1 Introduction

The dispersion of particulate pollutants around buildings raises significant environmental concerns due to the adverse health impacts of exposure to suspended urban particles (Alemayehu et al., 2020). Accurate prediction of particle dispersion is important for evaluating health risks associated with particulate pollutants in urban areas.

Field experiments provide sparse observations and are limited by high costs and difficulty in standardizing testing conditions across measurements (Antoniou et al., 2019; S. Liu et al., 2018; Stathopoulos et al., 2004; Zou et al., 2021). Wind tunnel simulations can provide reliable data under controlled conditions but still bring substantial costs for the tunnel infrastructure and instrumentation (Blocken et al., 2016a, b).

Large-eddy simulation (LES) models accurately predict turbulent urban winds by explicitly resolving turbulent eddies (Blocken, 2018; Mirzaei, 2021; Zheng & Yang, 2021). These resolved eddies also explicitly carry pollutants in a manner closer to actual transport phenomena. However, the fine resolution requires substantial computational resources. As a less demanding alternative for transient simulation, unsteady Reynolds-averaged Navier–Stokes (URANS) modeling reduces cost through ensemble-averaged modeling that filters out stochastic turbulence with turbulence models. However, URANS exhibits low-frequency unsteadiness with single-peak spectra for urban simulation, overlooking key multiscale flow behaviors, and significantly underestimating turbulence, especially within building arrays where fluctuations can disappear entirely (Tominaga, 2015; Tominaga & Stathopoulos, 2017). Hybrid URANS-LES can save approximately 20% in mesh and time versus pure LES with comparable accuracy for urban prediction (J. Liu & Niu, 2019). It is still resource-intensive, since the accuracy still relies heavily on resolved turbulence requiring fine mesh resolution, and the model reverts to URANS when under-resolved. Compared to these transient modeling strategies, steady RANS (SRANS) modeling significantly reduces expenses by avoiding expensive time-resolved calculations with further simplified handling of turbulence. However, the single-point closure and ill-conditioned governing equations derived from the time-averaging assumptions degrade accuracy even more (Brener et al., 2021; Girimaji & Abdol-Hamid, 2005; Hao & Gorlé, 2022; Wu et al., 2019; R. Zhao et al., 2022, 2023). Nevertheless, SRANS remains the main workhorse for industrial computational fluid dynamics (CFD) simulations (Blocken, 2014), benefiting from its computational efficiency and widely accepted guidelines (Blocken & Gualtieri, 2012; Blocken et al., 2012; Franke et al., 2011; Tominaga et al., 2008). As SRANS only provides steady-state eddy viscosity and turbulence kinetic energy (TKE) to represent turbulence, the turbulent dispersion of particles must also be modeled using additional sub-models based on analogous assumptions.

Two main approaches exist for outdoor particle dispersion modeling: Lagrangian particle tracking (LPT) (Bahlali et al., 2019; Haghighifard et al., 2018; Oettl, 2015; Trini Castelli et al., 2018) and Eulerian continuum models (Blocken et al., 2016a, b; Karttunen et al., 2020; Niu et al., 2018). Lagrangian particle tracking (LPT) methods model discrete particles subject to various forces (e.g., gravity and drag from computed flow fields), moving them according to Newton's laws. When paired with LES (termed LES-LPT), as resolved eddies transport the numerical particles, the LES-LPT method provides the most realistic particle tracking with the fewest modeling assumptions. When applied with SRANS simulations (termed SRANS-LPT), an additional sub-model, the discrete random walk (DRW) model, is required to incorporate stochastic effects of turbulence on particles based on the simulated airflow and turbulence statistics (Haghighifard et al., 2018; Oettl, 2015). The DRW model simply uses Gaussian random numbers to generate pseudo fluctuation sequences of the airflow velocity, based on the SRANS output of TKE. The reductionist assumptions made in the DRW model and underlying inaccuracy in SRANS introduce considerable uncertainty for SRANS-LPT modeling. Regardless of the airflow simulation approach, LPT by its nature enables analysis of trajectories, deposition, and retention times. However, the tracking process remains transient even when using steady-state airflow data, and the computational cost grows with the number of particle tracers. Aside from the Lagrangian view, the Eulerian view is another way to model the particle phase, assuming it as a continuum and establishing a convection–diffusion equation for particle concentrations, as for gases (B. Zhao et al., 2009). Despite significantly reducing computational time, one modeling challenge is accounting for particle inertia and gravity, which cause velocity differences between particle and carrier phases, i.e. particle drift. Using the airflow velocity directly in the convection–diffusion equation provides misleading particle concentration flux, requiring extra drift-flux (DF) models (E. M. A. Frederix et al., 2017). Additionally, for application with SRANS (SRANS-DF), turbulence transport of particles can only be based on molecular diffusion assumption (B. Zhao et al., 2009).

