An Empirical Theoretical Model for the Turbulent Diffusion Coefficient in Urban Atmospheric Dispersion

Abstract

Turbulent diffusion plays a critical role in atmospheric pollutant dispersion, particularly in complex environments such as urban areas. This study proposes a novel theoretical approach to enhance the calculation of the turbulent diffusion coefficient in pollutant dispersion models. We propose a new expression for the turbulent diffusion coefficient (KC), which incorporates both hydrodynamic and turbulence-related time scales. This formulation links the turbulent diffusion coefficient to pollutant travel time and turbulence intensity, offering more accurate predictions of pollutant concentration distributions. By addressing the limitations of existing empirical models, this approach improves the parameterization of turbulence and reduces uncertainties in predicting maximum individual exposure under various atmospheric conditions. The study presents a theoretical model designed to advance the current understanding of atmospheric dispersion modeling. Experimental validation, while recommended, is beyond the scope of this work and is suggested as a direction for future empirical research to confirm the practical utility of the model. This theoretical formulation could be integrated into urban air quality management frameworks, providing improved estimations of pollutant peaks in complex environments.

1. Introduction

Atmospheric dispersion of pollutants presents a significant challenge, particularly in urban and industrial environments, where emissions from point sources can lead to short-term, high-concentration exposure events [1,2,3]. Predicting maximum individual exposure (MIE) in such scenarios is critical for risk assessment, due to the potentially severe health impacts associated with high pollutant concentrations [4,5,6]. Traditional approaches to estimating MIE often rely on statistical models and probability density functions to predict peak concentrations based on empirical data [7,8,9]. Although these models offer valuable insights, they often fail to capture the complex behavior of turbulent flows in real-world environments such as urban areas, where buildings and surface geometries greatly influence dispersion [10,11,12].

Turbulent diffusion is a critical mechanism in atmospheric dispersion, determining how pollutants spread in complex environments [13,14,15]. Accurate modeling of this process is essential for predicting pollutant concentration fields and assessing MIE to hazardous substances [16,17,18]. Over the years, a wide range of models have been developed, from basic empirical approaches to advanced Computational Fluid Dynamics (CFD) methods. One of the earliest and most widely used models for pollutant dispersion is the Gaussian plume model, which offers a simplified steady-state solution for predicting pollutant concentrations from continuous point sources under the assumption of a homogeneous atmosphere. This model remains useful for basic scenarios but is significantly limited when applied to complex environments such as urban areas, where obstacles like buildings lead to localized turbulence that it fails to account for.

In addition to Gaussian plume and CFD methods, Lagrangian models represent a third approach commonly used in pollutant dispersion modeling. Lagrangian models track the motion of individual particles or “packets” of pollutants, offering unique advantages in complex environments such as urban areas, where the local flow field is strongly influenced by buildings and obstacles. These models are particularly effective in capturing pollutant dispersion in highly dynamic and geometrically complex settings, where Eulerian methods might face limitations.

Numerous studies in the field of atmospheric dispersion modeling have highlighted the significant impact of turbulence intensity and pollutant travel time on pollutant dispersion patterns. For instance, ref. [11] discusses how local turbulence levels in urban canyons strongly affect pollutant concentration distribution, resulting in complex patterns that are difficult to capture by steady-state models. Such findings align with the theoretical foundation of our model, which integrates pollutant travel time and local turbulence intensity to provide a more accurate prediction in a dynamic environment.

Similarly, ref. [19] demonstrated through large-eddy simulations that time-dependent turbulence parameters can substantially improve the accuracy of pollutant dispersion models, particularly in heterogeneous environments such as urban areas. Their findings underscore the importance of including time-variant factors in models that aim to predict pollutant concentration peaks in environments with high spatial and temporal variability.

Studies have also shown that traditional models, such as the Gaussian plume, often fail to account for the influence of buildings and other structures on local turbulence patterns. Research by [14] highlights the limitations of steady-state models in urban settings, where obstacles can create localized turbulence that alters dispersion significantly. Our model, by incorporating dynamic turbulence adjustments, aims to address these limitations, providing a theoretical advantage in settings where conventional models struggle to capture time-variant pollutant behavior.

