A grid nesting method for large-eddy simulation of planetary boundary-layer flows

We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.

Remote sensing technology and the resulting high resolution geospatial data now allow for a detailed description of urban landscape, advancing the development of raster-based flood models. Previous studies have highlighted the critical role of finely resolved and accurate terrain data (5m or less) in capturing flow patterns in urban areas. However, using a uniform fine grid resolution over a rectangular domain generally results in dense grids and leads to large computational costs. The small cell size is often an overspecification for rural regions where the flow processes are changing much less rapidly. Unstructured grid models resolve this issue and trade off more complex programming and slower operation against being able to represent a given problem with fewer computational elements. An alternative solution has been recently proposed to apply the non-uniform structured grid, with fine grids covering only the regions where this detail is a necessity, for example to capture the preferential flow paths influenced by small-scale topographic features or man-made structures (river channels, buildings, roads, defences, etc.). Without this, the smoothing effect of mesh coarsening upon input topographical data in urban areas leads to a uncertain prediction of the inundation extent and the timing of inundation due to the simplified wetting process. Flow connectivity formed by the river channels and the road network, which has a strong control on urban floodplain hydraulics, is also better represented by mixing grid resolutions. Considering the large consequences in terms of economic losses caused by urban flooding, here we develop a GPU-accelerated non-uniform sub-/super-grid channel model (river channels with width below or above the fine grid resolution) for accurate and efficient urban flood modelling. Urban areas and the river channel network are forced to keep fine resolution, while a coarse representation, depending on the terrain gradient, is allowed for rural regions. This model allows the utilization of available sub-/super-grid scale bathymetric information for 1D in-channel flow representation, and a 2D model for floodplain with variable grid resolution, minimising the computational costs and below water line data requirements in the river channel. Three tests are set up to validate the model performance, and the results show that modelling the urban area with fine resolution improved the model reliability and accuracy, and reduces computational cost in rural areas where a coarse grid may be used.

Flow fields in a centrifugal fan for an indoor unit of an air-conditioner were calculated with finite element method-based large eddy simulation (LES) with the aim of predicting fan performance and aerodynamic noise in this study. The numerical simulation code employed throughout the LES was called FrontFlow/blue (FFB). We compared 10M grid [coarse grid] and 60M grid [fine grid] calculation results for investigation of influence of grid resolution. In the fine grid, the number of grid elements in blade-to-blade direction, and of region between the shroud and the bell mouth increased in particular. By calculating with the fine grid, calculated distributions of absolute velocities at blade exit reasonably agreed with experimental results. Because of this, maximum absolute velocity by fine grid near hub decreased as compared to those by coarse grid. Calculated sound pressure level by fine grid was therefore smaller than that by coarse grid, and the overestimation of sound pressure was suppressed by calculating with fine grid. This decrease of the absolute velocity was a first factor for the improvement of calculation accuracy. Moreover, number of captured streaks on the blade, hub, and shroud surfaces by fine grid increased as compared to those by coarse grid. As a result, size of streak by fine grid became smaller than that by coarse grid. Static pressure fluctuations by fine grid on the blade, hub, and shroud surfaces therefore reduced as compared to those by coarse grid. Aerodynamic noise was related to static pressure fluctuations according to Curle’s equation. This reduction of static pressure fluctuations was therefore a second factor for improvement of calculation accuracy.

An error arises in large eddy simulations when the order of spatial differentiation and filtering operations is commuted on a non-uniform grid. The magnitude of the commutation error is controlled by spatial variations in the width of the filter function, which is conventionally defined in terms of the local grid size. We have previously found large commutation errors in the sub-grid scale (SGS) terms of a numerical dynamo model, particularly near boundaries where the grid size changes most rapidly. In the present study, we propose a correction for the commutation error which is specifically designed for use with the nonlinear gradient SGS model. The commutation error correction for each SGS term can be expressed as the product of the spatial derivative of the second-order moment of the filter function and the second spatial derivative of the SGS term. We test the correction using output from a fully resolved convection-driven dynamo simulation in a rotating plane layer. As an example, we evaluate the SGS heat flux on a coarser grid with and without the commutation error correction. The result is tested using the resolved dynamo solution on a finer grid. We find that the commutation error correction is comparable in magnitude to the SGS heat flux near the boundary. Addition of the correction term significantly improves the modeled SGS heat flux near the boundaries.