Given the SRANS inaccuracy and additional uncertainty in dispersion modeling, rigorously evaluating the performance of SRANS-LPT and SRANS-DF approaches is critical before engineering application. There have been extensive experimental and numerical studies for particle transport in indoor environments (Cao et al., 2018; S. Liu et al., 2022; S. Liu & Deng, 2023; B. Zhao et al., 2009). However, for urban studies, while experimental studies on gaseous tracer dispersion exist (Architectural Institute of Japan, 2008; COST ES, 1006, 2012; Leitl & Schatzmann, 2005), there is a lack of rigorous validation data using particulate tracers. Many prior studies rely on validating particle models against gas dispersion data (Bahlali et al., 2019; Haghighifard et al., 2018; Oettl, 2015; Trini Castelli et al., 2018), which questions the conclusions due to differences in transport physics. The validity of the numerical models for particle dispersion around buildings remains unknown, especially when large particles are considered (E. Frederix, 2016; E. M. A. Frederix et al., 2017).

To provide reliable validation data to evaluate outdoor particle dispersion simulation with SRANS, this study utilizes a combined experimental and high-fidelity simulation approach to generate a comprehensive dataset for particulate dispersion around a single building. First, a detailed wind tunnel experiment is conducted using particulate tracers for concentration measurements. These results are complemented by a validated LES of the same scenario to broaden the reference dataset. SRANS simulations paired with both Eulerian and Lagrangian particle tracking models are performed and validated against the obtained reference data.

2 Methods

This section describes the wind tunnel experiment, as well as the employed models for turbulence and dispersion.

2.1 Wind tunnel experiment

The wind tunnel experiment was conducted in an open-circuit wind tunnel at Tianjin University, using a test section with dimensions of 0.35 m width by 0.225 m height. To generate atmospheric boundary layer flow, four triangular spires designed based on Irwin's approach (Irwin, 1979) were installed upstream, along with seven thin horizontal bars attached to the spires to produce the required turbulence intensity, as inspired by the lattice method for simulating part-depth ABL flows (Lee et al., 2004). Downstream of the spires, a fetch of cube-shaped roughness elements with decreasing heights was arranged (Fig. 1a). Iterative adjustments were made to the horizontal bar spacings on the spires and layout of the roughness cubes until the desired ABL velocity profile was achieved. Measurements were conducted using a single-wire constant temperature anemometer (CTA) TSI IFA-300 (Fig. 1b) mounted on a traverse system with a precision of 10 μm (Fig. 1c). The TSI IFA-300 outputs a real-time, continuous analog voltage signal that correlates to the dynamic air velocity fluctuations across its calibrated velocity range. It has a frequency response greater than 250 kHz. For the given single-wire probe used, the IFA-300 was calibrated for velocities ranging from 0.01 to 30 m/s. Data was acquired at a 40 kHz sampling frequency, collecting 2,097,152 samples at each measurement location over one-minute durations.

Fig. 1
figure 1

a Spires and roughness elements; b Constant temperature anemometer; c Traverse system

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The vertical profile of the mean wind velocity of the formulated part-depth ABL flow followed a 0.25 power law (Eq. 1). The profile of the streamwise turbulence intensity \({{I}}_{{x}}\) matched the empirical equation (Eq. 2) suggested by the American Society of Civil Engineers (ASCE) (American Society of Civil Engineers, 1999), where \(\sigma_{{U}} ({{z}})\) is the standard deviation of the measured velocity and \({{z}}_{0} = 1.8 \times 10^{-4}\text{ m}\) is the regional roughness height. Although some measured turbulence intensity values deviated from the theoretical curve in the upper region, this was acceptable for the validation study, since using the measured profiles rather than theoretical atmospheric boundary layer (ABL) profiles as boundary conditions is common practice. The measured velocity and turbulence intensity profiles of the wind tunnel flow used as boundary conditions are shown in Fig. 2.

