Towards stochastic subgrid‐scale modeling of turbulent thermal convection in an under‐resolved off‐lattice Boltzmann method

A characteristic‐based Off‐Lattice Boltzmann Method (OLBM) and a stochastic One‐Dimensional Turbulence (ODT) model is utilized for numerical simulation of turbulent thermal convection. Standalone ODT results for low‐order statistics are compared with those from various eddy‐viscosity‐based subgrid‐scale models utilized in Large‐Eddy Simulations (LES) with OLBM. The predictive capabilities of both approaches are discussed by comparison with available reference Direct Numerical Simulation (DNS) results. All turbulence models are able to predicted the mean temperature, but fail to fully capture fluctuations. While the OLBM aims to represent large‐scale structures, it misses some constitutional small‐scale fluctuations. By contrast, the reduced‐order ODT model captures small‐scale fluctuations in the vicinity of the wall, but cannot resolve the organized bulk flow. Here, the modeling capabilities of both OLBM and ODT as standalone tools are discussed. On this basis, a strategy for the incorporation of ODT as wall model in OLBM is suggested.


INTRODUCTION
Nowadays, the Lattice Boltzmann Method (LBM) is used as an alternative to traditional Navier-Stokes-based solvers to simulate all kinds of fluid flows and heat transfer problems [1].LBM is based on the discretization of the Boltzmann equation.LBM is, hence, a kinetic-theory-based method that describes the evolution of particle distribution functions in phase space rather than transported Eulerian flow variables.Originally, LBM was restricted to uniform Cartesian coordinates, which limited the geometry of computing domains.However, combining finite-difference, finite-volume, and finite-element approaches with LBM eliminates the drawbacks associated with the grid and domain shape.Such extended LBM approaches are termed as Off-Lattice Boltzmann Method (OLBM).Similar to traditional Navier-Stokes solvers, turbulent flows can be simulated in the LBM framework by formulations for Direct Numerical Simulation (DNS), Reynolds-Averaged Navier-Stokes Simulation (RANS), and Large-Eddy Simulation (LES) [2,3].By incorporation of a subgrid-scale (SGS) model, an LBM-based LES can simulate turbulent flows on coarser grids without significantly compromising accuracy for the medium and large turbulent scales.In LBM simulations, the SGS treatment is accomplished using turbulence models that address the statistical closure problem for the unresolved (filtered) turbulent scales.The effect of SGS turbulence on the resolved scales can be estimated with conventional flow solvers [4].The inclusion of SGS models in non-fully resolved LBM simulations can help to reduce the computational cost while still giving a reasonable representation of the resolved flow structures and scales.
In the present study, turbulent convection is investigated for a planar periodic 3-D Rayleigh-Bénard set-up (see Figure 1).Rayleigh-Bénard convection hence denotes the flow that emerges when the fluid confined between the two horizontal walls is heated from below and cooled from above.This very idealized setting has surprisingly many scientific and engineering applications, such as thermal and geothermal engineering, solar collectors, cooling of electronic circuit boards, and even for crystal growth problems [5], as well as various geophysical phenomena, including convection in the ocean, the Earth's outer core, and the outer layer of the Sun [6].Even for mildly turbulent conditions at low Rayleigh () number turbulence modeling is a challenge.Depending on the turbulence model used, steady, periodic, and aperiodic solutions can be obtained highlighting standing limitations in the numerical prediction of wall-bounded convective flows.It was shown recently that utilization of a map-based stochastic One-Dimensional Turbulence (ODT) model [7] can be beneficial.Utilizing ODT as near-wall SGS model within a conventional LES framework allowed to significantly extended the range of resolved scales, which mitigated the bias in the onset of the transition of the convective boundary layer at affordable cost [8].As a first step in the sketched direction, the present study primarily aims to assess the performance of different SGS models in an OLBM framework for turbulent Rayleigh-Bénard convection.In addition, the standalone predictive capabilities of ODT are assessed by utilizing it for extrapolation to a low  regime.On this basis, a possible strategy for stochastically closed OLBM is proposed.
The rest of the paper is organized as follows.Section 2 covers the configuration.Section 3 provides an overview of the OLBM and Section 4 of the ODT model.Section 5 presents low-order statistics obtained with various turbulence models.Section 6 discusses the applicability of ODT as SGS model in OLBM.Last, Section 7 provides some concluding remarks and an outlook.

