Search

Fine-scale models for the transport of marine aerosols are of great interest for the study of micro-climates and air quality in areas of complex topography, such as in urbanized coastal areas. To this end, the MIO laboratory implemented the Meso-NH model in its LES version over the northwest Mediterranean coastal zone using a recent sea-spray source function. Simulated meteorological parameters and aerosol concentrations are compared to experimental data acquired in the Mediterranean coastal zone in spring 2008 on board the R/V Atalante. Key findings indicate that the large eddy simulation (LES) mode closely matches with the experimental data, enabling an in-depth analysis of the numerical model ability to predict variations in aerosol concentrations. These variations are influenced by different wind directions, which lead to various fetch distances typical of coastal zones.

Abstract A large‐eddy simulation tool is developed for simulating the dynamics of atmospheric boundary layers (ABLs) using lattice Boltzmann method (LBM), which is an alternative approach for computational fluid dynamics and proved to be very well suited for the simulation of low‐Mach flows. The equations of motion are coupled with the global complex physical models considering the coupling among several mechanisms, namely basic hydro‐thermodynamics and body forces related to stratification, Coriolis force, canopy effects, humidity transport, and condensation. Mass and momentum equations are recovered by an efficient streaming, collision, and forcing process within the framework of LBM while the governing equations of temperature, liquid, and vapor water fraction are solved using a finite volume method. The implementation of wall models for ABL, subgrid models, and interaction terms related to multiphysic phenomena (e.g., stratification, condensation) is described, implemented, and assessed in this study. An immersed boundary approach is used to handle flows in complex configurations, with application to flows in realistic urban areas. Applications to both wind engineering and atmospheric pollutant dispersion are illustrated.

Mesocale atmospheric flows that develop in the boundary layer or microscale flows that develop in urban areas are challenging to predict, especially due to multiscale interactions, multiphysical couplings, land and urban surface thermal and geometrical properties and turbulence. However, these different flows can indirectly and directly affect the exposure of people to deteriorated air quality or thermal environment, as well as the structural and energy loads of buildings. Therefore, the ability to accurately predict the different interacting physical processes determining these flows is of primary importance. To this end, alternative approaches based on the lattice Boltzmann method (LBM) wall model large eddy simulations (WMLESs) appear particularly interesting as they provide a suitable framework to develop efficient numerical methods for the prediction of complex large or smaller scale atmospheric flows. In particular, this article summarizes recent developments and studies performed using the hybrid recursive regularized collision model for the simulation of complex or/and coupled turbulent flows. Different applications to the prediction of meteorological humid flows, urban pollutant dispersion, pedestrian wind comfort and pressure distribution on urban buildings including uncertainty quantification are especially reviewed. For these different applications, the accuracy of the developed approach was assessed by comparison with experimental and/or numerical reference data, showing a state of the art performance. Ongoing developments focus now on the validation and prediction of indoor environmental conditions including thermal mixing and pollutant dispersion in different types of rooms equipped with heat, ventilation and air conditioning systems.

Abstract The goal of this study is to assess the performance of an innovative Lattice Boltzmann (LB) - Large Eddy Simulation (LES) approach in simulating neutral and stratified pollutant dispersion in complex urban environments. Different simulations are performed for the central area of Paris, accounting for continuous neutral or non-neutral gas releases from a circular source located in both channeled or confined flows. Predicted concentrations are compared with detailed wind tunnel measurements from the MODITIC project (FFI, 2016). Results exhibit a good qualitative and quantitative agreement between numerical and experimental data for the different configurations studied. All the estimated quality metrics match acceptance criteria. In addition, it is shown that the new LBM LES approach is able to capture and highlight the key turbulent mechanisms underlying dispersion process in and above urban areas. Hence, being based on extensive and detailed simulations and quality assurance studies, this paper highlights that the developed approach is well suited to address urban dispersion issues, including accidental chemical releases and short term exposure problems. Such results are particularly valuable to support the design and use of fast response dispersion models. Highlights • Application of a hybrid LBM-LES model for the prediction of pollutant dispersion. • Study of chanelled or confined source configurations, and neutral or dense gas dispersion. • Qualitative and quantitative assessment of the model performance. • Analysis of time averaged and instantaneous dispersion processes. • The developed model appears well suited to study complex urban pollutant dispersion. [ABSTRACT FROM AUTHOR]