To overcome the above-mentioned limitations, Reynolds-Averaged Navier–Stokes (RANS) models were developed. These models provide time-averaged solutions to the turbulent fluctuations in the flow, making them computationally efficient for large-scale atmospheric flows. Despite their efficiency, RANS models struggle to resolve small-scale turbulent structures critical for accurate pollutant dispersion, especially in areas with sharp geometrical changes, such as urban street canyons. More advanced techniques, such as Large Eddy Simulation (LES), have been introduced to address this issue. LES directly resolves larger turbulent eddies and models smaller-scale turbulence using sub-grid-scale models, providing a detailed representation of pollutant dispersion, particularly in urban environments. However, LES is computationally intensive and impractical for large-scale or real-time simulations [19,20,21].

Hybrid RANS-LES models have been introduced to balance the computational efficiency of RANS and the turbulence-resolving capabilities of LES. These models show promise in applications where both large-scale atmospheric flows and local dispersion patterns need accurate modeling. Nevertheless, even hybrid models face challenges in environments where pollutant travel time and turbulence intensity significantly affect dispersion, and traditional models, despite empirical corrections, often fail to capture time-dependent pollutant behavior in transient emissions.

In light of these limitations, this study proposes a novel theoretical framework aimed at improving the calculation of turbulent diffusion coefficients in RANS-based models. The new model integrates pollutant travel time and turbulence intensity to offer a more dynamic and accurate prediction of concentration fields and MIE under varying atmospheric conditions. By addressing the shortcomings of traditional models, this work contributes to the advancement of atmospheric dispersion modeling, particularly in urban and industrial environments where localized turbulence and short-term high-concentration exposure pose significant risks.

This study is theoretical in nature and aims to propose a new framework. No new simulations or experiments are performed, and the approach is demonstrated using published datasets. Empirical validation is left for future work.

2. Methodology

This study applies a CFD-RANS methodology to model atmospheric dispersion based on data from the Prairie Grass experiment, focusing on predicting mean concentration values in an open, neutral atmosphere. The methodology involves key parameters—such as the turbulent diffusion coefficient, concentration time scale, and hydrodynamic time scale—to enhance the accuracy of pollutant dispersion predictions.

2.1. Calculation of the Turbulent Diffusion Coefficient

The turbulent diffusion coefficient KC is dynamically adjusted to capture the dispersion characteristics under varying atmospheric conditions. The formulation for KC incorporates both the turbulent viscosity coefficient Km, the concentration time scale TC, and the hydrodynamic time scale Tu, expressed as

KC = Km × TC/Tu,

(1)

Equation (1) is proposed in this study, combining turbulent viscosity, concentration time scale, and hydrodynamic time scale, based on dimensional analysis and physical interpretation of turbulence–pollutant interaction.

The turbulent diffusion coefficient, KC, represents the effective rate of pollutant spread within a turbulent flow. By adjusting dynamically according to local turbulence characteristics, KC captures variations in pollutant dispersion that are otherwise challenging to model. This formulation integrates three key components—turbulent viscosity (Km), concentration autocorrelation time scale (TC), and hydrodynamic time scale (Tu)—to provide a more responsive measure of dispersion, particularly under fluctuating flow conditions.

The parameters of Equation (1) are defined as follows:

• Turbulent Viscosity Coefficient Km:

Km = Cμ × k²/ε,

(2)

where

• Cμ is an empirical constant;
• k is the turbulent kinetic energy, representing the intensity of turbulence;
• ε is the rate of dissipation of turbulent kinetic energy.

The turbulent viscosity coefficient, Km, is derived from the empirical constant Cμ, turbulent kinetic energy (k), and the dissipation rate of turbulent kinetic energy (ε). This coefficient quantifies the ability of the flow to transport pollutants away from the source and is particularly sensitive to variations in turbulence intensity. Higher values of Km correspond to enhanced pollutant transport, which is critical in environments with intense turbulence, such as urban areas with complex geometries.

The proposed turbulent diffusion coefficient KC could also be integrated into Lagrangian models to enhance their predictive accuracy. By incorporating pollutant travel time and turbulence intensity, this formulation could improve the representation of dispersion in environments with varying turbulence levels and transient conditions, making it applicable in both Lagrangian and Eulerian frameworks. Such integration might also benefit advanced Gaussian/Lagrangian cloud models, offering an improved theoretical basis for handling pollutant dispersion in urban and industrial environments.

2.

