Low-cost simulations providing accurate predictions of transport of airborne material in urban areas, vegetative canopies, and complex terrain are demanding because of the small-scale heterogeneity of the features influencing the mean flow and turbulence fields. Common models used to predict turbulent transport of passive scalars are based on the Lagrangian stochastic dispersion model. The Quick Environmental Simulation (QES) tool is a low-computational-cost framework developed to provide high-resolution wind and concentration fields in a variety of complex atmospheric-boundary-layer environments. Part of the framework, QES-Plume, is a Lagrangian dispersion code that uses a time-implicit integration scheme to solve the generalized Langevin equations which require mean flow and turbulence fields. Here, QES-Plume is driven by QES-Winds, a 3D fast-response model that computes mass-consistent wind fields around buildings, vegetation, and hills using empirical parameterizations, and QES-Turb, a local-mixing-length turbulence model. In this paper, the particle dispersion model is presented and validated against analytical solutions to examine QES-Plume’s performance under idealized conditions. In particular, QES-Plume is evaluated against a classical Gaussian plume model for an elevated continuous point-source release in uniform flow, the Lagrangian scaling of dispersion in isotropic turbulence, and a non-Gaussian plume model for an elevated continuous point-source release in a power-law boundary-layer flow. In these cases, QES-Plume yields a maximum relative error below

Rapid growth of urban populations around the world impacts all sectors of human activity, including industry and transportation. Additionally, growth is also increasing pressure on agricultural systems to boost yields. These trends raise concerns about the deterioration of the environment, a decline in quality of life, or worsening air quality

In response to these and other issues, a number of fast-response transport and dispersion models for urban areas and complex terrain have been developed. Fast-response models are characterized by their ability to keep computational costs low while providing realistic representations of the effects of buildings, canopies, and terrain on velocity distributions and the dispersion of scalars

Examples of operational LSDMs are QUIC-PLUME

Another area of study where fast-dispersion models can have a big impact is aerobiology, the study of the aerial dispersion of biological particles in the atmosphere

Numerical integration of the equations used in LSDMs is challenging due to the stiffness of the model's mathematical formulation

Several methods have been presented to integrate LSDMs.

To address the issues presented above, the Quick Environmental Simulation (QES) framework was developed to provide high-resolution wind and concentration fields in complex urban and agricultural environments. The framework is composed of QES-Winds, a 3D fast-response model that computes mass-consistent wind fields around buildings and vegetation using empirical parameterizations

In the following, we first describe the mathematical formulation of the LSDM in Sect.

The motion of passive tracers in turbulent flow can be described by a random-walk model

For stationary, homogeneous, and isotropic turbulence, a simplified model is obtained by writing

The model in Eq. (

The GLEs are considered stiff because of the presence of the wide range of timescales. In particular, a particle may travel a significant distance over a small time step because of the existence of instabilities in the numerical solution. The SLEs drastically reduce the number of terms in the GLEs and are considered less unstable, however, numerical instabilities still exist for the SLEs

On the ground and at building surfaces, reﬂecting boundary conditions are used. At the domain top, an outlet condition is used, indicative of particles traveling past the top of the domain. Yet, the top boundary conditions can be more complex if the top of the atmospheric boundary layer is considered (see

To calculate the trajectory of fluid particles, the QES framework is composed of three main modules: (1) QES-Winds, a mass-consistent wind model; (2) QES-Turb, a turbulence model; and (3) QES-Plume, an LSDM. Figure

Simulation workﬂow of the QES framework.

The wind field is obtained using QES-Winds. This model is based on the framework introduced by

To accurately represent particle motion in turbulent flows, the LSDM needs the stress tensor and the dissipation rate of TKE. The QES framework calculates turbulence variables using a local-mixing model based on Prandtl’s mixing-length and the Boussinesq eddy-viscosity hypotheses

From the eddy-viscosity model, the TKE can be defined as

The strain-rate tensor from Eq. (

For boundary-layer flows, there is ample evidence suggesting that the normal stresses are anisotropic and that for flow aligned with the

Summary of

Because the turbulence model is a local-mixing model, it relies heavily on the magnitude of the local velocity gradients to estimate the stress tensor. This is problematic in regions where the velocity gradients are small and leads to the model predicting negligible turbulence, for example at the core of a street canyon. While multiple methods exist to address these issues (e.g.,

A common approach is the use of an explicit scheme to integrate the GLEs. The scheme is conditionally stable with a condition on the time step related to the velocity variance and mean dissipation rate. However, due to the stiffness of the equation, extremely small time steps are required to maintain stability and numerical errors can inject more energy than the viscosity can dissipate. As a consequence, the particle velocity can become arbitrarily large, leading to rogue trajectories

Finally, a forward Euler scheme is used to update the particle position. The position

Workflow of the QES-Plume model using the implicit scheme to solve the 3D GLE. The particle position is advanced from its position

The workflow of the QES-Plume model is presented in Fig.