FLOW CONFIGURATION AND NUMERICAL SETUP
Figure 1 shows a sketch of the Rayleigh-Bénard setup investigated.The domain dimensions are taken as 2 ×  × 2 in the -, -, and -direction, respectively, where  is the height of the domain that is taken as the characteristic length scale.The top wall is cooled and maintained at the low temperature   , while the bottom wall is heated and kept at a higher temperature  ℎ .Homogeneous no-slip boundary conditions are prescribed for the velocity at these walls.Periodic boundary conditions are imposed in the  and  directions.The flow simulations are characterized by the Rayleigh number  = ( 3 Δ)∕(), and the Prandtl number  = ∕, where  represents the volumetric expansion coefficient and  is the acceleration due to gravity, and Δ =  ℎ −   is the prescribed temperature difference across the domain height .
The molecular fluid properties  and  have been introduced above.The simulations discussed below are conducted at  = 6.3 × 10 5 (weakly turbulent) and  = 0.71 (air).In OLBM, a uniform grid is used in the periodic horizontal (, ) directions of the domain, while a non-uniform grid is employed in the wall-normal direction.Initial perturbations are prescribed following Schoppa et al. [9] in order to expedite the transition of the flow from the laminar to the turbulent state.Standalone ODT utilizes an adaptive mesh and resolves the flow only along a single vertical line (ODT line in sketch).

OLBM governing equations
In LBM, instead of solving the Navier-Stokes equations, the flow velocity and thermal fields are simulated using two discrete Boltzmann distribution function equations.The Boltzmann equations [10] for the density and temperature distribution functions,   and   , respectively, are considered in the discrete momentum space.The D3Q27 and D3Q7 lattice models [11] are used for the density and temperature distribution function equations, respectively.The discrete-velocity Boltzmann equation with the collision term modeled using the single-relaxation-time (SRT) Bhatnagar-Gross-Krook (BGK) approximation for thermal convection for the linearized equation of state (Oberbeck-Boussinesq limit) are given by where  denotes the lattice direction,  = 3 ( −   ) ( , − )    is the buoyancy force that acts as momentum source under globally unstable conditions, where   = ( ℎ +   )∕2 is the bulk mean temperature, and    and    the equilibrium distribution functions for the lattice velocity and temperature, respectively.The distribution functions   and   denote the probability of finding a molecule (fluid parcel) at location  with a velocity   = ( , ,  , ,  , ) at time instant .The parameters  ℎ and   denote the hydrodynamic and thermal relaxation rates, which relate the macroscopic flow parameters with the distribution functions.They are given as  ℎ = ∕ 2   and   = ∕ 2  [12], where  is the kinematic viscosity,  is thermal diffusivity of the fluid and   is the lattice speed of sound that is equal to 1∕ √ 3. The density and thermal equilibrium distribution functions are given by The macroscopic flow properties are then related to the density and temperature distribution functions such that where  is the mass density,  the momentum density, and  the temperature of the fluid.Notice that  = (, , ) T is the macroscopic fluid velocity vector with Cartesian components , , .
The characteristic-based OLBM extension was proposed by Lee et al. [13] and Bardow et al. [14].It preserves all physically required and algorithmically appealing advantages of the classical LBM while extending it to unstructured meshes and with better stability properties.The present work employs a characteristic-based OLBM formulated in a finite-difference framework for non-isothermal flows on the Oberbeck-Boussinesq regime.The collision term is treated implicitly in order to improve numerical stability of the scheme.In addition, a characteristic-based discretization schemes can work with larger CFL number and, hence, a larger ratio of the relaxation time  ℎ and the selected time step Δ in comparison to a naive discretization scheme.A variable transformation technique is used to address the implicitness from the equations.Characteristic-based treatment is addressed by utilizing a Lax-Wendroff scheme for time integration of the LBM equations.An explicit second-order central-difference scheme is used for the discretization of the advection terms.A sixth-order compact spatial filter is employed to remove oscillations caused by non-dispersive central difference schemes.The detailed explanation of the present OLBM is given in reference [15].
Boundary conditions are need to be prescribed to the distribution functions at the domain boundaries.The no-slip walls are treated as the sum of the equilibrium and the non-equilibrium distribution.The equilibrium distribution functions can be computed from the macroscopic properties using Equations ( 3) and ( 4).The non-equilibrium distribution functions at the isothermal no-slip walls are determined using the following approximation,  2    ∕ 2 = 0 and  2    ∕ 2 = 0, where ∕ represents the wall-normal derivative, calculated using a fourth-order finite-difference approximation.At the internal nodes, the non-equilibrium distributions are given by the excursions,    =   −    and    =   −    .The periodic boundary conditions at lateral non-wall boundaries are straightforwardly implemented using a "jump index" (modulus operation) for the derivatives that act across the domain boundary that is located staggered to the lattice points.