Fig. 2
figure 2

Profiles of a velocity and b turbulence intensity of the formulated part-depth ABL flow in the wind tunnel

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$$\frac{{{U}}\left({{z}}\right)}{{{U}}_{{H}}}\text{=}{\left(\frac{{z}}{{{H}}}\right)}^{0.25}$$
(1)
$$I_{x} (z) = \frac{\sigma_{U}(z)}{U(z)} = \frac{1}{\text{ln} (z/z_{0})}$$
(2)

After generating the desired atmospheric boundary layer (ABL) flow, a 2:2:1 building model was positioned where the incident ABL flow measurements were taken, with its windward face perpendicular to the wind direction. The building model dimensions were 0.09 m length, 0.09 m width, and 0.045 m height (H), so the reference wind speed of UH = 14.5 m/s ensured a building Reynolds number of \(Re = U_{H} H/v = \text{43,500}\). Hot-wire anemometer measurements were taken along nine vertical sample lines surrounding the building model, as illustrated in the schematic in Fig. 3. The measurements around the building were repeated three times on different dates and averaged to obtain the final values used.

Fig. 3
figure 3

a Schematic of the wind tunnel experiment; b measurement locations for velocity and turbulence

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After establishing reproducible atmospheric boundary layer (ABL) flow and building airflow patterns, an additional experiment was performed to measure particle dispersion from a roof stack. A diethylhexyl sebacate (DEHS) particle generator was used to produce a particle flow for the dispersion experiments (see Fig. 4a). The generator flow rate was controlled by an air pump and monitored with a flow meter. The DEHS generator output was connected via silicone tubing to a seamless steel tube (8 mm outer diameter, 6 mm inner diameter). This steel tube was inserted through an 8 mm diameter hole made in the center of the building model and protruded 6 mm above the roof level as a roof stack, as visualized in Fig. 4b. This particle injection setup enabled the DEHS particles from the generator to be emitted through the roof stack with manually controlled flow rate and thus a constant injection velocity We from the roof stack.

Fig. 4
figure 4

a DEHS generator; b visualization of the particle injection; c Aerodynamic Particle Sizer APS 3321 and Aerosol Diluter 3302A; d Particle collection tube mounted on the traverse system

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Particle concentration measurements were conducted using an Aerodynamic Particle Sizer (TSI APS 3321) and an Aerosol Diluter (TSI 3302A) set to a 100:1 dilution ratio. The APS 3321 simultaneously counts particles from 0.5 to 20 μm aerodynamic diameter in 32 size channels. Concentrations up to 1000 particles/cm3 can be measured at 0.5 μm with a coincidence of less than 2% and at 10 μm with a coincidence under 6%. Usable data are provided up to 10,000 particles/cm3. The size resolution is 0.02 μm at 1.0 μm and 0.03 μm at 10 μm. The minimum detectable concentration is 0.001 particle/cm3. The accuracy is ± 10% plus counting variations. The maximum processing rate exceeds 200,000 particles/s. The accompanying 3302A diluter offers 100:1 and 20:1 sample dilution at a 5 L/min flow rate drawn by the spectrometer. For respirable particles, it has a penetration efficiency over 93%, with lower efficiency for larger sizes. Efficiency corrections are built into the software. Particles were sampled iso-kinetically using a 3 mm diameter seamless steel tube with an angled tip at the lower end. The upper end of the sampling tube was connected to 19 mm silicone tubing and mounted on a traverse system, allowing controlled movement of the sampling assembly (Fig. 4d). This measurement setup enabled diluted particle sampling with precise position control to obtain concentration measurements around the building model. A schematic of the dispersion experiment and particle sampling locations is given in Fig. 5. Two We values (We = UH and We = 2UH) were considered in the measurements.