Large-eddy simulations based on the Lattice-Boltzmann method of the flow in a realistic, full scale urban area are performed to compare several wind comfort criteria. It is observed that popular criteria for pedestrian comfort lead to very different conclusions, due to the access to high spatio-temporal resolution data. Different mixed strategies based on the combination of several criteria are proposed and compared to enhance pedestrian wind comfort assessment in practical cases. [ABSTRACT FROM AUTHOR]

Urban traffic-related air pollution is a major source of environmental and health damage and is difficult to quantify due to its inherent physical complexity. We construct a CFD-based simulation framework coupling an efficient numerical method for turbulent fluid flows with a microscopic traffic model and an emissions model to simulate road transport pollutant dispersion at the urban microscale. We improve the open-source Lattice-Boltzmann based CFD software OpenLB to overcome its original stability deficiencies for high Reynolds number flows. A stable recursive regularization procedure with a double distribution function approach is proposed to solve an advection diffusion equation for passive scalar transport at high Reynolds number. The code is successfully validated on three reference cases of increasing complexity and the traffic model SUMO along with a physical engine emissions model are coupled with OpenLB to simulate traffic-induced pollution from a road network in a realistic complex geometry. Transient flow features are analysed and the time-averaged concentration levels in different neighbourhoods of the considered geometry are evaluated: high concentration levels are observed close to the streets but also inside specific building infrastructures due to complex wind dynamics. Analyses of altitudinal concentration variations show that flow recirculations located close to traffic lights can drive pollutant over the buildings and increase concentration levels inside inner courtyards. Time-averaged concentration maps are constructed using both spatially uniform and non-uniform line sources and it is shown that using uniform sources leads to up to 20% local concentration overestimations inside the urban canopy. • The code OpenLB is used to simulate high-Reynolds scalar dispersion with DDF LBM-LES • Traffic emissions are simulated with a traffic simulator and a physical engine model • Spatial uniformization of the emissions can lead to 20% concentration underestimation [ABSTRACT FROM AUTHOR]

The feasibility and accuracy of large eddy simulation is investigated for the case of three-dimensional unsteady flows past an elastically mounted cylinder at moderate Reynolds number. Although these flow problems are unconfined, complex wake flow patterns may be observed depending on the elastic properties of the structure. An iterative procedure is used to solve the structural dynamic equation to be coupled with the Navier-Stokes system formulated in a pseudo-Eulerian way. A moving mesh method is involved to deform the computational domain according to the motion of the fluid structure interface. Numerical simulations of vortex-induced vibrations are performed for a freely vibrating cylinder at Reynolds number 3900 in the subcritical regime under two low mass-damping conditions. A detailed physical analysis is provided for a wide range of reduced velocities, and the typical three-branch response of the amplitude behavior usually reported in the experiments is exhibited and reproduced by numerical simulation. [ABSTRACT FROM AUTHOR]

Abstract: This paper addresses a still unknown effect of a high downward gradient of the square of the magnetic flux density on a laminar methane diffusion flame established over an axisymmetric co-flow burner. Because the magnetic force is expected to act mainly on paramagnetic oxygen, a parametric study varying the co-flowing stream oxygen content, the oxidizer velocity and the magnitude of the gradient of the square of the magnetic flux density was performed. CH∗ radicals spontaneous chemiluminescence was captured to track the flame sheets. Despite a low Reynolds number at the fuel exit and oxidizer flow velocities slightly larger than the fuel exit velocity, the flame flickering was triggered by downward gradients of the square of the magnetic flux density of magnitudes from 0.4 and 1.4T2/m for O2 contents of 0.75 and 0.55, respectively. This flickering is shown to result from an axisymmetric mode of a convective instability. Qualitative speculations based on previous works suggest that the magnetic body force may alternately strengthen buoyancy and shear stress, therefore hastening the instability onset. Indeed, at relatively low Grashof numbers, a region of flickering conditions shows up and grows with a magnetic Reynolds number. The sensitivities of the flickering frequency and amplitude are finally addressed, namely showing that over the investigated range of parameters the frequency spreads from 10.5 to 14Hz. [ABSTRACT FROM AUTHOR]