Running a fine grid model with 107 - 109 of cells is possible using a supercomputer with 103 - 106 of CPUs but may not be always cost-effective. The most cost-effective way is to use a coarse grid model that is much smaller but with static/dynamic profiles very close to the fine grid model. This paper proposes a new layer optimization and upscaling method with the aim for creating a consistent coarse grid model. Unlike the industry's existing layer optimization and upscaling methods, the proposed method performs layer optimization and upscaling fully integrated with the Lorenz coefficient and curves (LCC). Coarse grid layers and their permeabilities are created by minimizing the difference between fine and coarse grid LCCs. The process consists of static and dynamic optimizations. The former is measured by LCC while the latter by pressure, GOR, and water-cut. A new LCC-based permeability upscaling method is developed to preserve the fine grid multiphase flow behaviors. A satisfactory coarse grid model is achieved when both static and dynamic criteria are met. The proposed method has been successfully applied to a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within and between the matrix and fracture/karst features. The reservoir drive mechanisms are fluid expansion, miscible gas injection and aquifer drive. The reservoir is undersaturated and has an abnormally high initial reservoir pressure. The fine-grid static model contains 104 million cells (370×225×625×2) and the optimized upscaled coarse-grid dynamic model has 8.3 million cells (370×225×50×2). The upscaled model can be run efficiently on the company's existing HPC infrastructure with a maximum of 64 CPUs. Excellent matches of the Lorenz coefficient maps for reservoir total/zones and Lorenz curves at all wells between the fine and coarse grid models have been achieved. Matches on the dynamic variables, e.g., pressure, gas breakthrough time, and GOR growth, in all producers are within the defined acceptable tolerances. The high quality of the static and dynamic matches between the coarse- and fine-grid models confirms that the reservoir properties of the coarse-grid model is very close to the fine-grid model and can be used a base model for history matching and uncertainty analysis.

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

This paper presents a multidomain (MD) computational approach to multiscale (MS) compositional reservoir simulation amenable to parallel processing. The main objective is to minimize the upscaling process and preserve the pertinent flow characteristics of the fine-scale geological features of the reservoir. The preservation of the fine-scale reservoir characteristics yields higher-accuracy solution compared to an upscaled model. The computing algorithm involves breaking down the fine-grid (FG) computing mesh into an underlying coarse-grid (CG) subdomain. Unlike other MS techniques, the CG subdomain only serves as an assemblage of the FG cells and not as a computational CG overlaying the FG cells. The flow equations are then restructured into a global pressure equation which is solved on the FG using a large time step without solving for the compositions, saturations, etc. The convergence tolerance for the global pressure solution is soft (no iterations) to reduce computing time. The pressure solution is then used to calculate the molar flux of components only at the FG boundaries of the CG subdomain cells. Finally, each CG subdomain subject to its boundary conditions is solved independently using a smaller time step to compute cell pressures, compositions, and phase saturations. This procedure is amenable to parallel processing. This MS scheme was applied to time domain, in which the global pressure solution is obtained with a larger time step than in the local FG solution. This means that the molar flux boundary condition at the CG subdomain boundaries is not updated every time step, thus, buying additional computational advantage. This new technique was tested on several volatile oil and gas-condensate systems and excellent agreement was obtained with FG simulation with about 60% reduction in computing time using a single processor. Much greater time reduction is expected in a multi processing environment.

The effects of subgrid scale (SGS) motions on the dispersion of heavy particles raise a challenge to the large-eddy method of simulation (LES). As a necessary first step, we propose the use of a stochastic differential equation (SDE) to represent the SGS contributions to the relative dispersions of heavy particles in LES of isotropic turbulence. The main difficulty is in closing the SGS-SDE model whilst accounting for the effects of particle inertia, filter width and gravity. The physics of the interaction between heavy particles and SGS turbulence is explored using the filtered direct numerical simulation method. It is found in the present work that (i) the ratio of the SGS Lagrangian and Eulerian timescales is different from that of the full-scale Lagrangian and Eulerian timescales. The ratios are also dependent on filter widths. (ii) In the absence of gravity, the SGS timescale seen by heavy particles non-monotonically changes with particle Stokes number and has a maximum at particle Stokes number (St = τp/δTE) near 0.5. (iii) In the presence of gravity, a similarity law exists between the SGS Lagrangian correlation function seen by a heavy particle within a time-delay τ and the SGS spatial correlation function with the displacement 〈w〉τ, where 〈w〉 is the average settling velocity of a heavy particle. The joint effects of particle inertia and gravity are accounted for using the elliptic model for pair correlation of SGS velocity seen by heavy particles. The SGS timescale seen by heavy particles is extracted from the elliptic model and used to close the SGS-SDE model. The validations of the model against direct numerical simulation show that the SGS-SDE model can improve the performance of LES on relative dispersions especially when their initial separations are in the inertial subrange. Furthermore, we assess the performance of the SGS-SDE model by comparing the results with the approximate deconvolution method. The results show that the SGS-SDE model is more suitable for particles with small Stokes numbers, StK < 2. The model developed here provides a basis for the development of a more advanced SGS model for particles in non-homogeneous and anisotropic turbulent flows in pipes or channels.