Concentration Autocorrelation Time Scale TC:

TC = LC/√k,

(3)

where LC is the turbulent length scale related to concentration fluctuations, indicating the distance over which the concentration remains correlated.

The concentration autocorrelation time scale, TC, represents the temporal scale over which pollutant concentration remains correlated in the flow. Defined as the ratio of the turbulent length scale for concentration (LC) to the characteristic velocity fluctuation (u’), TC indicates how quickly concentration variations dissipate over time. This parameter is essential for capturing the local dispersion characteristics, especially in cases of transient or time-dependent pollutant releases.

3.

Hydrodynamic Time Scale Tu:

Tu = Lu/√k,

(4)

where Lu represents the turbulent length scale related to velocity fluctuations, indicating the distance over which momentum fluctuations remain correlated.

The hydrodynamic time scale, Tu, describes the temporal scale of momentum fluctuations in the turbulent flow, with Lu denoting the turbulent length scale associated with velocity fluctuations and v’ representing the characteristic velocity fluctuation. Tu provides insight into the persistence of momentum structures within the flow, which influences the pollutant transport patterns, particularly in stable and calm flow regimes.

These relationships allow KC to adjust according to local turbulence levels, providing more accurate mean concentration predictions over varied distances in the Prairie Grass experiment setup.

The KC formulation is intended to be implemented within existing RANS solvers or as a post-processing correction in LES. Its dynamic structure allows real-time adaptation to local turbulence characteristics, making it suitable for urban-scale CFD studies.

2.2. Governing Equations for Mean Concentration Prediction

The mean concentration $\overline{C}$ is computed based on time-averaged flow properties obtained through the RANS equations. The k-ζ turbulence model is employed to accurately represent the turbulent kinetic energy and dissipation rate, enabling the model to capture pollutant dispersion in open terrains effectively.

2.3. Validation Using Prairie Grass Experimental Data

The comparison is performed using existing data from the Prairie Grass experiment, which serves as a historical benchmark for model demonstration purposes, not as a formal validation. Dimensionless scatter plots of mean concentration values demonstrate the model’s effectiveness, showing good alignment with experimental data.

This study relies on a theoretical framework rather than empirical testing. As such, while parameters and theoretical assumptions are detailed in the model development, practical application details (e.g., data requirements, conditions) are not fully expanded. Future studies should outline specific steps for applying this model under different atmospheric scenarios, including parameter calibration guidelines.

While this study primarily applies the proposed model within an RANS framework, future work could explore its applicability in Lagrangian-based models, including modern Gaussian/Lagrangian cloud models. This extension could potentially enhance these models’ flexibility and accuracy in simulating pollutant dispersion in complex environments. The addition of pollutant travel time and turbulence intensity within a Lagrangian framework could make these models more responsive to local flow variations, which are often critical in urban and industrial settings.

The Prairie Grass experiment was a landmark field study conducted in the 1950s in Nebraska, USA, to investigate atmospheric dispersion from a continuous ground-level point source under relatively flat terrain and neutral atmospheric conditions. Sulfur dioxide (SO₂) was released, and its concentration was measured at multiple downwind locations and heights under varying meteorological conditions. This dataset is widely used for validating dispersion models due to its controlled setup and publicly available measurements [22]. In this study, we used data from this experiment to evaluate the ability of our model to predict mean pollutant concentration profiles.

3. Results and Discussion

The development of the proposed turbulent diffusion coefficient (KC) offers a theoretical advancement in understanding pollutant dispersion in complex environments. While no new experiments or simulations were conducted for this study, the formulation of the KC provides a framework for enhancing existing models of atmospheric dispersion, particularly in urban and industrial settings.

3.1. Theoretical Implications of the Model

The proposed model, by integrating both hydrodynamic and pollutant-specific time scales, represents a significant shift from traditional approaches that rely on empirical adjustments. The concentration autocorrelation time scale (TC) and the turbulent velocity time scale (Tu) offer dynamic parameters that adjust to the variations in turbulence intensity and pollutant travel time, thus addressing key limitations of steady-state models.

In traditional models such as the Gaussian plume and RANS-based approaches, the assumption of constant turbulence across the flow domain leads to oversimplifications, especially in complex environments like urban areas. The introduction of time-dependent parameters allows the proposed model to dynamically capture the time-varying nature of pollutant dispersion, offering more accurate predictions, theoretically speaking, in scenarios involving short-term, high-concentration exposures. This aspect is critical for assessing potential health risks in urban street canyons or industrial facilities.