Fig. 5
figure 5

a Schematic of the dispersion experiment; b sampling locations for particle concentration, with Lines D-I and points 0 and 1 defined

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2.2 Governing equations and turbulence modeling

The previous subsection outlined the wind tunnel experiment, this subsection describes the SRANS and LES approaches for the reproduction of airflow and turbulence around the building model.

Alongside the continuity equation for velocity (Eq. 3), the governing equations for velocity and turbulence can be written in a general form (Eq. 4), where \({{U}}_{{i}}, {{U}}_{{j}}\) are the mean (SRANS) or filtered (LES) velocity components, in the \({{x}}_{{i}}\), \({{x}}_{{j}}\) directions (\({{i , j} = 1, 2, 3}\)). \(\phi\) is the transported quantity, \(\varGamma\) is the diffusion coefficient of \(\phi\), and \({S}_{\phi }\) is the source term. Table 1 summarizes the model equations for SRANS with Kato-Launder (KL) k-ε model (Kato & Launder, 1993) and LES with wall-adapting local eddy-viscosity (WALE) model (Nicoud & Ducros, 1999). Previous studies have provided comprehensive evaluations comparing the performance of different SRANS turbulence models for urban airflow and gaseous concentration prediction (Gousseau et al., 2011; Toja-Silva et al., 2015; Tominaga & Stathopoulos, 2007, 2010; Vardoulakis et al., 2011; Yoshie et al., 2007; R. Zhao et al., 2022, 2023). Though the various SRANS models differ, their discrepancies were small relative to differences with LES or experimental data, since all SRANS models showed poor accuracy, especially for TKE. This study selects the KL model as representative of SRANS performance. It introduces a production limiter to the standard k-ε model to mitigate the unphysical generation of in impingement regions. For LES, provided a sufficiently fine resolution mesh is used, the choice of sub-grid scale model contributes only negligible differences, according to Okaze et al. (Okaze et al., 2021). Thus, this study employs the numerically stable WALE model.

Table 1 Coefficients and source terms
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$$\frac{\partial {U}_{j}}{\partial {x}_{j}}=0$$
(3)
$$\frac{\partial \phi }{\partial t}+{U}_{j}\frac{\partial \phi }{\partial {x}_{j}}=\varGamma \frac{{\partial }^{2}\phi }{\partial {x}_{j}\partial {x}_{j}}+{S}_{\phi }$$
(4)

In Table 1, \(v\) and \(v_{t}\) are the kinematic viscosity and turbulence viscosity. \(p\) is the pressure, and \(\rho\) is the density. \(k\) is TKE, \({{P}}_{{k}}\) is the turbulence production term, \(\varepsilon\)  is the turbulence dissipation rate, \({S}_{ij}\) is the strain rate tensor, \({\Omega }_{ij}\) is the rotation rate tensor, \({\delta }_{ij}\) is the Kronecker Delta function.

2.3 Dispersion modeling

Based on the airflow and turbulence solved using the approaches described above, this subsection describes the LPT and DF approaches for the simulation of particle dispersion.

For LPT, the position of a particle \({x}_{P,i}\) and its velocity \({u}_{P,i}\) are tracked by the following equation system (Eqs. 5-6), where \({\rho }_{P}\) is the particle density, \({d}_{P}\) is the particle diameter, \({C}_{D}\) is the drag coefficient, \(R{e}_{P}\) is the particle Reynolds number. \({U}_{i}\) and \({U}_{i}^{\prime}\) are the mean and fluctuating velocity of air. Here only the drag force and gravitational force are considered.

$$\frac{d{x}_{P,i}}{dt}={u}_{P,i}$$
(5)
$$\frac{d{u}_{P, i}}{dt}=\frac{18\rho \nu }{{\rho }_{P}{d}_{P}^{2}}\frac{{C}_{D}R{e}_{P}}{24}\left[\left({U}_{i}+{U}_{i}^{\prime}\right)-{u}_{P, i}\right]+\left(1-\frac{\rho }{{\rho }_{P}}\right){g}_{i}$$
(6)

\(R{e}_{P}\) is calculated using Eq. 7 and \({C}_{D}\) is obtained using Eq. 8 (Putnam, 1961).