Statistical subgrid-scale modeling in OLBM LES
In OLBM-LES simulations, the distribution functions   and   in Equations ( 1) and ( 2) are formally replaced with the spatially filtered distribution functions   and   , respectively.The SGS effects are incorporated into the governing equations simply by adding a relaxation time parameter  ℎ, or  , to the molecular relaxation time scale in the collision term parameterizations.The time scales  ℎ and   in Equations ( 1) and ( 2) will be replaced with  * ℎ ,  *  , respectively. * ℎ ,  *  are total relaxation times and are given by where  ℎ and   depend on the molecular viscosity and thermal diffusivity, respectively, and  ℎ, and  , are eddy relaxation times for the momentum and temperature equations, which are related to the eddy viscosity   and eddy diffusivity   , respectively.Conventional statistical turbulence models can hence be utilized to calculate first the eddy viscosity and eddy diffusivity from the resolved flow state, and then the relaxation time scales.Various statistical SGS models have been implemented in the in-house OLBM solver.These comprise the constant coefficient Smagorinsky model [16], the Smagorinsky model with buoyancy production modification model [17], the Wale model [18], and the Vreman model [19].Moreover, a Van-Driest wall function [18] is incorporated with the constant coefficient Smagorinsky model to reduce the value of   near the walls.The Smagorinsky LES model employs a Smagorinsky constant   = 0.1, while the Vremann model employs a constant of 0.158, and the WALE model employs a constant of 0.5.Implicit LES simulations are also conducted, where the grid itself acts as a filter and the second-order finite-difference schemes employed in the OLBM yield numerical dissipation terms that are re-interpreted as physical SGS dissipation terms.In Implicit LES, the inherent numerical dissipation acts as an implicit subgrid model, forming a natural form of LES [20].No additional eddy viscosity is introduced in Implicit LES simulations, but a coarse grid and low-order (or stabilized) scheme are needed to provide sufficient small-scale dissipation.