Accuracy and reliability of large-eddy simulation data in a really complex industrial geometry are invesigated. An original methodology based on a response surface for LES data is introduced. This surrogate model for the full LES problem is built using the Kriging technique, which enables a low-cost optimal linear interpolation of a restricted set of large-eddy simulation (LES) solutions. Therefore, it can be used in most realistic industrial applications. Using this surrogate model, it is shown that (i) optimal sets of simulation parameters (subgrid model constant and artificial viscosity parameter in the present case) can be found; (ii) optimal values, as expected, depend on the cost functional to be minimized. Here, a realistic approach, which takes into account experimental data sparseness, is introduced. It is observed that minimization of the error evaluated using a too small subset of reference data may yield a global deterioration of the results. [ABSTRACT FROM AUTHOR]

This paper presents the first numerical investigation via direct numerical simulation of the tone noise phenomenon occurring in the flow past laminar airfoils. This phenomenon corresponds to the radiation of discrete acoustic tones in some specific flow conditions, and has received much attention since the 1970s, and several experimental studies have been carried out to identify and understand the underlying physical mechanisms. However, several points remain to be clarified in order to provide a complete explanation of its origin. The flow around a two-dimensional NACA0012 airfoil is considered in order to have a deeper understanding of the tone noise phenomenon. Consistently with previous experimental studies, it is shown that depending on the Reynolds number and angle of attack, two different types of acoustic spectrum are observed: one which exhibits a broadband contribution with a dominant frequency together with a sequence of regularly spaced discrete frequencies, while the other one is only characterized by a simple broadband contribution. The first configuration is typical of the tone noise phenomenon. The present work shows that in this case, the mean flow on the pressure side of the airfoil exhibits a separation bubble near the trailing edge and the main tone frequency is close to the most amplified frequency of the boundary layer. The mechanism proposed in previous works for the main tone generation ? which implies the existence of a separation bubble at the pressure side ? is therefore validated by numerical simulation. On the other hand, the analysis of the suction side boundary layer reveals that there is no separation and that the most amplified frequency is different from the main tonal one. However, the suction side boundary layer is highly receptive to the tone frequency. Finally, an original explanation for the existence of the secondary discrete frequencies observed in the radiated pressure spectrum is given. They are associated to a bifurcation of the airfoil wake from a symmetric to a non-symmetric vortex pattern. A possible explanation for the existence of this bifurcation is the interaction between the disturbances which are the most amplified by the suction side boundary layer and those originating in the forcing of the suction side flow by the main tone noise mechanism. [ABSTRACT FROM AUTHOR]

A Kriging-based method for the parametrization of the response surface spanned by uncertain parameters in computational fluid dynamics is proposed. A multiresolution approach in the sampling space is used to improve the accuracy of the method. It is illustrated considering the problem of the computation of the corrections needed to recover equivalent free-flight conditions from wind-tunnel experiments. Using the surface response approach, optimal corrected values of the freestream Mach number and the angle of attack for the compressible turbulent flow around the RAE 2822 wing are computed. The use of the response surface to gain an insight into the sensitivity of the results with respect to other parameter is also assessed. [ABSTRACT FROM AUTHOR]

In order to assess the capability of the Sutton model to evaluate aero-optical effects in a turbulent boundary layer, large-eddy simulation (LES) evolving spatially and Reynolds averaged Navier-Stokes (PANS) computations are carried out at Mach number equal to 0.9. First aerodynamic fields are proved to compare favorably with theoretical and experimental results. Once validated, the characteristics of the boundary layer allow us to obtain information concerning optical beam degradation. On the one hand, the density field is used to compute phase distortion directly and, on the other hand, by means of the Sutton model. Therefore, LES and PANS simulations allow us to study optical models and the validity of their assumptions. Finally, LES is proved to be considered as a reference tool to evaluate aero-optical effects. [ABSTRACT FROM AUTHOR]