$$R{e}_{P}= \frac{\left|{(U}_{i}+{U}_{i}^{\prime})-{u}_{P,i}\right|{d}_{P}}{\nu }$$
(7)
$${C}_{D}=\left\{\begin{array}{c}\frac{24}{R{e}_{P}}\left(1+\frac{1}{6}{Re}_{P}^\frac{2}{3}\right), R{e}_{P}<1000\\ 0.44, R{e}_{P}>1000\end{array}\right.$$
(8)

LES directly simulates the fluctuating velocity \({U}_{i}^{{\prime}}\) in both space and time, while SRANS can only obtain \({U}_{i}\) with turbulence represented by \(k\) and \(\varepsilon\). The discrete random walk (DRW) model (Eqs. 9-10) is a viable approach to construct spatial and temporal sequence of \({U}_{i}^{\prime}\) from the turbulence statistics using normal random numbers \({\lambda }_{i}\) (OpenCFD Ltd., 2020).

$${U}_{i}^{\prime}={\lambda }_{i}\sqrt{\frac{2k}{3}}$$
(9)
$${t}_{inter}={\text{min}}\left(\frac{k}{\varepsilon }, 0.164\frac{{k}^{1.5}}{\varepsilon \left|{(U}_{i}+{U}_{i}^{\prime})-{u}_{P,i}\right|}\right)$$
(10)

The particle interacts with the fluid over a particle interaction timescale \({t}_{inter}\) with a constant \({U}_{i}^{\prime}\) value. After each interaction time has elapsed, a new \({U}_{i}^{\prime}\) value is obtained by generating new \({\lambda }_{i}\) values.

In an Eulerian perspective, the particle concentration follows a species transport equation (Eq. 11). For particles, the diffusion is only related to turbulence and \({Sc}_{t}\)=1 (B. Zhao et al., 2009).

$$\frac{\partial \left[\left({U}_{j}+{u}_{drift,j}\right)C\right]}{\partial {x}_{j}}=\frac{\partial }{\partial {x}_{j}}\left(\frac{{\nu }_{t}}{{Sc}_{t}}\frac{\partial C}{\partial {x}_{j}}\right)$$
(11)

To obtain the inertial drift velocity \({u}_{drift,i}={u}_{P,i}-{U}_{i}\), the local equilibrium assumption, as described by Manninen et al. (Manninen et al., 1996), can be adopted. Under this assumption, the inertial drift velocity can be modeled using an algebraic drift-flux model (ADF) as (E. M. A. Frederix et al., 2017):

$${u}_{drift,j}=\tau \left(1-\frac{\rho }{{\rho }_{P}}\right){g}_{i}$$
(12)
$$\tau =\frac{{\rho }_{P}{d}_{p}^{2}}{18\rho \nu }$$
(13)

However, the assumption does not necessarily hold for particles with large Stokes numbers (E. Frederix, 2016), for which the Stokes drift-flux model (SDF) should be applied for accurate modeling of the drift velocity (E. M. A. Frederix et al., 2017).

$$\left({u}_{drift,j}+{U}_{j}\right)\frac{\partial \left({u}_{drift,j}+{U}_{j}\right)}{\partial {x}_{j}}=-\frac{1}{\tau }{u}_{drift,i}+\left(1-\frac{\rho }{{\rho }_{P}}\right){g}_{i}$$
(14)

2.4 Case setup

This subsection describes the computational domain, mesh generation, and numerical setups of the CFD simulations.

The computational domain was set following the AIJ guidelines (Tominaga et al., 2008). The distances from the building to the inlet, the sides, and the top of the domain were all set as 5H, and the distance to the outlet was 10H. Computational grids of the three cases were generated by the open-source mesh generation tool snappyHexMesh. The cells adjacent to the building surface had a size of H/40. A refinement box extending streamwise from -2H to 4H, spanwise from -2H to 2H, and vertically from 0 to 2H was designated to match the dimensions of the measurement region. Within this refinement box, 20 levels of boundary layers were generated from the building surface, with the first layer having a thickness of H/1500 and a growth rate of 1.15. The mesh maintained a size of H/40 and was then coarsened to H/10 in a larger refinement box extending streamwise from the inlet plane to 6H, spanwise from -3H to 3H, and vertically from 0 to 3H. The background mesh outside of the refinement boxes had a cell size of H/2. The mesh sizes and boundary layer settings are essentially identical to the finest LES setup in Z. Liu et al. (Z. Liu et al., 2020) for a same 2:2:1 model. The mesh details are shown in Fig. 6.

