In general, computational fluid dynamics (CFD) models incur high computational costs when dealing with realistic and complicated flows. In contrast, the massconsistent flow (MASCON) field model provides a threedimensional flow field at reasonable computational cost. Unfortunately, some weaknesses in simulating the flow of the wake zone exist because the momentum equations are not considered in the MASCON field model. In the present study, a new set of improved algebraic models to provide initial flow fields for the MASCON field model are proposed to overcome these weaknesses by considering the effect of momentum diffusion in the wake zone. Specifically, these models for the wake region are developed on the basis of the wake models used in wellrecognized Gaussian plume models, ADMSbuild and PRIME. The MASCON fields provided by the new set of wake zone models are evaluated against windtunnel experimental data on flow around a wallmounted rectangular obstacle. Each MASCON field is compared with the experimental results, focusing on the positions of the vortex core and saddle points of the vortex formed in the nearwake zone and the vertical velocity distribution in the farwake zone. The set of wake zone models developed in the present study better reproduce the experimental results in both the wake zones compared to the previously proposed models. In particular, the complicated recirculation flow which is formed by the union of the sidewall recirculation zone and the nearwake zone is reproduced by the present wake zone model using the PRIME model that includes the parameterization of the sidewall recirculation zones.
Predicting the threedimensional flow fields around surfacemounted rectangular obstacles is important in applications such as urban air quality research and tracking the plumes of accidentally released toxic air pollutants. For these applications, various types of computational fluid dynamics (CFD) models have been applied to predict the flow fields around obstacles. However, CFD models have the drawback that the computational cost is substantially high.
A diagnostic massconsistent flow (MASCON) field model is often used to predict the flow field in complex terrains [
The MASCON field model was developed by Röckle [
QUICURB is part of the Quick Urban and Industrial Complex (QUIC) dispersion modeling system containing an “urbanized” randomwalk model [
MCAD, which is also based on Röckle’s MASCON field model, is a model that can calculate the flow fields in a greater variety of geometric configurations compared to the previously proposed models [
These models are typical algebraic models for estimating the initial flow fields of the flow around buildingshaped obstacles on land. The physics and formulations of these algebraic models were well explained in various papers. However, most of the model validations were based on element tests using obstacles with a simple aspect ratio, such as cubes [
In the present study, improved algebraic models for the initial flow fields are proposed in which the effect of momentum diffusion in the wake region is considered. Each MASCON field is compared with the experimental results, focusing on the positions of the vortex core and saddle points of the vortex formed in the nearwake zone and the vertical velocity distribution in the farwake zone. Additionally, the parameters contained in the algebraic models are improved such that the MASCON field better reproduces the experimental results compared to the conventional models.
Unlike the CFD model, the MASCON field model does not solve the NavierStokes equations. Instead, flow velocities are calculated from the continuity equation in such a way that the mass conservation should be met by a variational approach. The MASCON field model is a massconsistent model that is often used for predicting the flow fields in complex threedimensional terrain [
The initial flow field
Typical existing algebraic models for the initial flow fields are the modified Röckle model and the shelter model, which are shown in detail below.
Most algebraic models for the diagnostic MASCON field models for calculating the threedimensional flow field around obstacles are based on the model presented by Röckle [
Here, the algebraic models related to the near and farwake zones are presented. For the upstream recirculation zone, the parametrization proposed by Gowardhan et al. [
Rooftop recirculation zone
Gowardhan et al. [
The following velocity parameterization [
Nearwake zone
A cavity zone, which is called the nearwake zone in the present study, is produced in the downwind of the obstacle. In the Röckle model, the nearwake zone is an ellipsoid with the length of
The nearwake zone described in Equations (5) and (6) is confined vertically to the height of the obstacle and laterally to its width. The length of the nearwake zone
Farwake zone
The length of the farwake zone
The length of the farwake zone
Within the farwake zone
The MASCON field obtained from the modified Röckle model exhibits an excessively strong velocity gradient confined to a narrow area behind the obstacle [
In the above equations,
The functions
The coefficient
Note that the shelter model applies only to the downstream region, including the nearwake and farwake zones in the Röckle model. The shelter model does not include the parametrization of the nearwake and rooftop recirculation zones and the velocity parameterization in these zones. In the present study, the velocity parameterization in the zones proposed by Röckle in Equations (2)–(8) is incorporated into the shelter model.
The results of the MASCON field using the modified Röckle and shelter models are shown in
The MASCON field based on existing algebraic models inadequately reproduces the experimental results in the near and farwake zones. In the present study, two algebraic models are proposed to resolve the issues associated with these existing models. The flow field estimation schemes used in the two different plume models are improved for use as algebraic models in the MASCON field.
The ADMSbuild wake model is a module in the urbanized version of the Gaussian plumedispersion model ADMS [
The ADMSbuild wake model is applicable only when
Note that, if
The upper limit of the nearwake zone
The flow components in the farwake zone are as follows [
The normalized crosswind dimensions
The virtual origin
As shown in
Similar to the ADMS model, the PRIME model [
