Coarse Grid Research Articles

A hybrid numerical framework of potential and viscous flows for simulations of free running surface ship maneuvering in waves

This study presents a hybrid numerical framework for the accurate and efficient simulation of free-running surface ship maneuvering in waves. The proposed hybrid framework combines the MOUM (Modified Osaka University Method) body-force method to model the propeller and a hybrid approach of potential and viscous flows for wave-ship interactions. The MOUM body-force method, based on BEMT (Blade Element Momentum Theory), considers the three-dimensional viscous effects of propellers. The hybrid approach of potential and viscous flows decomposes the total physical field into the incident field and the complementary field, where the incident wave is solved by the potential flow theory to ensure the accuracy of wave propagation with coarse grids, and viscous and nonlinear effects are incorporated through the adapted URANS (Unsteady Reynolds Averaged Navier-Stokes) with refined grids. The 6DOF (Degrees of Freedom) motion equations of the rigid body and the dynamic structured grid with the overset technology are adopted to simulate the 6DOF motions of ship maneuvering in waves. Firstly, the applicability of the computational domain, domain-moving strategy, and ship motion-solving method for effectively simulating ship maneuvering in wave conditions are thoroughly examined. Then the reliability of present hybrid approach is validated using turning benchmark model tests of KCS model in regular waves. Simulations of turning circle and zig-zag maneuvers in regular waves are carried out by the traditional viscous CFD (Computational Fluid Dynamics) based on the boundary wave-generation approach and the proposed hybrid framework. The trajectories, motions and forces of ship maneuvering in waves predicted by both methods, as well as the computational efficiency, are compared. Numerical results indicated that the proposed method offers high accuracy and efficiency for predicting ship maneuvering in waves and could serve as a highly credible alternative to existing methods.

A time two-grid algorithm for two-dimensional nonlinear time-fractional partial integro-differential equations

In this paper, a temporal second order two-grid difference scheme is proposed for the two-dimensional nonlinear time-fractional partial integro-differential equations with a weakly singular kernel. The first-order backward difference and L1 formula are used in the temporal direction to estimate the first level of time, the L2−1σ formula and L1-type formula are used in the temporal direction for later time steps, and the central difference formula is used in the spatial directions. To improve the computational efficiency of nonlinear system, an efficient time two-grid algorithm is proposed. This algorithm firstly solves a nonlinear system on the coarse grid, and then the Lagrangian linear interpolation is applied on the coarse grid to estimate the function values on the fine grid. The stability and convergence of the two-grid difference scheme are analyzed by the energy method. The convergence order of the two-grid difference scheme is O(τF2+τC4+hx2+hy2), where τF and τC are the time step sizes of fine grid and coarse grid respectively, while hx and hy are the space step sizes. Numerical experiments show that the accuracy of the theoretical analysis and the efficiency of the two-grid algorithm.

Improved matrix model of sequence grid partition based on vector space sampling

In information technology and data science predictive analysis work, the goal is to infer guess values as accurately as possible. A more densely spaced grid of points should be set as a tool for operating measurement models. It has a wide range of applications in psychology, education and other social science research fields. Although the grid seems to meet the requirements of further improving the accuracy of reasoning and guessing from a certain point of view, it also means complex calculations. When multiple false values fall between another grid point, the traditional Data Oriented Architecture fails. To guarantee the accuracy and efficiency of the calculation, it is necessary to solve the problem of excessive noise in the prediction of samples in meta-learning to solve the uncertainty caused by the noise database data or the assumption of the matrix model. This paper proposes a grid partition movable inference least squares method, that is, the sparse Bayesian least squares method based on the grid minimum matrix. The method is used for solving the problem that the prediction noise of the sample in the meta-learning is too large, to solve the uncertainty brought by the generated noise database data or the array matrix model assumption. According to the Symplectic Bayesian learning of the sequence matrix model, the basic parameters for processing the input sample database data are determined. The sparse Bayesian algorithm solution is used on the grid, combined with the off-grid inference guess of the sequence matrix model and the grid division. We set a coarse grid of points around the grid of false values. The rough point grid division is set, and the grid division around the false value is divided in detail. Then select more appropriate meta-learning parameters to clean up data. In the experiment analysis, we take the small sample study as the scene and carry on the concrete analysis to the experiment. The test proof in the fitting degree of the algorithm adopted in the paper surpasses the Support Vector Machine (SVM) and the k-Nearest Neighbor (KNN) algorithm by 3.5% and 6.4% respectively. The convergence effect surpasses the contrast plan above 10 and has a bigger superiority in the inference factor thrust. It proves that the optimization method in this paper has a strong promotion effect for the application of data forecasting systems in various industries, and has theoretical value as well as practical significance.