In closed rooms with limited convection human motion can considerably affect the airflow and thus the dispersion of pollutant. However, in Computational Fluid Dynamics (CFD) simulations on air quality and safety for human beings this effect is generally not considered, which is mainly due to a lack of a well-founded and detailed estimation of the human behavior and the high computational cost of taking into account moving objects in CFD meshes. This work addresses this issue by coupling multidisciplinary methods to allow for a more realistic simulation of pollutant dispersion by taking into account the influence of human movements. A Social Force Model predicts trajectory and speed of each person moving in a complex environment. A lattice Boltzmann-based CFD tool provides a Large Eddy Simulation of the unsteady turbulent airflow with pollutant dispersion and thermal effects. And an Actuator Line Model supplies the CFD tool with body forces that mimic the impact of moving objects on the airflow, thus, avoiding computationally expensive dynamic meshing. The capability of the coupled model is demonstrated on three realistic evacuation scenarios with various pollutant sources and a wide range of scales (dimension from 10 to 100 m, occupation from 10 to 6000 persons). The results allow to access instantaneous environmental parameters like pollutant concentration for each person during the course of the evacuation, enabling the assessment of associated health risks. • Multidisciplinary coupled model based on lattice-Boltzmann CFD and Social Force Model. • Large Eddy Simulation of pollutant dispersion under the influence of moving persons. • Actuator Line Model to simulate the wake of walking humans. • Social Force Model to predict evacuation path and speed of individual agents. [ABSTRACT FROM AUTHOR]

The numerical instability of the lattice Boltzmann method (LBM) at high Mach or high Reynolds number flow is well identified, and it remains a major barrier to its application in more complex configurations such as moving geometries. This work combines the compressible lattice Boltzmann model with rotating overset grids (the so-called Chimera method, sliding mesh, or moving reference frame) for high Mach flows. This paper proposes to use the compressible hybrid recursive regularized collision model with fictitious forces (or inertial forces) in a noninertial rotating reference frame. Also, polynomial interpolations are investigated, which allow fixed inertial and rotating noninertial grids to communicate with each other. We suggest a way to effectively couple the LBM with the MUSCL-Hancock scheme in the rotating grid, which is needed to account for thermal effect of compressible flow. As a result, this approach is demonstrated to have an extended Mach stability limit for the rotating grid. It also demonstrates that this complex LBM scheme can maintain the second-order accuracy of the classic LBM by appropriately using numerical methods like polynomial interpolations and the MUSCL-Hancock scheme. Furthermore, the method shows a very good agreement on aerodynamic coefficients compared to experiments and the conventional finite-volume scheme. This work presents a thorough academic validation and error analysis of the LBM for simulating moving geometries in high Mach compressible flows.

Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.

A barotropic counterpart of the well-known convected vortex test case is rigorously derived from the Euler equations along with an athermal equation of state. Starting from a given velocity distribution corresponding to an intended flow recirculation, the athermal counterpart of the Euler equations are solved to obtain a consistent density field. The present initialization is assessed on a standard lattice Boltzmann solver based on the D2Q9 lattice. Compared to the usual isentropic initialization, a much lower spurious relaxation toward the targeted solution is observed, which is due to the spatial resolution rather than approximated macroscopic quantities. The amplitude of the spurious waves can be further reduced by including an off-equilibrium part in the initial distribution functions.