For example, Gaussian models do not include time-dependent turbulence or pollutant-specific time scales, which renders them ineffective for predicting short-term peaks in complex urban topographies. In contrast, the KC model can accommodate these factors in a unified formulation, linking time-resolved turbulence effects with pollutant behavior.

As an example, in this study, a key simulation (CFD-RANS) based on the Prairie Grass experiment was performed to evaluate pollutant dispersion in a flat terrain scenario in the absence of obstacles, with sulfur dioxide (SO₂) emitted from a point source in the atmospheric boundary layer. The study compared two cases using a RANS-based modeling framework:

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Case 1: the proposed dynamic KC model, dependent on pollutant travel time and turbulence intensity.

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Case 2: a traditional RANS approach assuming full mixing, with constant turbulence and KC independent of travel time.

The simulations were used to produce a scatter plot of the dimensionless concentration values for varying distances from the source (Figure 1), highlighting the agreement between the model’s predictions and experimental data under neutral atmospheric conditions. These results were critical in demonstrating the new model’s accuracy in capturing both the mean and maximum pollutant concentrations in such configurations.

This simulation framework offers valuable insights for future studies in similar environmental contexts, where pollutant dispersion is influenced primarily by atmospheric turbulence without complex surface structures.

Although the model shows good theoretical alignment with known dispersion behavior and is supported by comparison with Prairie Grass data, a comprehensive validation using a larger range of experimental datasets and modern CFD simulations (e.g., urban LES studies) is essential. This will be addressed in future research. The current study focuses on the theoretical formulation and a proof-of-concept application.

Although the Prairie Grass experiment was conducted under relatively simple terrain conditions, the enhanced accuracy of Case 1 suggests that the dynamic KC formulation is inherently more responsive to real atmospheric conditions, even when geometry is not complex. This supports its expected value in more intricate environments.

3.2. Comparative Evaluation of Case 1 and Case 2

To evaluate the theoretical advantages of the proposed turbulent diffusion coefficient (KC), two modeling scenarios were considered. Case 1 employed the new KC formulation, which dynamically adjusts based on pollutant travel time and turbulence intensity. Case 2 followed a traditional RANS approach, assuming full mixing and constant turbulence, with a fixed KC value independent of time-dependent dispersion characteristics.

The RANS simulations were performed using a structured mesh consisting of approximately 270,000 cells. Boundary conditions included a fixed velocity inlet, a pressure outlet, and no-slip conditions on solid walls. Convergence was achieved when all residuals dropped below 10⁻⁵.

Figure 1 presents a scatter plot of model-predicted concentrations (y-axis) versus experimental values (x-axis), using data from the Prairie Grass experiment. Perfect agreement would correspond to points lying along the diagonal. Case 1 demonstrates improved agreement with measured concentrations, especially under varying downwind distances, indicating the model’s ability to better capture the evolving dispersion profile. In contrast, Case 2 showed less accurate performance, particularly in regions where turbulence-driven variability plays a dominant role.

This comparison highlights a key theoretical strength of the proposed model: its adaptability to local turbulence conditions. By incorporating hydrodynamic and pollutant-specific time scales, the model can respond to variations in turbulence intensity and pollutant transport. This is particularly important in urban environments with complex building layouts, where pollutant behavior is highly sensitive to localized flow structures.

In high-turbulence scenarios—such as urban street canyons with irregular geometries—the proposed KC model is expected to predict higher mixing rates and more accurate peak concentrations due to its sensitivity to turbulence intensity. Conversely, in low-turbulence environments such as rural areas or calm conditions, the model can reflect reduced dispersion through longer travel times, where traditional models might overpredict mixing.

From a practical standpoint, the model’s dynamic formulation has potential utility in environmental risk assessment and urban air quality management. Its theoretical ability to capture pollutant hotspots and time-varying exposure could support more targeted mitigation strategies. Although this study does not include full experimental validation, the comparison with historical data provides a solid theoretical foundation for further development. Future work should focus on applying the model in complex real-world urban domains, with detailed simulations and empirical data to quantify its predictive performance. The improved alignment of Case 1 is consistent with findings from the literature that stress the importance of turbulence-resolving parameters in capturing near-field dispersion accurately. The incorporation of time scales in KC enhances the model’s responsiveness, unlike static RANS approaches.