Modeling of flow and transport in multiscale digital rocks aided by grid coarsening of microporous domains

Many subsurface porous media such as soils, carbonate rocks, and mudstones possess multiscale porous structures that play an important role in regulating fluid flow and transport therein. A pore-network-continuum hybrid model is promising for numerical studies of a multiscale digital rock. It is, however, still prohibitive to the REV-size modeling because tens of millions of microporosity voxels may exist. In this work, we develop a novel and robust algorithm for coarsening microporosity voxels of a multiscale digital rock. Then, we combine coarsened microporosity grids with the pore network of resolved macropores to form efficient computational meshes. Furthermore, a pore-network-continuum simulator is developed to simulate flow and transport in both a synthesized multiscale digital rock and a realistic Estaillades carbonate rock. We show that the coarsening algorithm can reduce computational grids by about 90%, which substantially reduces computational costs. Meanwhile, coarsening microporosity has a minor impact on the predictions of absolute permeability, gas production curves, and breakthrough curves of solute transport. We illustrate the mechanisms of flow and transport in multiscale porous media induced by microporosity. Finally, the efficient hybrid model is used to predict the absolute permeability of an Estaillades digital rock. The numerical prediction matches well with the reported experimental data. We highlight the importance of characterizing mean pore-size distributions in microporosity for the prediction of rock permeability and local flow fields. The developed pore-network-continuum hybrid model aided by grid coarsening of microporosity serves as a useful numerical tool to study flow and transport in multiscale porous media.

Enforcing accurate volume conservation in VOF‐based long‐term simulations of turbulent bubble‐laden flows on coarse grids

AbstractThis study proposes two different strategies to enforce accurate volume conservation in volume‐of‐fluid (VOF)‐based simulations of turbulent bubble‐laden flows on coarse grids. It is demonstrated that, without a correction, minimal volume errors on a time‐step level, caused by the under‐resolution of the interface, can accumulate to significant deviations from the intended flow conditions despite the comparably good volume conservation properties of the geometric VOF method. In particular, large volume errors are observed for challenging setups combining coarse grid resolutions and comparably high Reynolds and Eötvös numbers. The problem is reinforced for long‐term simulations in periodic domains, which are often performed to collect flow statistics of bubbly flows. The first proposed volume conservation method simply corrects the volume error of a bubble by uniformly adding or removing the respective amount of gas volume in the interface cells. The second proposed method performs an additional reconstruction and advection step of the VOF field using a non‐divergence‐free velocity field, which can be interpreted as a slight dilatation or contraction of the bubble. A comparison between the global flow statistics as well as the individual bubble dynamics for both volume conservation methods reveals that the results are quasi‐identical for a number of challenging test cases, while the gas volume is accurately conserved. The proposed methods allow to perform numerical simulations of freely deformable bubbles in turbulent flows for setups that have previously been out of reach for this numerical framework.