ODT governing equations
The ODT computational domain is a statistically representative, vertical coordinate (see Figure 1).There are no sidewall boundaries so that ODT simulations are performed for a notionally infinite layer of fluid.Molecular fluxes are described by a diffusion equation and fully resolved in the vertical direction, whereas turbulent advection is represented by a stochastic process that is influenced by velocity shear and buoyancy forces.Utilizing the Oberbeck-Boussinesq approximation analogous to the case of OLBM described above, the temporal ODT equations for thermal convection take the form [21] where   () and   denote a stochastic process that depends on the current flow state.  () models turbulent advection [7], buoyancy-velocity couplings [22], and pressure fluctuations [23], while   represents turbulent advection.The efficiency of pressure-velocity-buoyancy couplings is adjustable by the model parameter .Notice that the buoyancy force  is formally absent in the ODT equation for  since the effects are represented by   ().Further details on the stochastic terms are provided below.Spatial derivatives in Equations ( 7) are discretized with a finite-volume method on an adaptive grid and temporal derivatives with a low-order explicit time-marching scheme [24].This allows straightforward treatment of isothermal no-slip walls by Dirichlet boundary conditions  = 0 at  = 0 and  = , and  =  ℎ ( =   ) at  = 0 ( = ), respectively.The initial condition is a linear conductive solution with quiescent flow that quickly approaches a statistically stationary state.

Stochastic eddy events
Equations ( 7) describe a relaxing laminar-diffusive flow evolution in response to stochastic "kicks" representing turbulence.The stochastic terms   and   are implemented with the aid of a point process that generates a sequence of eddy events.These events are formulated with the aid of a physical mapping operation that is based on scale-locality and conservation principles.The former induces a turbulence-like forward cascade and the latter is assured by the measure preserving property.Map applications can be represented as permutations of fluid parcels along a selected interval of the ODT line coordinate but are more generally described by the triplet map [7].For a selected spatial line interval, the triplet map (i) compresses flow profiles to a third of their length, (ii) pastes two copies to fill the line interval, and (iii) flips the central copy in order to not introduces discontinuities.This algorithm automatically increase the resolution in turbulent regions of the flow and allows to coarsen the resolution in laminar regions [24].Eddy events are described by three random variables: size   , location   , and time of occurrence   .Scheduling is done with a Poisson process with ha high sampling rate  −1  .The momentary rate of a size-  eddy event is given by the unknown eddy-rate distribution (  ,   ; ) =  −2  −1  (  ,   ; ), which depends on the flow state and is therefore unknown.A thinningand-rejection method is used in practice to avoid the costly construction of the distribution function [7].The alternative approach estimates the momentary rate  −1  from the flow state by considering the eddy energy  2  ∕ 2  yielding where   and   represent the specific kinetic energy and the specific potential energy of an eddy event covering the interval [  ,   +   ], and   a specific viscous penalty energy that is used to cut-off unphysically small eddy events from the sampling procedure [7].Explicit expressions corresponding to the symbolic terms above are summarized in the literature [25].Furthermore,  is the eddy-rate parameter and  the small-scale (viscous) suppression parameter discussed below.Eddy events are unphysical and rejected when  −1  is imaginary.Otherwise over-sampling assures that physically plausible eddy events are accepted with probability  −1  ∕ −1  ≪ 1.No large-scale suppression is used so that eddy events can be as large as .
Both  and  have to be calibrated for generic setup for thermal convection for a selected value of .For  = 0, the potential energy is released to and taken from the vertical velocity component  [22].For  > 0, a corresponding fraction of released potential energy is instantaneously distributed also to the  and  velocity components modeling pressurevelocity-buoyancy couplings.In the single-parameter buoyancy formulation used here, kinetic energy is taken from all velocity components in proportion to the current inter-component distribution, ignoring  in order to avoid unphysical exceptions.However,  = 0 [22] and  = 2∕3 (present) are the two fixed points of the buoyancy parameterization described in the literature [25], that are consistent with the slightly different buoyancy formulation used here.Here,  = 60,  = 220, and  = 2∕3 is used utilizing the calibration for turbulent thermal convection with  ≃ 0.7 [21].