An extended version of the hybrid recursive regularized lattice-Boltzmann model which incorporates external force is developed to simulate humid air flows with phase change mechanisms under the Boussinesq approximation. Mass and momentum conservation equations are solved by a regularized lattice Boltzmann approach well suited for high Reynolds number flows, whereas the energy and humidity related equations are solved by a finite volume approach. Two options are investigated to account for cloud formation in atmospheric flow simulations. The first option considers a single conservation equation for total water and an appropriate invariant variable of temperature. In the other approach, liquid and vapor are considered via two separated equations, and phase transition is accounted for via a relaxation procedure. The obtained models are then systematically validated on four well-established benchmark problems including a double diffusive Rayleigh Bénard convection of humid air, two- and three-dimensional thermal moist rising bubble under convective atmospheric environment, as well as a shallow cumulus convection in the framework of large-eddy simulation.

Large-scale coherent structures emerging in two-dimensional flows can be predicted from statistical physics inspired methods consisting in minimizing the global enstrophy while conserving the total energy and circulation in the Euler equations. In many situations, solid obstacles inside the domain may also constrain the flow and have to be accounted for via a minimum enstrophy principle. In this work, we detail this extended variational formulation and its numerical resolution. It is shown from applications to complex geometries containing multiple circular obstacles that the number of solutions is enhanced, allowing many possibilities of bifurcations for the large-scale structures. These phase change phenomena can explain the downstream recombinations of the flow in rod-bundle experiments and simulations.

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from 10^{4} to 10^{6}, and where a thorough analysis of the case at Re=3×10^{4} is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

This study focuses on grid refinement techniques for the direct simulation of aeroacoustics, when using weakly compressible lattice Boltzmann models, such as the D3Q19 athermal velocity set. When it comes to direct noise computation, very small errors on the density or pressure field may have great negative consequences. Even strong acoustic density fluctuations have indeed a clearly lower amplitude than the hydrodynamic ones. This work deals with such very weak spurious fluctuations that emerge when a vortical structure crosses a refinement interface, which may contaminate the resulting aeroacoustic field. We show through an extensive literature review that, within the framework described above, this issue has never been addressed before. To tackle this problem, we develop an alternative algorithm and compare its behavior to a classical one, which fits our in-house vertex-centered data structure. Our main idea relies on a directional splitting of the continuous discrete velocity Boltzmann equation, followed by an integration over specific characteristics. This method can be seen as a specific coupling between finite difference and lattice Boltzmann, locally on the interface between the two grids. The method is assessed considering two cases: an acoustic pulse and a convected vortex. We show how very small errors on the density field arise and propagate throughout the domain when a vortical flow crosses the refinement interface. We also show that an increased free stream Mach number (but still within the weakly compressible regime) strongly deteriorates the situation, although the magnitude of the errors may remain negligible for purely aerodynamic studies. A drastically reduced level of error for the near-field spurious noise is obtained with our approach, especially for under-resolved simulations, a situation that is crucial for industrial applications. Thus, the vortex case is proved useful for aeroacoustic validations of any grid refinement algorithm.

In this paper, a variant of the acoustic multipole source (AMS) method is proposed within the framework of the lattice Boltzmann method. A quadrupole term is directly included in the stress system (equilibrium momentum flux), and the dependency of the quadrupole source in the inviscid limit upon the fortuitous discretization error reported in the works of E. M. Viggen [Phys. Rev. E 87, 023306 (2013)PLEEE81539-375510.1103/PhysRevE.87.023306] is removed. The regularized lattice Boltzmann (RLB) method with this variant AMS method is presented for the 2D and 3D acoustic problems in the inviscid limit, and without loss of generality, the D3Q19 model is considered in this work. To assess the accuracy and the advantage of the RLB scheme with this AMS for acoustic point sources, the numerical investigations and comparisons with the multiple-relaxation-time (MRT) models and the analytical solutions are performed on the 2D and 3D acoustic multipole point sources in the inviscid limit, including monopoles, x dipoles, and xx quadrupoles. From the present results, the good precision of this AMS method is validated, and the RLB scheme exhibits some superconvergence properties for the monopole sources compared with the MRT models, and both the RLB and MRT models have the same accuracy for the simulations of acoustic dipole and quadrupole sources. To further validate the capability of the RLB scheme with AMS, another basic acoustic problem, the acoustic scattering from a solid cylinder presented at the Second Computational Aeroacoustics Workshop on Benchmark Problems, is numerically considered. The directivity pattern of the acoustic field is computed at r=7.5; the present results agree well with the exact solutions. Also, the effects of slip and no-slip wall treatments within the regularized boundary condition on this pure acoustic scattering problem are tested, and compared with the exact solution, the slip wall treatment can present a better result. All simulations demonstrate that the RLB scheme with the AMS method is capable of accurately simulating 2D and 3D acoustic generation, propagation, and scattering at zero viscosity.