Fig. 6
figure 6

a Overview and b regional view of mesh

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The simulations were conducted in the open-source code OpenFOAM-v2012 (OpenCFD Ltd., 2020). The inlet boundary conditions for velocity (U) and turbulence kinetic energy (k) were directly imposed by interpolating the measured profiles of the incident wind tunnel flow. The inlet turbulence dissipation rate (ε) profile was calculated from U and k using empirical equations suggested by Mochida et al. (Mochida et al., 2002). For LES simulation, the prescribed profiles were used as inputs for the inlet turbulence generator, a divergence-free synthetic eddy method (DFSEM) boundary condition, as provided in OpenFOAM. Symmetry boundary conditions were applied at the side and top boundaries. For building surfaces, standard wall functions were used. For ground surfaces, z0-type wall functions as suggested in Richards and Hoxey (Richards & Hoxey, 1993) were used in SRANS simulations while standard wall functions were for LES. The roughness length (z0) values were estimated from the measured U and k values closest to the ground in the incident flow (Mochida et al., 2002). Other boundary conditions are listed in Tables 2 and 3 for clarity.

Table 2 Boundary conditions for LES and SRANS
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Table 3 Boundary conditions for DF and LPT
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To solve for LES in combination with LPT, the kinematicParcelFoam solver based on the PIMPLE algorithm was employed. Since LES directly resolves the fluctuating velocity, no DRW model was used. The central linear scheme was applied for gradient terms and Laplacian terms. For the divergence terms, A filtered linear scheme was used for U. To replicate the measurement process using a single-wire probe placed perpendicular to the y = 0 plane, the effective velocity \({U}_{eff}\) is approximated as shown in Eq. 15, following the approach of Fischer et al. (Fischer et al., 2022). Since \({U}_{y}\) is aligned parallel to the wire, it has a minimal effect on the cooling of the hotwire. In contrast, \({U}_{z}\) is aligned perpendicular to the wire and therefore influences the cooling similarly as the primary flow component \({U}_{x}\). The simulations were run for six flow-through times in the computational domain with a temporal resolution of 40 kHz, matching the hot-wire measurements. This small time step of 2.5 × 10−5 s ensures an average Courant–Friedrichs–Lewy (CFL) number well below 0.1. The first pass allowed for a spin-up of the flow field. The airflow and concentration data from the subsequent five passes were averaged to obtain the mean and variance statistics.

$${U}_{eff}=\sqrt{({U}_{{{x}}}^{2}+ 0.01{U}_{{{y}}}^{2}+{U}_{{{z}}}^{2})}.$$
(15)

The simpleFoam solver based on the SIMPLE algorithm was employed to solve for SRANS. For the divergence terms, the second-order upwind scheme was used for U, the Van Leer scheme for k and ε. The simulations were of second-order accuracy. The convergence criterion was 10–4 for the residuals of all the variables. Based on the steady-state results for the airflow and turbulence field, the DF models were solved using an implemented drift-flux solver. LPT was performed using kinematicParcelFoam with a transient time-averaging process as was done in LES, but with a fixed airflow field, and a DRW model (Eq. 9) to estimate the fluctuating velocity.

3 Results

This section presents the airflow, turbulence, and particle concentration measurements from the wind tunnel experiment, along with validation of LES and SRANS against this experimental data.

3.1 Validation of airflow and turbulence

Figure 7 compares the effective velocity \(\overline{{U}_{eff}}\) and variance of effective velocity \(\sigma ({U}_{eff})\) measured using the single-wire CTA and simulated using LES with the numerical probe (Eq. 15). LES accurately simulated the mean statistics of airflow and turbulence, except for minor differences above the roof height.

Fig. 7
figure 7

Comparison of a mean effective velocity \(\overline{{U}_{eff}}\) and (b) variance of effective velocity \(\sigma ({U}_{eff})\) measured (points) and predicted by LES (solid line)

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Since the effective velocity cannot be resolved in SRANS, the validated LES data was used as a reference to reconstruct the full velocity components. Figures 8–10 compare the velocity components and TKE obtained from SRANS and LES. In Fig. 8a, the SRANS streamwise velocity differs from LES results mainly in the rooftop and far-wake regions, indicating an inaccurate prediction of the separation. This is more clearly observed in Fig. 9, with SRANS showing a significantly overpredicted wake reattachment length. Figure 8b shows SRANS underpredicting TKE on the roof and in the far-wake region, also evident in Fig. 10.