RESULTS
The OLBM solver used has been verified previously with several benchmark cases [15].The current study extends those preliminary results by an assessment of various SGS models for turbulent Rayleigh-Bénard convection in a low-Rayleighnumber regime.As introduced above, the Smagorinsky model, its extension by a Van-Driest wall-function (VD) and with buoyancy production modification (Buo), the wall-adapting local eddy-viscosity (WALE) model, the Vreman model, and implicit LES without a turbulence closure model and only numerical dissipation are considered.Present OLBM results are compared with reference DNS results of Wörner [26].Temperature is normalized with Δ, while the velocity components are normalized using the free-fall velocity as bulk velocity scale,   = √ Δ such that  = ∕  ,  = ∕  , and  = ∕  .In OLBM, statistics are computed by averaging over the horizontal directions  and , and over a time interval of ≈ 700 bulk time units ∕  in the statistically stationary state.In ODT, statistics are computed solely by temporal averaging over ≈ 10 5 eddy events accumulated during ≈ 75,000 bulk time units requiring ≈ 1.5 h for simulation on a single core with ≈ 200 non-equidistant cells in the adaptive grid.

Flow visualization based on LES results from OLBM
Figures 2A and 2B show contours of instantaneous isotherms and streamlines on a spanwise plane obtained with OLBM using the Smagorinsky SGS model.The temperature contours exhibit larger patches and elongated structures.The instantaneous streamlines are colored based on the non-dimensional velocity magnitude and reveal large-scale vortical flow structures.These flow structures correspond with rising and descending thermal plumes due to which cooler or warmer fluid may occupy the entire domain height at some horizontal locations.

Low-order temperature statistics in OLBM and ODT with comparison to DNS
Figure 3A shows the normalized mean temperature ⟨⟩ profiles over the normalized coordinate  = ∕.These profiles exhibit anti-symmetry about the midplane located at  = 0.5.The highest temperature gradients are observed in the proximity to the top and bottom walls within a distance of 0.15, corresponding to the thermal boundary layer region.
The mean temperature profile remains almost flat in the center of the domain.Remarkably, all the SGS models and ODT exhibit good agreement with the reference DNS [26].
Figure 3B shows the normalized temperature root-mean-square (RMS) fluctuation   = √ ⟨ 2 ⟩ − ⟨⟩ 2 across the domain height.The profile of   is symmetric about  = 0.5 and exhibits two peaks each of which is located close to a wall.From there, the RMS gradually decreases towards the center of the domain.All SGS models fail in predicting   , but not as much as standalone ODT.Interestingly, the error is lowest for the RMS peaks in OLBM and has an insignificant dependence on the SGS model.ODT captures the near-wall behavior rather well but only half of the RMS in the bulk, capturing foremost the small-scale fluctuations.Artificial local minima next to the RMS peak locations are a known modeling artifact [21,24,27].
The Nusselt numbers for reference DNS, all SGS models used in OLBM, and standalone ODT are reported in Table 1.The Nusselt number is computed as the areal average of  = − ⟨⟩  ∕( Δ  ) over the top and bottom walls in OLBM and TA B L E Nusselt () number predicted by reference DNS, reference LES, 3D OLBM LES using various SGS models, and 1D standalone ODT for  = 6.3 × 10 5 ,  = 0.71 in a large-aspect-ratio Rayleigh-Bénard configuration.
Reference DNS [26] Reference LES [28]   by temporal averaging in ODT.All models have been applied without re-calibration.Notably, the Smagorinsky model with Van-Driest damping, as well as the WALE, Vreman, and ILES SGS models, closely reproduce the reference DNS within a 1% margin.Present standalone ODT predictions are ≈ 27% lower than the reference DNS, but it is worth to emphasize at this point that the model is built for fully developed turbulence [22].Furthermore, in its present set-up, ODT was calibrated for  ≃ 10 10 using reference data for a cylindrical Rayleigh-Bénard configuration with aspect ratio 0.5 [21].Those preliminary results suggest that ODT exhibits classical Malkus scaling ( ∼  1∕3 ) also for  < 10 8 .The extrapolation down to  = 6.3 × 10 5 hence suffers from a not fully captured transition to the Shraiman-Siggia scaling ( ∼  2∕7 ) as shown in the literature [21].The regime transition occurs for  ∼  (10 9 ) so that  can be expected ≈ 45% less for the selected .Present ODT results demonstrate that the actually observed difference is only half of that estimated discrepancy, hence less severe.⟩ that are aligned with the horizontal  and the vertical  coordinate, respectively.The mean velocity vanishes (⟨⟩ = 0) in both OLBM and ODT.  and   are symmetric to  = 0.5.In OLBM and reference DNS,   is bimodal, whereas   is unimodal, and both quantities exhibit different wall gradients.In standalone ODT,   and   are identical.They are bimodal like the reference   profile but the wall gradient reproduces that of the reference   profile.This is a consequence of the map-based advection modeling approach due to which the velocity components are only representative of the kinetic energy while they do not take part in the transport of fluid.Similar to   discussed above, only ≈ 50% of the bulk velocity fluctuations are captured.