Hybrid RANS/LES modeling of hypersonic turbulent boundary layers with cold walls

Various strategies for hybrid RANS/LES modeling of hypersonic turbulent boundary layers subjected to strong wall-cooling are developed and analyzed. By defining a turbulent thermal length scale, the eddy viscosity and diffusivity of the hybrid model are calculated by the same formula, ℓ2ω. This allows a similar blending of the length scales and the development of analogous dynamic models for turbulent stress and heat flux. The dynamic procedure is applied to heat transfer by equating norms, rather than invoking least-squares with a lower bound, which improves robustness. The shielding function, as used in detached eddy simulations models, is revised to prevent a spurious early switch from the RANS to the LES mode. This revision significantly improves the accuracy of hybrid models on coarse grids. Finally, the proposed models are tested and validated for Mach numbers between 2.5 and 14, and wall-to-recovery temperature ratios between 0.18 and 1.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

Fast deconvolved beamforming For arbitrary arrays based on beam-domain sparse Bayesian learning

Aiming at the problem that the conventional deconvolved beamforming methods cannot be directly applied to the specific array with a shift-variant point spread function and also have considerable computational workload, a deconvolved beamforming method for arbitrary arrays based on beam-domain sparse Bayesian learning(SBL) is proposed. First, generalized convolution model for arbitrary arrays in beam-domain is derived. Conventional beamforming is used to obtain several complex output beams. Then, to improve the accuracy of direction of arrival(DOA) estimation, the off-grid SBL method which adopts a coarse grid and takes the sampled positions in the coarse grid as the adjustable parameters is applied to achieve deconvolution of complex output beams. Controlling the number of output beams from the conventional beamforming can accelerate the off-grid SBL method while maintaining reasonable accuracy. The simulation results show that the proposed method provides enhanced recovery performance and higher DOA estimation accuracy comparable to the traditional SBL beamforming method in element domain at the same grid interval. Especially for short and dense arrays, it can achieve a decrease in computational complexity by one to two orders of magnitude with the same accuracy.

An efficient two-grid high-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

QLB-NET: A Dense Soil Moisture and Freeze–Thaw Monitoring Network in the Qinghai Lake Basin on the Qinghai–Tibetan Plateau

Abstract Soil moisture (SM) and soil freeze–thaw (FT) are two relatively active surface parameters that are significant to the sustainable development of the water–land–air–plant–human nexus. Over time, regional or global SM and FT datasets with different spatial resolutions have been developed. In response to the requirements of multiscale product validation and multisource uncertainty tracking, a soil moisture and soil temperature (ST) monitoring network in the Qinghai Lake Basin (QLB-NET) was established in September 2019. The QLB-NET is characterized by densely distributed in situ sites (82 sites) measuring SM and ST at 5-, 10-, and 30-cm depths, with 60 sites in a large-scale network covering an area of 36 km × 40 km and 22 sites evenly distributed across two small-scale 1 km × 1 km networks. Quantitative analyses of the in situ measurements show that the QLB-NET can provide stable and reliable ground truth for SM and FT over coarse grid scales, e.g., 36 km × 36 km, 25 km × 25 km, and 0.25° × 0.25°. When statistics are correspondingly performed over 50 out of 54, 25 out of 29, and 25 out of 28 sites, the results are described as follows: 1) the standard deviation of the mean SM varies between 0.0127 and 0.0196 m3 m−3, with the corresponding difference between the upper and lower quartiles being less than 0.02 m3 m−3; 2) the ground freeze–thaw state can be correctly identified with high probabilities ranging from 85.3% to 100% on two freeze–thaw transitional dates. The QLB-NET observed datasets are distributed online and will be continuously updated through cooperation with the National Tibetan Plateau Data Center (http://data.tpdc.ac.cn), facilitating product validation and uncertainty tracking, spatiotemporal analysis of SM change and FT transition, optimization of the SM and FT retrieving algorithms and scaling methods, and development of the mountainous microwave radiative transfer model.