Fig. 8
figure 8

Comparison of time-averaged a stream-wise velocity component, and b TKE predicted by SRANS (dashed) and LES (solid)

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Fig. 9
figure 9

Comparison of contours of time-averaged stream-wise velocity predicted by a SRANS and b LES

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Fig. 10
figure 10

Comparison of contours of TKE predicted by a SRANS and b LES

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3.2 Validation of dispersion models

Figure 11 compares the simulated particle concentrations for dP = 1 μm to the measured values under the condition We/UH = 1. The concentrations were normalized by the value at the nozzle opening (C0 at point 0). Overall, LES was the most accurate in this case, while the other SRANS-based models also showed reasonable agreement with the experimental data. Results from SRANS-ADF and SRANS-SDF were identical and deviated most from measurements, especially downstream of the building (Lines G-I). The overpredicted concentrations indicate insufficient particle dispersion, explainable by the underpredicted TKE and thus \({\nu }_{t}\) (see Eq. 11). However, SRANS-LPT aligned well at Lines G and H, despite slightly underpredicting at Line I, suggesting the DRW suitably compensated for the low TKE here. All models lacked accuracy at Lines E and F.

Fig. 11
figure 11

Comparison of normalized particle concentration using LES-LPT and SRANS paired with LPT, algebraic drift-flux (ADF), and Stokes drift-flux (SDF). We/UH = 1, dP = 1 μm

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Figure 12 compares the simulated particle concentrations for dP = 1 μm and 10 μm to the measured values under the condition We/UH = 2. Due to uncertainties in the inlet concentration measurement at the nozzle caused by the high particle count, the concentration above the nozzle opening (C1 at point 1) was used as the reference for normalization. Again, all the approaches, except for SRANS-ADF, showed a reasonable prediction of the concentration for both particle diameters. In Fig. 12a, SRANS-ADF and SRANS-SDF results remained identical and aligned best with experiments in terms of peak concentrations and positions. Overall, LES-LPT tended to underpredict concentrations with a higher peak position, while SRANS-LPT overpredicted with a lower peak position. The estimated nozzle concentration by LES-LPT in this case was C0 =4.5C1, significantly larger than the measured C0. SRANS-ADF and SRANS-SDF also predicted the same nozzle concentration value while SRANS-LPT overpredicted it.

Fig. 12
figure 12

Comparison of normalized particle concentration using LES-LPT, SRANS-LPT, as well as RANS with algebraic drift-flux (ADF) or Stokes drift-flux (SDF). a We/UH = 2, dP = 1 μm and b We/UH = 2, dP = 10 μm

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In Fig. 12b, SRANS-ADF strongly overpredicted the concentration, with similar values obtained for the large particles as for the small particles in Fig. 12a. SRANS-SDF continued to closely match the experimental value. LES-LPT also predicted well, while SRANS-LPT was the most inaccurate, generally underpredicting the concentration. The estimated nozzle concentration by LES-LPT and SRANS-LPT in this case was close to the measured C0. SRANS-ADF and SRANS-SDF overpredicted the value.

Figure 13 provides a visual comparison of the particle distributions obtained from LPT using SRANS versus LES. The SRANS-LPT leads to a smooth particle distribution that follows an overall Gaussian profile, lacking detailed resolution. Many particles rise to unrealistic elevations in unimpeded random walks. In contrast, the LES-LPT particle distribution appears more realistic. Particles are shown to be carried by coherent turbulent eddies. The dispersion follows a more constrained distribution at lower elevations. This showcases the ability of LES to simulate particle interaction with large flow structures more realistically. Although SRANS-LPT can reasonably simulate the overall particle concentration in a statistical sense (Figs. 11, 12), the individual particle trajectories and reachability are questionable due to the simplified random-walk modeling.

