This study describes the numerical implementation, verification and validation of an immersed boundary method (IBM) in the atmospheric solver Meso-NH for applications to urban flow modeling. The IBM represents the fluid–solid interface by means of a level-set function and models the obstacles as part of the resolved scales.

The IBM is implemented by means of a three-step procedure: first, an explicit-in-time forcing is developed based on a novel ghost-cell technique that uses multiple image points instead of the classical single mirror point. The second step consists of an implicit step projection whereby the right-hand side of the Poisson equation is modified by means of a cut-cell technique to satisfy the incompressibility constraint. The condition of non-permeability is achieved at the embedded fluid–solid interface by an iterative procedure applied on the modified Poisson equation. In the final step, the turbulent fluxes and the wall model used for large-eddy simulations (LESs) are corrected, and a wall model is proposed to ensure consistency of the subgrid scales with the IBM treatment.

In the second of part of the paper, the IBM is verified and validated for several analytical and benchmark test cases of flows around single bluff bodies with an increasing level of complexity. The analysis showed that the Meso-NH model (MNH) with IBM reproduces the expected physical features of the flow, which are also found in the atmosphere at much larger scales. Finally, the IBM is validated in the LES mode against the Mock Urban Setting Test (MUST) field experiment, which is characterized by strong roughness caused by the presence of a set of obstacles placed in the atmospheric boundary layer in nearly neutral stability conditions. The Meso-NH IBM–LES reproduces with reasonable accuracy both the mean flow and turbulent fluctuations observed in this idealized urban environment.

Urbanization impacts the physical and dynamical structure of the atmospheric boundary layer, influencing both the local weather and the concentration and residence time of pollutants in the atmosphere, which in turn impact air quality. While the physical mechanisms driving these interactions and their connections to climate change are well understood (the urban heat island effect, anthropological effects), their precise quantification remains a major modeling challenge. Accurate predictions of these interactions require modeling and simulating the underlying fluid mechanics processes to resolve the complex terrain featured in large urban areas, including buildings of different sizes, street canyons and parks. For example, it is well known that pollution originates from traffic and industry in and around cities, but the actual dispersion mechanisms are driven by the local weather. Furthermore, fine-scale flow fluctuations influence nonlinear physicochemical processes. The present study addresses these issues by focusing on the numerical aspects of the problem.

With the progress in metrology, it is now possible to obtain reliable measurements of the atmospheric conditions over a city. For example, during the Joint Urban experiment (JU2003), scalar dispersion was measured experimentally over the streets of Oklahoma City (

In order to use these experimental data in the future for model validation, the numerical models need to first be verified for academic test cases and simplified scenarios representative of atmospheric turbulent boundary layer flows. In particular, flow interaction with buildings or any generic obstacles plays a crucial role in urban flow modeling. The range of scales of objects acting as obstacles is huge in an urban setting, encompassing large buildings and small vegetation scales, and so is the range of the corresponding flow–obstacle interactions. Covering all possible cases is obviously impossible but we can rely on the invariance of certain flow characteristics at different scales. For example, the von Kármán streets are observed in the wake of a centimeter-scale cylinder as well as in the cloud layout behind an island. Following this, a wide selection of benchmark flows can be analyzed to verify and validate the numerical treatment of fluid–obstacle interaction with a view to atmospheric applications.

The physical application considered in this work is the atmospheric mesoscale reaction to perturbations induced by urban areas; the more the obstacles are considered part of the scales numerically resolved, the higher the accuracy of the results. To access this resolution, this study presents the development, implementation, verification and validation of an immersed boundary method (IBM)

Meso-NH scientific documentation:

In this study, a discrete forcing approach is adopted wherein the fluid–solid interface is modeled by means of a level-set function

The paper is organized as follows. Section

MNH is an atmospheric non-hydrostatic research model. Its spatiotemporal resolution ranges from the large meso-alpha scale (hundred of kilometers and days) down to the micro-scale (meters and seconds). It is massively parallel on the nested and structured grids adapted on most international hosting computer platforms. Several parameterizations are available: radiation, turbulence, microphysics, moist convection with phase change, chemical reactions, electric scheme and externalized surface scheme. In the present study, only two subgrid parameterizations are approached: turbulence and surface schemes.

The spatial discretization

The core of the MNH dynamic in its dry version is based on the resolution of the Euler and thermodynamic equations (energy preserving). The anelastic approximation

The bottom, lateral wall and top surfaces take a free-slip, impermeable and adiabatic behavior without the call of an externalized surface scheme. The open boundary condition is a Sommerfeld equation defined as wave radiation

The wind of the resolved scales has to satisfy the continuity equation

The horizontal part of the operator to invert in the elliptic problem is treated in the Fourier space

To execute LES, the Reynolds stress

The numerical domain is divided into two regions: where the equations of continuum mechanics hold and a solid region embedding the obstacle where they do not. After comparing the methods (Fig.

The

The right-hand-side (RHS) first term of Eq. (

Note that

The forced points are called ghost points and are renamed ghosts. To estimate the variable

In classical GCT

Quadratic interpolations of two analytical profiles

The accuracy of an interpolation depends on the

For truncated cells (at least one corner node is in the solid region),

The

Finally, the third component is

An interface condition depending on the characteristics of the surrounding fluid such as

Profile normal to the interface of two points of fluid information

In practice, three

First looking at the RHS of Eq. (

The elliptic problem (Eq.