In regulatory and safety frameworks, accurately predicting high-exposure zones can inform zoning laws, emergency planning, and industrial design. A model that accounts for turbulence evolution offers a step forward compared to static approximations.

3.3. Limitations and Future Research Directions

While the proposed model offers theoretical improvements over traditional models, it has yet to be validated through simulations or empirical data. Further research is needed to test the model’s predictions against real-world data and refine its parameters under various atmospheric conditions. Additionally, the model assumes non-reactive pollutants, limiting its applicability in environments where chemical reactions are significant. Future work should aim to extend the model to include pollutant chemistry, particularly in urban environments where secondary pollutants are commonly formed.

Although this study introduces a novel theoretical framework for calculating turbulent diffusion coefficients, no new experimental validation was conducted. Future research should aim to empirically test the proposed model to further establish its accuracy across various atmospheric conditions. Additionally, conducting controlled field experiments similar to the Prairie Grass experiment could provide more robust data for model validation, particularly in complex urban environments.

3.4. Model Applicability and Data Requirements

The proposed model is designed for applications in atmospheric dispersion scenarios where pollutant travel time and turbulence intensity significantly influence concentration fields. Typical use cases include near-field dispersion in urban or industrial environments. To apply the model, the following input data are required:

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Mean wind velocity field (from simulations or field measurements);

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Turbulent kinetic energy (k) and its dissipation rate (ε);

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Turbulent length scales for velocity and concentration (Lu and Lc);

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Turbulence intensity or standard deviation of velocity fluctuations.

These parameters can be obtained from RANS or LES or derived from observational data. The main sources of uncertainty in applying the model include variability in atmospheric conditions, measurement inaccuracies, and assumptions related to the turbulence length scales. While the current formulation is deterministic, future work will incorporate stochastic approaches to address such uncertainties.

4. Conclusions

This study introduces a novel theoretical framework for improving the calculation of the turbulent diffusion coefficient (KC) in atmospheric pollutant dispersion models. By incorporating both hydrodynamic and pollutant-specific time scales, the proposed model offers a dynamic and more accurate representation of how pollutants disperse in complex environments such as urban and industrial areas.

Unlike empirical models, which often rely on calibration, the proposed KC formulation is based on first-principles dimensional analysis and physical scaling, making it more generalizable across scenarios.

The proposed model’s theoretical contributions are applicable in various atmospheric and industrial scenarios where traditional dispersion models face limitations. By incorporating time-dependent turbulence characteristics, this model can better predict pollutant behavior in environments with high spatial and temporal turbulence variations, such as urban street canyons and industrial facilities. Its adaptability to varying turbulence intensities renders it suitable for use in risk assessment, environmental policy planning, and urban air quality management, where precise concentration predictions are essential.

Future research should explore several directions to build on the theoretical contributions of this study with respect to the model validation, the extension to reactive pollutants, and the integration with hybrid models. Model validation through simulations or field experiments is essential to test the accuracy of the predictions and refine the empirical constants involved. Furthermore, extending the model to account for chemical reactions, especially in urban environments where secondary pollutants are commonly formed, will broaden its applicability. Finally, the integration of this model with hybrid CFD approaches, such as RANS-LES models, could enhance its accuracy while maintaining computational efficiency.

Despite these promising theoretical advancements, this study deliberately limits itself to a theoretical framework. The aim is to propose a physically grounded formulation for the turbulent diffusion coefficient, which future research can evaluate using comprehensive numerical or experimental validation. As such, this work should be viewed as a necessary theoretical foundation rather than a validated modeling tool.

This study presents a theoretical model aimed at advancing current understanding in atmospheric dispersion modeling. Experimental validation, while recommended, is beyond the scope of this work and is suggested as a direction for future empirical research to confirm the practical utility of the model.

The next step in the model’s development is to integrate it within CFD solvers and validate it against simulations in real urban settings, including LES and hybrid RANS-LES approaches.

Its flexibility to integrate with both RANS and Lagrangian frameworks opens new pathways for practical use in environmental engineering tools and policy-making instruments, bridging the gap between high-fidelity modeling and decision-making needs.