Low-order velocity statistics in OLBM and ODT with comparison to DNS
The error of all SGS models with respect to   ,   , and   are illustrated in Figure 4C, with error percentages calculated relative to the DNS data [26].The maximum local error values for   and   occur at the central point ( = 0.5).Notably, advanced SGS models exhibit superior performance compared to the Smagorinsky SGS model in terms of the average Nusselt number.Given the known limitations of the Smagorinsky model near walls, a larger discrepancy in the overall Nusselt number error is observed.Conversely, the ILES, Smagorinsky with Van-Driest damping, Vreman, and WALE SGS models demonstrate errors of less than 1% for the average Nusselt number.In terms of   and   local errors at the central point, the Smagorinsky model performs better than other models.Remarkably, the results obtained with ILES are surprisingly comparable to those obtained with other SGS models.This similarity can be attributed to the low turbulence intensity resulting from the low  considered in this study.

DISCUSSION OF THE UTILIZATION OF ODT AS STANDALONE AND WALL MODEL
Klein and Schmidt [21] have previously demonstrated excellent agreement of the standalone application of ODT with DNS results at higher Rayleigh numbers in a cylindrical cell after parameter calibration.Their ODT parameters are calibrated at  ≈ (10 10 ) and demonstrate a robust correspondence up to  ≈ (10 16 ).Notably, they observed a Nusselt value offset for  < 10 8 .Even though the problem domain is slightly different in this study, the same parameters are taken into account for the ODT.The ODT model is utilized to simulate turbulent Rayleigh numbers lower than its originally calibrated range.Consequently, the Nusselt number in the extrapolated-to-low- 1D model is lower compared to the results obtained from 3D models in the present study.Schmidt et al. [29] have previously integrated ODT as a wall model within a conventional LES framework based on the Navier-Stokes solver.Freire [8] employed a Navier-Stokes LES code coupled with the ODT wall model to simulate a convective boundary layer under various conditions.The ODT model offers a vertically refined flow field near the wall with small-scale fluctuations and vertical fluxes resolved by a stochastic approach, forming a complementary extension to the LES resolved large-scale coherent structures.This approach proved to be a valuable tool for studying atmospheric boundary-layer problems with relevant near-wall phenomena.Remarkably, such LES with ODT wall-modeling (LES-ODT) has yielded low-order statistical results similar to traditional wall-function LES but showcased improvements in velocity variances and turbulence spectra in the near-wall region.The methodology described in the literature utilizes ODT in the first layers of LES cells adjacent to a wall, complemented by conventional LES for the bulk.Notably, ODT as a wall model distinguishes itself by preserving turbulent fluctuations, unlike conventional wall models, and excels at resolving the up-scale energy cascading phenomenon near walls.However, existing hybrid LES-ODT models are still rather costly [8], where a hybrid OLBM-ODT approach offers additional potential for cost reduction due to its fundamentally different formulation.