This paper investigates the problem of sound source localization from acoustical measurements obtained by an array of microphones. The sound propagation medium is assumed to be randomly inhomogeneous, being modelled by a random function of space. In this case, classical source localization methods (e.g., beamforming, near-field acoustical holography, and time reversal) cannot be used anymore. Therefore, an approach based on the statistical moments of acoustical measurement is proposed to solve the aforementioned problem. In this work, a Karhunen-Loève expansion is used so that the random medium can be represented by a small number of uncorrelated and identically distributed random variables. The statistical characteristics of the measurements in terms of probability density function and statistical moments are also studied. Then, the sound source is localized by minimizing the error of statistical moments between the real measurements obtained from the microphone array and the measurements simulated from an assumed source. Finally, a numerical example is introduced to justify the proposed method. This experiment shows that the random field can be replicated by a very small number of random variables, the statistical moments of measurements guarantee the convergence, and the source location can be accurately estimated using the proposed source localization method.

In this paper the sound source identification problem is addressed with the use of the lattice Boltzmann method. To this aim, a time-reversed problem coupled to a complex differentiation method is used. In order to circumvent the inherent instability of the time-reversed lattice Boltzmann scheme, a method based on a split of the lattice Boltzmann equation into a mean and a perturbation component is used. Lattice Boltzmann method formulation around an arbitrary base flow is recalled and specific applications to acoustics are presented. The implementation of the noise source detection method for two-dimensional weakly compressible (low Mach number) flows is discussed, and the applicability of the method is demonstrated.

Trajectory of an optical vortex has been identified for its propagation in atmospheric turbulence using numerical simulations. An analytical expression has been found, relating the radial departure of the vortex in plane perpendicular to the direction of propagation, to the refractive index structure function parameter and the inner scale of turbulence. The angular orientation of the vortex in the same transverse plane is found to be related to the anisotropy of the medium. The obtained results provide an alternative way to find turbulent parameters with the help of optical vortices.

The present paper provides an up-to-date survey of the use of large eddy simulation (LES) and sequels for engineering applications related to aerodynamics. Most recent landmark achievements are presented. Two categories of problem may be distinguished whether the location of separation is triggered by the geometry or not. In the first case, LES can be considered as a mature technique and recent hybrid Reynolds-averaged Navier-Stokes (RANS)-LES methods do not allow for a significant increase in terms of geometrical complexity and/or Reynolds number with respect to classical LES. When attached boundary layers have a significant impact on the global flow dynamics, the use of hybrid RANS-LES remains the principal strategy to reduce computational cost compared to LES. Another striking observation is that the level of validation is most of the time restricted to time-averaged global quantities, a detailed analysis of the flow unsteadiness being missing. Therefore, a clear need for detailed validation in the near future is identified. To this end, new issues, such as uncertainty and error quantification and modelling, will be of major importance. First results dealing with uncertainty modelling in unsteady turbulent flow simulation are presented.

An exact relationship for the local skin friction is derived for the compressible turbulent wall-bounded flow (channel, pipe, flat plate). This expression is an extension of the compressible case of that derived by Fukagata [Phys. Fluids 14, L73 (2002)] in the case of incompressible wall-bounded flows. This decomposition shows that the skin friction can be interpreted as the contribution of four physical processes, i.e., laminar, turbulent, compressible, and a fourth coming from the interaction between turbulence and compressibility. Compressible numerical simulations show that, even at Mach number M=2 , the main contribution comes from the turbulence, i.e., the Reynolds stress term.