The new implementation of the LSDM has been tested following the procedure proposed by

The method presented eliminates the possibility of the calculation of a rogue trajectory during particle advection. However, some physical processes, such as reflection or deposition, require the particle to travel only one Eulerian grid cell (i.e., QES-Winds velocity grid) at a time. To control the total distance traveled, a Courant-number-based algorithm is used. The time step is reduced as the particle moves close to a wall. The new time step is calculated following the procedure presented in Algorithm

Courant-Number-based time step reduction.

This procedure is used to divide the user-defined time step into smaller time increments to ensure that the particle travels small distances. Hence, other algorithms requiring that the particle only travels one grid cell can be executed correctly.

Perfect rebound is used as wall boundary conditions for the particles

Wall reflection.

Find closest face along trajectory

Find intersection with face

Example of double reflection in a corner from particle position

This specular rebound approach is the most common treatment of the wall boundary condition for LSDM. However,

In this study, fluid particles are considered massless, and, therefore, the Eulerian concentrations are obtained by counting the number of particles

The performance of QES-Plume has been evaluated against two idealized test cases and a wind-tunnel test case for a

The normalized concentration profiles from the 3D-GLE model computations have been compared to a classical Gaussian solution for an elevated continuous point-source release in a steady-state, horizontally homogeneous, neutrally stable atmosphere with constant wind speed and constant eddy diffusivity

For this test case, the flow was prescribed by a horizontally and vertically uniform wind speed and friction velocity of

The particles were continuously released from a point source at height

Simulation parameters for continuous release in a horizontally and vertically uniform flow test case.

Figure

Profiles of the normalized concentration for the classical Gaussian plume model (line) and the QES-Plume model (diamonds) at three different

In addition, this test case corresponds to the dispersion from a point source in statistically stationary isotropic turbulence presented in

The timescale,

Scaling of trajectories in the spanwise direction for homogeneous isotropic turbulence showing the linear spread for

The next test case examines the performance of the QES-Plume model against an existing analytical solution for a continuous point-source release in a boundary-layer flow. The source was relatively close to the ground (

To obtain near-statistically stationary concentration estimates, 420 000 particles were continuously released from a point source at an emission rate of 200 particles per second with a time step of 1 s (

Simulation parameters for continuous release in a power-law boundary-layer flow test case.

Figure

Profiles of the normalized concentration for the non-Gaussian plume analytical solution (line) and the QES-Plume model (diamonds) at four different

The idealized test cases are useful to evaluate QES-Plume's GLEs solution methodology because they control for external factors that impact the quality of the dispersion model including turbulence parameterization, mean velocity specification, and boundary conditions. The limitation is that they do not fully engage the GLEs because they do not activate all components of the stress and velocity gradient tensors. To more fully examine the performance of QES-Plume’s GLE implementation, it is compared to dispersion data from a wind-tunnel experiment for a

Simulation setup for the

For this test case, QES-Plume was driven using the flow field computed by QES-Winds. Following

A total of 1 950 000 particles were released continuously for 3900 s (i.e.,

Simulation parameters for the array of cubical buildings test case.

Non-dimensional concentration field from QES-Plume:

Cross sections of the concentration results of dispersion through the building array are shown in Fig.

To begin a quantitative comparison, QES-Plume results are plotted with respect to the concentrations from the wind-tunnel dataset in Fig.

Comparison of QES-Plume concentration (lines) with wind-tunnel data (squares) for the

In the first street canyon (first graph of Fig.

In the second street canyon (

The middle panel (Fig.

Lateral measurements of the concentration are presented in Fig.

Some of the discrepancies found in Fig.

Summary of QES-Plumes's RMSEs and relative RMSEs at various locations for the

Signed relative error (RE) between the concentration field from QES-Plume and the wind-tunnel measurements where positive values (red markers) represent an overestimation by the model. The panels in the figure are

Figure

Finally, the local-mixing turbulence model relies heavily on the magnitude of the local velocity gradients (see Eqs.

