reduced scale can enrich our understanding on the basic physics of turbulent plume behaviors in CBLs and help to explain the commonly observed pollutant plume behaviors in the ABL.
3. Results and discussions In this paper, statistical means are denoted by parenthesis /S that represent the spatial (in the
spanwise direction) and temporal averaged values. The double primes 00 denotes the deviation from the statistical mean c00 ¼ c hci. The mean plume height z¯ ðxÞ and the dispersion coefficient sz ðxÞ are calculated in R H R H accordance with z¯ ðxÞ ¼ 0 cðx; zÞ z dz= 0 cðx; zÞ dz R H R H and s2z ðxÞ ¼ 0 cðx; zÞ ½z z¯ ðxÞ 2 dz= 0 cðx; zÞ dz, respectively, where c is the pollutant concentration. A rapid plume rise is observed for the pollutant emitted at the lower part of the channel at zs =H ¼ 0:25 (Fig. 1a) where zs is the pollutant emission height. The pollutant travels near the wall-level in the near field ðx=Hp4Þ that leads to high pollutant concentration there. Afterward, the plume ascends (at 4ox=Ho5Þ that drifts the mean plume height from the lower part to the center core of the channel. This plume rise also causes a rapid increase in pollutant concentration at the upper part of the channel. The plume is carried aloft with a maximum mean plume height z¯ =H ¼ 0:62 at x=H ¼ 8:5 before converges to the channel center thereafter. Engineering practice often assumes Gaussian-shape plume model in the crosswind direction that calculates the spatial distribution of pollutant. Using the same dispersion coefficient determined by the current DNS output and assuming Gaussian-shape pollutant dispersion, the pollutant distributions on the vertical plane for zs =H ¼ 0:25 shows neither plume rise nor descent (Fig. 1b). As a result, the plume generally travels horizontally near the wall-level along the streamwise direction and does not switch the ground-level maximum pollutant concentration to the top throughout the channel. A rapid plume descent is observed for the scalar emitted at the upper part of the channel at zs =H ¼ 0:75 (Fig. 1c). The plume rises slightly in the near field before x=Hp4 then descends rapidly from the upper part to the center core and lower parts of the channel at 4ox=Ho5. The descending plume eventually develops a local maximum of pollutant concentration traveling at the wall level in the far field ðx=HX5Þ. The descending mean plume height passes through the channel center at x=H ¼ 5 followed by an overshoot of minimum mean plume height z¯ =H ¼ 0:4 at x=H ¼ 8:5. The Gaussian-shape pollutant dispersion for an emission height zs =H ¼ 0:75 is illustrated in Fig. 1d. Unlike the plume behavior determined by the current DNS model, no significant plume meandering is calculated by Gaussian plume model and the plume generally travels toward the downstream outlet around its emission height. It is worth mentioning that the aforementioned pollutant plume meandering has been confirmed by a series of convective water tank experiments (Willis and Deardorff 1976, 1978, and 1981), the empirical secondorder closure model of the ABL (Liu and Leung 2001a, b), and the LES of CBLs (Dosio et al. 2003). In
ARTICLE IN PRESS C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
2997
z/H
1 0.5
(a) 0 z/H
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x/H Fig. 1. Comparison of the spatial contours of the dimensionless mean pollutant concentration hci=C calculated by the current DNS model and the conventional Gaussian plume model. Dimensionless mean pollutant concentration for emission heights zs =H ¼ 0:25: (a) DNS model and (b) Gaussian model; and zs =H ¼ 0:75: (c) DNS model and (d) Gaussian model. Also shown are the mean plume height z¯ =H: dashed line and the mean plume coverage ¼ 2sz =H: white bars.
particular, the mean plume heights and dispersion coefficients calculated by the LES and second-order closure models are close to those determined from the water tank experiments. In fact, the current DNS results are inline with the findings from various approaches discussed above that signifies the accuracy of the DNS FEM model and its representative of the physical processes in the realistic ABL. In the following discussion, the plume meandering mechanism will be explained using the current DNS results from a mathematical point of view. Turbulent transport (or mixing) of pollutant in the vertical direction is represented mathematically by the mean vertical pollutant flux hc00 w00 i (w is the vertical velocity) which also accounts for the correlation between vertical velocity and pollutant concentration. Because of plume development and widening, the mean vertical pollutant fluxes are generally positive and negative, respectively, above and below the mean plume height. This configuration signifies that upward flow ðw00 40Þ results in an increase in pollutant concentration ðc00 40Þ while downward flow ðw00 o0Þ results in a decrease in pollutant concentration ðc00 o0Þ, for region above the mean plume height. On the other hand, for region below the mean plume height, upward flow results in a decrease in pollutant concentration while downward flow results in an increase in pollutant concentration.