CONCLUSION
In this paper, various SGS models have been evaluated in the OLBM framework using turbulent Rayleigh-Bénard convection as test case.The results obtained showcase that the spatio-temporal variability of large-scale highly-organized flow structures dominates the temperature and velocity fluctuations in the weakly turbulent regime investigated.On the one hand, small-scale fluctuations are not well represented by the statistical SGS models so that a systematic under-prediction of ≈ 2% with respect to   up to ≈ 20% with respect to   has been observed.These results do not show a significant dependence on the statistical SGS model used.Hence, fundamental improvements in OLBM LES for application to turbulent thermal convection may therefore only be obtained with an advanced SGS modeling strategy.On the other hand, the effects of large-scale flow structures are at most partly captured in standalone ODT resulting in a systematic underprediction of temperature and velocity fluctuations by ≈ 50% in the bulk.Nevertheless, the relevant features of low-order temperature and velocity statistics are very well captured by ODT in the vicinity of the wall at rather low numerical cost.Improvements of OLBM LES may be achieved first if ODT is used as wall model combining the approach of Schmidt et al. [29] and Malaspinas et al. [30].

F I G U R E 1
Schematic drawing of the Rayleigh-Bénard setup investigated.

F
I G U R E 2 (A) Contours of the instantaneous temperature, (B) streamlines of the instantaneous velocity field for the mid-spanwise plane obtained with OLBM.OLBM, Off-Lattice Boltzmann Method.

F
I G U R E 4 (A) Horizontal and (B) vertical RMS velocity vertical profiles.The line styles and symbols are the same as in Figure3.(C) Error of temperature, vertical velocity fluctuations at  = 0.5 and overall Nusselt number for various SGS models in OLBM.OLBM, Off-Lattice Boltzmann method; RMS, root-mean-square; SGS, subgrid-scale.

Figures
Figures4A and 4Bshow the normalized RMS fluctuation velocities   = √ ⟨ ′ 2 ⟩ and   = √ ⟨ ′ 2 ⟩ that are aligned with the horizontal  and the vertical  coordinate, respectively.The mean velocity vanishes (⟨⟩ = 0) in both OLBM and ODT.  and   are symmetric to  = 0.5.In OLBM and reference DNS,   is bimodal, whereas   is unimodal, and both quantities exhibit different wall gradients.In standalone ODT,   and   are identical.They are bimodal like the reference   profile but the wall gradient reproduces that of the reference   profile.This is a consequence of the map-based advection modeling approach due to which the velocity components are only representative of the kinetic energy while they do not take part in the transport of fluid.Similar to   discussed above, only ≈ 50% of the bulk velocity fluctuations are captured.The error of all SGS models with respect to   ,   , and   are illustrated in Figure4C, with error percentages calculated relative to the DNS data[26].The maximum local error values for   and   occur at the central point ( = 0.5).Notably, advanced SGS models exhibit superior performance compared to the Smagorinsky SGS model in terms of the average Nusselt number.Given the known limitations of the Smagorinsky model near walls, a larger discrepancy in the overall Nusselt number error is observed.Conversely, the ILES, Smagorinsky with Van-Driest damping, Vreman, and WALE SGS models demonstrate errors of less than 1% for the average Nusselt number.In terms of   and   local errors at the central point, the Smagorinsky model performs better than other models.Remarkably, the results obtained

Smagorinsky ILES Smagorinsky +Van-Driest SGS Buoyancy WALE Vreman ODT
: DNS, direct numerical simulation; LES, Large-Eddy Simulations; ODT, one-dimensional turbulence; OLBM, Off-Lattice Boltzmann method; SGM, subgrid-scale. Abbreviations This research is supported by the German Federal Government, the Federal Ministry of Education and Research and the State of Brandenburg within the framework of the joint project EIZ: Energy Innovation Center (project numbers 85056897 and 03SF0693A) with funds from the Structural Development Act (Strukturstärkungsgesetz) for coal-mining regions.Open access funding enabled and organized by Projekt DEAL.