In the PRIME model, the length scale in Equation (2) is used to estimate the rooftop recirculation zone. The value used for
When
The horizontal boundary of the nearwake zone
In the present study, the envelope height is assumed to be
The parameterizations for the rooftop recirculation zone and nearwake zone in Equations (2)–(8) proposed by Röckle are also applied as the zones of the PRIME model. Equation (7) can be expressed as
The wake boundary,
These equations indicate the
In the present study, the velocity parameterization proposed by Röckle for the farwake zone in Equations (10) and (11) is applied to the farwake zone of the PRIME model. Equation (10) can be expressed as
The physics and formulation of some MASCON field algebraic models are well explained in various research papers. These models were validated on the basis of element tests using obstacles with simple aspect ratios, such as cubes [
In the experiment by Wang et al. [
The inflow conditions used in the present study were obtained from the mean velocity profile and turbulence intensity. The logarithmic law of the velocity distribution for neutral atmospheric stability is given by
In the present study, the friction velocity and surface roughness were, respectively, obtained as
The Working Group of the Architectural Institute of Japan (AIJ) has published guidelines [
In the present study, the friction velocity and surface roughness were, respectively, obtained as
The MASCON fields obtained from the algebraic models were compared with the experimental results. The experimental results obtained by Wang et al. [
It is worth noting that the streamlines in
The recirculation flow is generally characterized by the locations of the vortex cores and saddle points in the nearwake zone. The locations of the saddle points and vortex cores of the recirculation flow inside the nearwake zone obtained by the MASCON field model were compared with the experimental results. The locations of the saddle points and vortex cores in the nearwake zone could be identified from the streamlines in
The contours of the velocity component
The calculation results for the MASCON fields based on the algebraic models were evaluated by comparison with the experimental results of Meng et al. [
The location
The locations included in the nearwake zone were
As shown in the profiles in
In the modified Röckle model, the farwake zone was defined by a 1/4 ellipsoid that extended the nearwake zone to the downstream direction, resulting in a steep velocity gradient on the ellipsoidal surface. Considering that momentum diffusion is one method to overcome this defect, in the MASCON field using the shelter model, the velocity distribution in the farwake zone insufficiently reproduced the experimental results. It is shown that, using the PRIME model proposed in the present study, the MASCON field could reproduce the experimental results well not only in the nearwake zone, but also in the farwake zone velocity field.
The MASCON field model provides a threedimensional flow field at a low computational cost, which is very useful for predicting realistic and complicated flow fields for applications such as environmental risk assessments. However, some weaknesses in the flow of the wake zone exist because the momentum equations are not considered in the MASCON field scheme. The MASCON field obtained using the shelter model, which is a typical algebraic model for considering momentum diffusion, insufficiently reproduced the experimental results in the wake zone.
In the present study, an algebraic model in the wake zone to provide the initial velocity for the MASCON field model was developed on the basis of the wake models implemented in two wellrecognized Gaussian plume models, ADMSbuild and PRIME. The conclusions are summarized as follows:
The new set of wake zone models based on ADMSbuild and PRIME wake models can provide the initial velocity in the nearwake zone and take into consideration the effect of momentum diffusion in the farwake region.
Streamlines obtained from the experiment around the obstacle representing the flow field on the midheight horizontal plane show the complicated recirculation flow formed by the union of the sidewall recirculation zone and the nearwake zone. The present wake zone model based on the PRIME model that includes the parameterization of the sidewall recirculation zones can reproduce such a recirculation flow.
In the farwake zone, the flow fields according to the present models considering the effect of momentum diffusion are all in general agreement with the experimental results. In particular, the wake zone model based on the PRIME model provides the excellent flow field that precisely reproduces the profile of the vertical velocity distribution of the experimental results.
Therefore, these findings suggest that the MASCON field provided by the wake zone model based on the PRIME model proposed in the present study best reproduces the entire flow field from relatively close to the obstacle to the downstream.
Conceptualization, M.A. and H.O.; methodology, M.A., A.K. and H.O.; formal analysis, M.A.; resources, M.A., A.K. and H.O.; writing—original draft preparation, M.A.; writing—review and editing, A.K. and H.O.; supervision, H.O.; project administration, H.O.; funding acquisition, M.A. All authors have read and agreed to the published version of the manuscript.
This work was supported by JSPS KAKENHI (Grant Number JP16K00580).
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The authors declare no conflict of interest.
The initial flow field
Suppose that the function that minimizes the functional
If
Here, the term of second order
On the domain surface on which a boundary condition such as walls or inflow on
On the boundary where other types of boundary conditions such as symmetry planes or outflows are given,
Since
The above equation is a Poisson equation for
This show that the Lagrange multiplier
Schematic diagram of the flow patterns around a surfacemounted obstacle.
Plan view (
Plan view (
Computational domain on the symmetry plane at
Streamlines around the obstacle representing the flow fields on the obstacle symmetry plane at (
Contours of velocity components
(
The coordinate information of Vortex core and Saddle point in the nearwake zone.
Aspect Ratio of the Obstacle  Experiment/MASCON Model  Central Vertical Plane  MidHeight Horizontal Plane  

Vortex Core 
Saddle Point 
Vortex Core 
Saddle Point 

1:1:2  Experiment [ 





MASCON  Röckle 





Shelter 





ADMS 





PRIME 