Estimating multivariate ecological variables at high spatial resolution using a cost‐effective matching algorithm

AbstractSimulation models are valuable tools for estimating ecosystem response to environmental conditions and are particularly relevant for investigating climate change impacts. However, because of high computational requirements, models are often applied over a coarse grid of points or for representative locations. Spatial interpolation of model output can be necessary to guide decision‐making, yet interpolation is not straightforward because the interpolated values must maintain the covariance structure among variables. We present methods for two key steps for utilizing limited simulations to generate detailed maps of multivariate and time series output. First, we present a method to select an optimal set of simulation sites that maximize the area represented for a given number of sites. Then, we introduce a multivariate matching approach to interpolate simulation results to detailed maps for the represented area. This approach links simulation output to environmentally analogous matched sites according to user‐defined criteria. We demonstrate the methods with case studies using output from (1) an individual‐based plant simulation model to illustrate site selection, and (2) an ecosystem water balance simulation model to illustrate interpolation. For the site selection case study, we identified 200 simulation sites that represented 96% of a large study area (1.12 × 106 km2) at a ~1‐km resolution. For the interpolation case study, we generated ~1‐km resolution maps across 4.38 × 106 km2 of drylands in North America from a 10 × 10 km grid of simulated sites. Estimates of interpolation errors using cross validation were low (<10% of the range of each variable). Our point selection and interpolation methods, which are available as an easy‐to‐use R package, provide a means of cost‐effectively generating detailed maps of expensive, complex simulation output (e.g., multivariate and time series) at scales relevant for local conservation planning. Our methods are flexible and allow the user to identify relevant matching criteria to balance interpolation uncertainty with areal coverage to enhance inference and decision‐making at management‐relevant scales across large areas.

A Two-Grid Algorithm of the Finite Element Method for the Two-Dimensional Time-Dependent Schrödinger Equation

In this paper, we construct a new two-grid algorithm of the finite element method for the Schrödinger equation in backward Euler and Crank–Nicolson fully discrete schemes. On the coarser grid, we solve coupled real and imaginary parts of the Schrödinger equation. On the fine grid, real and imaginary parts of the Schrödinger equation are decoupled, and we solve the elliptic equation about real and imaginary parts, respectively. Then, we obtain error estimates of the exact solution with the two-grid solution in the H1-norm and carry out two numerical experiments.

Suitability of ERA5-Land reanalysis dataset for hydrological modelling in the Alpine region

Study region: The region of interest is the South-Tyrol in the southeastern Alps, Italy. A comparison of meteorological forcing is performed with reference to this region while hydrological simulations are conducted in the Passirio river basin.Study focus: The objective of this work is to evaluate the suitability of ERA5-Land reanalysis product as a reference dataset for hydrological modelling in topographically complex Alpine regions. ERA5-Land is compared with observational gridded dataset, obtained by means of the Kriging interpolation, available at two contrasting spatial resolutions a coarse grid size of 11 × 8 kilometres and a high-resolution grid size 1 × 1 kilometres. The hydrological coherence of the three datasets (ERA5-Land and the two observational datasets) with observed streamflow time series is then assessed with refer the Passirio/Passer river basin and to its nested and snow-dominated sub-catchment Plan/Pfelders.New hydrological insights for the region: The study assesses ERA5-Land’s suitability for hydrological simulations in the South-Tyrol region by comparing streamflow outcomes. At the annual scale, an overestimation of precipitation of about 40mm/month and a temperature underestimation of about 2.1 °C is reported. These discrepancies influence hydrological simulations, resulting in an overestimation of both streamflow and snow water equivalent. It is thus crucial to consider the limitations of ERA5-Land when using it as forcing for hydrological application in watersheds characterized by complex terrains, such as Alpine region.