Fig. 13
figure 13

Particle distribution at t = 200t0 by a SRANS-LPT b LES-LPT. Colored by particle age normalized by t0 = H/UH

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4 Discussion

The raw measurements could not directly evaluate SRANS models due to instrumentation constraints. However, the high-fidelity LES reconstructed flow fields and particle concentrations beyond the experiments' limitations in a complementary manner. Due to single-wire CTA instrumentation limitations, only positive velocity values could be measured, so the full three-dimensional velocity vector was unobtainable. Also, instrumentation limitations in the particle experiments prevented high-quality measurements of particle concentrations and size distributions in some scenarios. The We/UH = 1 condition lacked data for 10 μm particles since the injection velocity was low for enough generation of DEHS particles and large particle concentrations were negligible. For We/UH = 2, the DEHS generator's doubled flow rate caused particle concentrations to exceed the APS's upper detection limit at the nozzle opening (point 0), even with 100:1 dilution. Therefore, the second nearest point from the nozzle opening (point 1) was chosen instead for normalization. Ideally, the We/UH = 1 case should apply consistent normalization using the same point. However, this is not possible since the measured concentration at point 1 for We/UH = 1 is a near-zero value not suitable as a denominator. Despite the inconsistency in this normalization between cases, the normalization within each case is consistent for the evaluated models, allowing a fair evaluation. This further stresses the importance of the LES data providing a highly feasible baseline to determine an appropriate normalization approach, enhancing the usability of the experimental data.

SRANS-ADF was not able to correctly model the particle drift for large particles. The calculated drift velocity is very small using Eq. 12. The result is almost identical to a passive scalar modeling of gas dispersion. So the conclusions in previous studies hold only for fine particles since tracer gas data were employed for validation (Bahlali et al., 2019; Haghighifard et al., 2018; Oettl, 2015; Trini Castelli et al., 2018).

LES-LPT demonstrated consistent performance across the simulated cases, whereas SRANS-LPT and SRANS-SDF showed varying tendencies between scenarios. This discrepancy stemmed from differences in the tracking mechanisms. LES-LPT did not employ a DRW model. Particles were dispersed solely by explicitly resolved eddies. Uncertainties mainly arose from resolved airflow and turbulence interactions between the jet, inflow, and building-induced flow. For SRANS-LPT, uncertainties originated not only from inaccurate mean airflow and underpredicted TKE but also from the reductionist DRW model approximating eddy strength and duration. For the simplified models, errors from multiple sources could cancel out or strengthen each other in a case-dependent manner. These compensations were not generalizable. Nevertheless, the simplified models were computationally efficient and can be employed for rapid prediction of particle concentrations.

In this experiment, particles were injected from a high-speed roof-stack jet. Whether the conclusions hold for other scenarios, such as ground-level releases in the building wake, remains an open question. Future studies will conduct additional experimental investigations across more scenarios.

5 Conclusions

This study conducted wind tunnel measurements of airflow, turbulence, and particle dispersion around a single low-rise building. Large eddy simulation (LES) validated by the measurements was used to expand the data. Eulerian and Lagrangian methods paired with steady Reynolds-averaged Navier–Stokes (SRANS) were evaluated for predicting outdoor particle dispersion. This study led to the following findings:

LES had superior accuracy for modeling airflow, turbulence, and particle dispersion around a single building. The high time-resolution transient simulation enabled replicating the measurement process of the single-wire CTA and helped infer the high concentrations around the injection nozzle that could not be directly measured in the experiment.

SRANS had lower accuracy for predicting airflow and turbulence around the low-rise building, featuring an extended wake recirculation zone and underpredicted TKE. However, in this case, these inaccuracies did not prevent SRANS from generating accurate particle concentration predictions when Lagrangian particle tracking (LPT) was used. The predicted concentration was close to the LES prediction. However, the individual particle trajectories and reachability are questionable due to the simplified random-walk modeling.

For the Eulerian drift-flux simulations based on the SRANS results, the Stokes drift-flux model accurately predicted the dispersion of both small 1-micron and large 10-micron particles downstream of the stack. The algebraic drift-flux model accurately predicted the dispersion of the small particles but failed for the large particles. Proper drift modeling was essential for Eulerian prediction in this case, especially for the particles with large Stokes numbers.