According to the Green–Ostrogradski theorem, the

The

The four encountered cases correspond to a pure fluid cell

Knowing

It is known that

The turbulent characteristics are highly affected by surface interaction. As a consequence and for LES, the subgrid turbulence scheme (Sect.

Illustration of the unresolved physical processes near a nonidealized solid wall (black line) in an atmospheric context: the length scale based on the viscous effects (grey line) is drastically smaller than the roughness length. The roughness length approaches the scale of smallest eddies and governs the log-law profile.

Finally, the pragmatic limitation

Isolated from the rest of the code, the resolution of the pseudo-Poisson Eq. (

Potential flow around a cylinder:

Figure

Potential flow around a cylinder:

The Richardson (RICH) and the residual conjugate gradient iterative (RESI) methods are tested (Sect.

With a change of Galilean reference frame, this study corresponds to a uniform body acceleration

Potential solution around a sphere:

A pure dynamic and well-documented case that naturally follows previous ones is studied here. This physical case is the wake past a circular cylinder (non-stratified flow) at two moderate Reynolds numbers

Eddy structure in a viscous fluid: steady (left,

The standing eddies at

Recirculating region at

The

Description of the standing eddies in the wake of the solid cylinder (

The focus is on the unsteady mode at

This section is devoted to turbulent flows approaching our perspective: the simulation of an atmospheric flow over a city. The turbulent flows around a cubic body vertically confined in a channel and over an urban-like roughness (set of obstacles) are here described. MNH–IBM is explicitly compared to experimental investigations in the two cases. Comparisons to other LESs from the literature will be mentioned.

Using static pressure measurements, as well as laser-sheet and oil-film visualizations,

The mean flow around the cube presents a set of five recirculating regions (Fig.

Vertical symmetry plane of the mean flow:

Figure

Mean vertical profiles of velocities (top), turbulent kinetic energy and Reynolds stress (bottom). The lines correspond to the MNH–IBM results. The symbols are the

Still in the vertical symmetry plane of the mean flow, Fig.

To conclude this section and despite the uncertainties of the inlet condition, MNH–IBM is in good agreement with the experiments of

The MUST is an experimental campaign organized during early autumn 2001 in Utah's West Desert

Mean vertical profiles in the MUST experiment:

The distance between the horizontal limit of the computational domain and the array is about 20 times the container height. The large-scale flow is forced by an open boundary condition. A mean horizontal angle of

Wind at the horizontal cut at

Figure

Kinetic energy, turbulent kinetic energy and friction velocity obtained by the experimental

The mean kinetic energy

Spectrum of the measured (blue line) and modeled (green line)

The discrete Fourier transform of the

The MNH–IBM results are consistent with the experimental observations below

This study details the first implementation of an immersed boundary method (IBM) in the atmospheric Meso-NH (MNH) model currently based on mathematical formulations written for structured grids. The MNH–IBM aim is to explicitly model the fluid–solid interaction in the surface boundary layer developed over grounds presenting complex topographies such as cities or industrial sites.

A level-set function

The pressure solver, adapted to the IBM and isolated from the rest of MNH, is used to model potential flows around several obstacles. Compared to analytical and theoretical solutions, the numerical results demonstrate the ability of the IBM adaptation to ensure that the momentum is preserved and the continuity equation is respected. Non-dissipative flows are simulated to test the IBM forcing of the wind advection scheme (the impact of interpolations collected in

The immersed boundary method has been implemented in the 5.2 version of the Meso-NH code. This reference version is under the CeCILL-C license agreement and freely available at

The source files dedicated to IBM and the input files for the simulations in Sects.

Array of vortices around a cylinder:

Potential flow function of the space resolution (color code) plotting the pressure contours around two bells (

For most atmospheric applications, the region size for which the fluid molecular viscosity

The vorticity equation for a 2-D inviscid flow reveals no production in time. The numerical vorticity production at the immersed surface of a cylindrical body is studied here by initializing the simulation with the potential solution. To fit the potential solution, a nontrivial condition is employed on the tangent velocity

Solving the Euler equations:

Solving the Euler equations:

Figure

Figure

Summary of the mean wind advection scheme used (WENO: weighted essentially non-oscillatory).

Figure

The supplement related to this article is available online at:

All authors contributed to the development of the source code. The execution and the exploitation of the presented simulations were conducted by FA and GR. All authors contributed to the writing and editing of the paper.

The authors declare that they have no conflict of interest.

We thank the following for stimulating and fruitful discussions: Jacques Magnaudet (Institut de Mécanique des Fluides, IMFT, Toulouse), Yannick Hallez (Laboratoire de Génie Chimique de Toulouse, LGC, Toulouse), Jean-Lou Pierson (Institut Francais du Pétrole Energies Nouvelles, IFPEN, Lyon), Jean-Luc Redelsperger (Laboratoire de Physique des Océans, IFREMER, Brest), Jean-Pierre Pinty and Juan Escobar (Laboratoire d'Aérologie, LA, Toulouse), Odile Thouron, Thibaut Lunet, Luc Giraud, Isabelle d'Ast and Gerard Dejean (Centre Européen de Recherche et Formation Avancée en Calcul Scientifique, CERFACS, Toulouse). The simulations were performed on the Neptune/Nemo-CERFACS supercomputers on the Occigen-CINES Bull cluster (c2016017724 and a0010110079 GENCI projects).

Part of this work was supported by Région Midi-Pyrénées funding.

This paper was edited by Ignacio Pisso and reviewed by two anonymous referees.