A filtering approach for applying the two-fluid model to gas-liquid flows on high resolution grids

The two-fluid model is usually combined with closure forces designed for applications on coarse grids, i.e. bubbles (or particles) are typically assumed to be smaller than a grid cell. Practical applications however include situations where the mesh is comparatively fine, e.g. when meshing the wall boundary layer or in cases with growing bubbles. This may lead to non-convergent behaviour in mesh studies or to void fraction oscillations. To tackle this problem, a filtering approach is proposed, based on an additional diffusion term in the continuity equation. This approach increases the robustness of the results in regions of high spatial resolution, significantly reducing mesh-dependency of the simulation results. The implementation is straightforward, without a need to solve any additional system of equations. It is analysed in four different bubbly flow cases with varying characteristics: 2D/3D, wedge, square, and cuboid computational domains, with resolutions up to 32 cells per bubble diameter, laminar and turbulent flows, and several ways of gas injection. The additional computational effort varies, but is moderate. The proposed approach is applicable in multi-field two-fluid models for which a stable Euler-Euler behaviour on fine meshes is required, for example to prepare the transfer to an interface-resolving volume-of-fluid representation in morphology-adaptive approaches.

TWO-GRID VIRTUAL ELEMENT DISCRETIZATION OF QUASILINEAR ELLIPTIC PROBLEM

In this paper a two grid algorithm for quasilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a quasilinear elliptic problem on a fine grid is reduced to the solution of a quasilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimate in H1 norm is derived. Numerical experiments are carried out to illustrate the theoretical findings.

An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows

AbstractNumerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method.

Development of a Fourier‐expansion based differential quadrature method with lattice Boltzmann flux solvers: Application to incompressible isothermal and thermal flows

AbstractThis paper presents a high‐order Fourier‐expansion based differential quadrature method with isothermal and thermal lattice Boltzmann flux solvers (LBFS‐FDQ and TLBFS‐FDQ) for simulating incompressible flows. The numerical solution in the present method is approximated via trigonometric basis. Therefore, both periodic and non‐periodic boundary conditions can be handled straightforwardly without the special treatments as required by polynomial‐based differential quadrature methods. The incorporation of LBFS/TLBFS enables the present methods to efficiently simulated various types of flow problems on considerably coarse grids with spectral accuracy. The high‐order accuracy, efficiency and competitiveness of the proposed method are comprehensively demonstrated through a wide selection of isothermal and thermal flow benchmarks.

Development of 3D flow diagnostic numerical simulator and rock property modelling for a homogeneous faulted reservoir

Research has shown that faults, especially sealing faults, have become a barrier to flow. Flow diagnostics, on the other hand, are straightforward and controlled numerical flow experiments that are performed to examine a reservoir model, build links and offer rudimentary volume estimates as well as to quickly produce a qualitative representation of the reservoir’s flow patterns. To evaluate, assess, and validate reservoir models and production scenarios, the ’diagnostics’ module in the Matlab Reservoir Simulation Toolbox (MRST) provides a computationally less expensive alternative to running fully featured multiphase simulations. This is done to determine qualitatively regions for well placement to maximize hydrocarbon recovery. Monitoring flow fields, timelines, and tracers—neutral particles in the fluid that flow with the fluid—are all part of flow diagnostics. These techniques are used to determine volumetric communication between a pair of injector-producer wells, as well as flow patterns between injection-production wells and the arrival time between them. To create the grid, the simulation schedule, the wells, upscaling, arrival time computation, and flow diagnostics (drainage and sweep regions) were all done using the Matlab reservoir simulation toolbox (MRST) package by “Makemodel 3” on the data obtained from “SPE 10th comparative solution project” on the MRST package. This was done for a homogeneous anisotropic faulted porous medium. The simulation runs for about 473,739 milliseconds on a fine grid having 20040 active cells while in the coarse model with faster computational time of about 200 seconds, 112 milliseconds. At the early stages of the reservoir bottom-hole pressure for injector I1(about 7000psia) and three producers P1, P2 and P3 remains constant over time (about 3600psia) thanks to pressure support by I1. The extent of residence time after 10 years for the fine, coarse and more coarser model is that fluid element find it easier to transverse through the fault line from I1 to P1, P2 then P3 which takes lesser time than crossing across two intersecting faults, It can also be seen that the sweep efficiency of the fine grid is more effective than the coarse grid. Finally, it has been seen clearly that the fine grid model gives more accurate details but slower computational time while the coarse model which is less accurate but with faster computational time.