Paired scatterplot of the wind-tunnel concentration data and QES-Plume concentrations for the

In summary, QES-Plume is capable of reproducing concentration levels in this complex mock-urban setting despite weak performance in the first street canyon. In particular, the results show the expected linear behavior near the source and constant region farther away as reported by

Summary statistics comparing QES-Plume results with the wind-tunnel data for the

Finally, QES-Plume exhibits excellent computational performance even without parallelization. The

The method implemented to partially solve the stiffness problem from the GLEs (Sect.

The test cases presented by

The local-mixing turbulence model relies heavily on the magnitude of the local velocity gradients (see Eqs.

Due to the presence of a large number of terms in the GLEs, SLEs are employed in most mainstream dispersion models. The presence of numerical instabilities due to the stiffness of the GLEs has also been a problem for the explicit integration of the GLEs into Lagrangian dispersion models. Although commonly used numerical methods for solving the SLEs are still numerically unstable, the SLEs are considered slightly more stable compared to the GLEs due to the drastic reduction in the number of terms. This paper discussed the implementation of the GLEs with the implicit time-integration method from

QES-Plume has been validated against analytical solutions in idealized conditions where the model yielded good results. In particular, the overall maximum relative error was under

Finally, QES-Plume was implemented from the ground up as an object-oriented C++ code and has demonstrated excellent computational performance. Future versions of QES-Plume are likely to use a GPU-based implementation to enable much faster than real-time simulations. The potential use cases of a model like QES-Plume are numerous. For example, the model can be used to run simulations for decision makers for tabletop exercises or to study particulate dispersion in complex environments, such as spore or smoke transport in agricultural fields, as well as urban pollution and air quality.

In this section, the performance of QES-Winds and QES-Turb is briefly discussed in the context of the

Figure

Figure

Profiles at three locations in the first street canyon are presented in Fig.

Figure

Overall, the mean flow along the centerline of the array is in good agreement with the experimental data and illustrates that the model proposed by

Comparison of QES-Winds (lines) with wind-tunnel data (squares) for the streamwise velocity

Comparison of QES-Winds (lines) with wind-tunnel data (squares) for the streamwise velocity

Comparison of QES-Winds (lines) with wind-tunnel data (squares) for the streamwise velocity

Comparison of QES-Winds (lines) with wind-tunnel data (squares) for the streamwise velocity

In this section, the velocity variances from the QES-Turb model and the wind-tunnel data are compared at selected locations to qualitatively evaluate the performance of the model.

Figure

Figure

Figures

In summary, the turbulence model used to drive QES-Plume shows good agreement with the wind-tunnel data. Importantly, the local-mixing model needed the addition of a constant

Comparison of QES-Turb (lines) with wind-tunnel data (squares) for the streamwise-velocity variance

Comparison of QES-Turb (lines) with wind-tunnel data (squares) for the streamwise-velocity variance

Comparison of QES-Turb (lines) with wind-tunnel data (squares) for the streamwise-velocity variance

Comparison of QES-Turb (lines) with wind-tunnel data (squares) for the streamwise-velocity variance

The Quick Environmental Simulation (QES) software has been developed as part of a collaboration between the University of Utah, the University of Minnesota Duluth, and Pukyong National University (

The data from the EPA for the

Conceptualization was done by RS and ERP. The code was written by JAG, LA, and FM. Test cases were developed by BS and FM. The analysis and interpretation of the data were carried out by FM. The figures were produced by FM. The original draft of the paper was written by FM, with edits, suggestions, and revisions provided by JAG, BS, RS, and ERP. Grants supporting the project leading to this publication were awarded to RS and ERP.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to acknowledge the funding provided by the National Institute of Environment Research (NIER), funded by the Ministry of Environment of the Republic of Korea; the United States Department of Agriculture National Institute for Food and Agriculture (USDA-NIFA) Specialty Crop Research Initiative (SCRI); and the United States Department of Agriculture Agricultural Research Service (USDA-ARS) through a research support agreement.

This research has been supported by the National Institute of Environmental Research (grant no. NIER-SP2019-312), the United States Department of Agriculture National Institute for Food and Agriculture (grant nos. USDA-NIFA-SCRI 2018-03375 and USDA-NIFA-SCRI 2020-02656), and the United States Department of Agriculture Agricultural Research Service (research support agreement no. 58-2072-0-036).

This paper was edited by Leena Järvi and reviewed by Bertrand Carissimo and Jérémy Bernard.