As there is no vertical mean flow ðhwi ¼ 0Þ, pollutant is transported in the vertical direction by diffusion in which most are by turbulence while an insignificant amount is by molecular scales. Looking into the structure of mean vertical pollutant flux helps elucidate our understanding on the mechanism of vertical plume meandering. Generally, in the near field ðx=Hp2Þ, the pollutant emitted from zs =H ¼ 0:25 (Fig. 2a) and 0:75 (Fig. 2b) shows positive and negative mean vertical pollutant fluxes above and below the mean plume height, respectively. This finding is inline with our aforementioned hypothesis of plume development. In the far field ðx=H42Þ, the pollutant emitted from the lower and upper parts of the channel exhibits different structures of mean vertical pollutant flux. For the pollutant emitted at zs =H ¼ 0:25, the positive mean vertical pollutant flux ascends from the emission height to the center core of the channel at 4ox=Ho5 (Fig. 2a). It covers the mean plume height and almost half of the vertical height of the channel. The domination of positive mean vertical pollutant flux signifies that the pollutant transport is in a single direction (upward) regardless of the position. It thus explains the mechanism of the rapid plume rise in CBLs. A similar rationale can be used to explain the rapid plume descent for the pollutant emitted at zs =H ¼ 0:75. After descending from the emission height and a slight rebound at 4ox=Ho5, a broad negative mean vertical pollutant flux is developed at the center core of the channel
ARTICLE IN PRESS C.-H. Liu, D.Y.C. Leung / Atmospheric Environment 39 (2005) 2995–2999
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z/H
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x/H Fig. 2. Dimensionless mean vertical pollutant flux hc00 w00 i=ðCUÞ calculated by the current DNS model for emission heights: (a) zs =H ¼ 0:25 and (b) zs =H ¼ 0:75. Also shown are the mean plume height z¯ =H: dashed line and the mean plume coverage ¼ 2sz =H: white bars.
Fig. 3. Comparison of the dimensionless mean plume height z=H calculated by the current DNS model and the conventional Gaussian plume model. Emission height zs =H ¼ 0:25: DNS model, solid line; Gaussian model, dotted line. Emission height zs =H ¼ 0:75: DNS model, dashed line; Gaussian model, dashed-dotted line.
covering the mean plume height (Fig. 2b). Hence, the mean pollutant transport is solely downward and leads to the rapid plume descent. The mean plume heights for emission heights zs =H ¼ 0:25 and 0:75 calculated by the current DNS FEM model and the conventional Gaussian plume models are contrasted in Fig. 3 for comparison. For the pollutant emitted at zs =H ¼ 0:25, the maximum mean plume height calculated by the current DNS FEM model is z¯ =H ¼ 0:6 while the value calculated by Gaussian plume model is around 0.3. Besides, the current DNS results show that the pollutant emitted at zs =H ¼ 0:75 descends down to z¯ =H ¼ 0:4. However, the corresponding maximum plume descent calculated by Gaussian plume model is z¯ =H ¼ 0:65. Hence, Gaussian plume model is over-simplified for handling the meandering of mean plume height in unstable stratification accurately. In particular, it is unable to determine the overshoots of pollutant plume ascent and descent. It is thus recommended that the conventional Gaussian plume models should be applied with caution to explain pollutant plume behaviors in CBLs.
4. Conclusion The plume transport phenomenon in an unstably stratified open channel flow is studied using DNS
technique. In contrasts to the conventional Gaussian plume model calculation, plume meandering is found in the flow which is due to the domination of a singledirection mean vertical pollutant flux beyond the near field. The plume rise with pollutant emitted at zs =H ¼ 0:25 is due to the broad positive mean vertical pollutant flux covering the mean plume height. Similarly, the plume descent with pollutant emitted at zs =H ¼ 0:75 is due to the broad negative mean vertical pollutant flux covering the mean plume height. The present work is targeted on the plume meandering in unstable stratification from a mathematical point of view. Future works will be focused on DNS of pollutant plume behaviors in unstable stratification at higher Reynolds numbers as well as LES of CBLs of realistic ABL scales so as to verify the current DNS findings.
Acknowledgements The authors would like to thank the Computer Centre (CC) of the University of Hong Kong (HKU) for permitting us to access to its High Performance Computing Clusters systems. In particular, we are grateful to Mr. Wing-Keung Kwan and Mr. Chat-Ming Woo of HKU CC for their kind assistance during the development of the DNS FEM code.
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