A physically constrained Monte Carlo-Neural Network coupling algorithm for BNCT dose calculation.

In boron neutron capture therapy (BNCT)-a form of binary radiotherapy-the primary challenge in treatment planning systems for dose calculations arises from the time-consuming nature of the Monte Carlo (MC) method. Recent progress, including the use of neural networks (NN), has been made to accelerate BNCT dose calculations. However, this approach may result in significant dose errors in both the tumor and the skin, with the latter being a critical organ in BNCT. Furthermore, owing to the lack of physical processes in purely NN-based approaches, their reliability for clinical dose calculations in BNCT is questionable. In this study, a physically constrained MC-NN (PCMC-NN) coupling algorithm is proposed to achieve fast and accurate computation of the BNCT three-dimensional (3D) therapeutic dose distribution. This approach synergizes the high precision of the MC method with the speed of the NN and utilizes physical conservation laws to constrain the coupling process. It addresses the time-consuming issue of the traditional MC method while reducing dose errors. Clinical data were collected from 113 glioblastoma patients. For each patient, the 3D dose distributions for both the coarse and detailed dose grids were calculated using the MC code PHITS. Among these patients, the data from 14 patients were allocated to the test set, 9 to the validation set, and the remaining to the training set. A neural network, 3D-Unet, was built based on the coarse grid dose and patient CT information to enable fast and accurate computation of the 3D detailed grid dose distribution of BNCT. Statistical evaluations, including relative deviation, dose deviation, mean absolute error (MAE), and mean absolute percentage error (MAPE) were conducted. Our findings suggested that the PCMC-NN algorithm substantially outperformed the traditional NN and interpolation methods. Furthermore, the proposed algorithm significantly reduced errors, particularly in the skin and GTV, and improved computational accuracy (hereinafter referred to simply as 'accuracy') with a MAPE range of 1.6%-4.0% and a maximum MAE of 0.3Gy (IsoE) for different organs. The dose-volume histograms generated by the PCMC-NN aligned well with those obtained from the MC method, further validating its accuracy. The PCMC-NN algorithm enhanced the speed and accuracy of BNCT dose calculations by combining the MC method with the NN algorithm. This indicates the significant potential of the proposed algorithm for clinical applications in optimizing treatment planning.

Variational calculus in hybrid turbulence transport models with passive scalar

Non-zonal methods of hybrid Reynolds Averaged Navier–Stokes (RANS) equations-Large Eddy Simulation (LES) with subfilter transport equations enable to simulate turbulent flows out of spectral equilibrium on relatively coarse grid resolution. They can operate from RANS to LES depending on a control parameter linked to the grid step size. Variational analysis gives a useful framework to specify the functional dependence of this parameter to the grid size. This is the case for the partially integrated transport modelling method originally established in spectral space for homogeneous turbulence. The partitioning control function then monitors the ratio of the subfilter part to the total turbulent energy and the same process is developed in the present work for the thermal or transported scalar variance as well. So, we demonstrate here that this control function mechanism can be derived by variational calculus from a mathematical physics formalism developed both for first-order and second-order moment closures. We show that the result is entirely consistent with the spectral model derivation made in previous papers and that this approach can be transposed to almost any subfilter transport model developed in hybrid RANS-LES methodology in full generality. As a result, it will be evidenced also that resolved turbulent diffusion terms play a significant role in the acting mechanisms of turbulence and cannot be therefore neglected, even if these terms do not explicitly appear in LES subfilter closure because they are computed by the simulation itself and not modelled by the subfilter model. In addition, numerical simulations have been performed for illustrating the theoretical results of the variational analysis.