The absence of sub-grid scale (SGS) motions leads to severe errors in particle pair dynamics, which represents a great challenge to the large eddy simulation of particle-laden turbulent flow. In order to address this issue, data from direct numerical simulation (DNS) of homogenous isotropic turbulence coupled with Lagrangian particle tracking are used as a benchmark to evaluate the corresponding results of filtered DNS (FDNS). It is found that the filtering process in FDNS will lead to a non-monotonic variation of the particle collision statistics, including radial distribution function, radial relative velocity, and the collision kernel. The peak of radial distribution function shifts to the large-inertia region due to the lack of SGS motions, and the analysis of the local flowstructure characteristic variable at particle position indicates that the most effective interaction scale between particles and fluid eddies is increased in FDNS. Moreover, this scale shifting has an obvious effect on the odd-order moments of the probability density function of radial relative velocity, i.e. the skewness, which exhibits a strong correlation to the variance of radial distribution function in FDNS. As a whole, the radial distribution function, together with radial relative velocity, can compensate the SGS effects for the collision kernel in FDNS when the Stokes number based on the Kolmogorov time scale is greater than 3.0. However, it still leaves considerable errors for $${ St}_\mathrm{k }<3.0$$ .

Coastal regions are dynamic environments influenced by various atmospheric and oceanic processes. Understanding the complex interplay of these forces is crucial for coastal management, navigation, and impact assessments. This becomes especially crucial when investigating climatological scale dynamics.In this study, two state-of-the-art third-generation wave models, WAM-Cycle 6 (Wave Action Model) and SWAN 41.45 (Simulating Waves Nearshore &amp;#8211; Version 41.45) are utilized for numerical wave projection. The methodology involves nested modeling approach by configuring SWAN within WAM to simulate waves at finer resolution near the Irish coast. This setup involves WAM operating on a coarser grid (1.0 degree), an intermediate WAM grid with a resolution of 0.5 degree, and SWAN running on a finer grid (0.025 degree). Wind forcing (10 m wind speed) from climate reanalysis produced by ECMWF (ERA5) is used to drive the wave model. A comparison with wave buoys at different locations around Ireland showed that models agreed on the significant wave height with bias and RMSE differing at most 0.6m.Statistical techniques are utilized to connect the Max Planck Institute Earth System Model to ERA5 wind data, which is employed as forcing for the wave models to predict waves on climatological scales. The current model setup not only focuses on the present wave conditions but extends to provide future predictions and projections utilizing input data from climate models. By combining the insights from the present with the predictions and projections for the future, the current study aims to provide valuable information for decision-makers in the near and long-term future.

: To illustrate Brandt's multigrid method this report considers the Helmholtz equation in cases where the frequency is close to an eigenfrequency. In a second part the eigenvalue problem is discussed. The basic idea for the treatment of the inhomogeneous problem is contained in Brandt's publications. Difficulties are encountered because the coarse grid eigenvalues differ from those of the finer grid. As a consequence, approximations to a solution in the coarse grid may differ widely from the corresponding approximation in a finer grid, if the frequency defined by the inhomogeneous problem is close to one of the eigenfrequencies. This may be detrimental to the aim of the coarse grid step; namely to remove from an approximation in a fine grid long waves which cause divergence of fine grid iterations. This is remedied by eliminating long wave contributions to the residual directly in the next finer grid. The success of this procedure still depends upon the choice of the subspace from which the long wave corrections in the finer grid are taken. Even this modified method fails if the frequency pertaining to the inhomogeneous term is too close to an eigenvalue, unless the subspace contains the pertinent eigenfunction.

Convergence analysis of two-grid fixed stress split iterative scheme for coupled flow and deformation in heterogeneous poroelastic media

We investigated the influence of spatial aggregation on modeled forest responses to climate change by applying the process-based Terrestrial Ecosystem Model (TEM) to a fine resolution spatial grid (100 km2) and to a coarse resolution spatial grid (2500 km2). Three climate scenarios were simulated: baseline (present) climate with ambient CO2 and 2 future climates derived from the general circulation models OSU and GFDL-Q with elevated atmospheric CO2. For baseline climate, the aggregation error of the national (U.S.) study area was very small, –0.4%. Forest-level aggregation error ranged from –1.6 to 11.8%, with the largest aggregation error occurring in boreal forest types. Coarse grid resolution inputs underestimated production for boreal and forested boreal wetland forests and overestimated net primary production (NPP) for temperate conifer, temperate deciduous, and temperate forested wetland forests. Aggregation error for coarse grid cells ranged between –25.6 and 27.3%. Aggregation errors were especially large in transition regions between temperate and boreal forest types. An analysis that homogenized inputs for the 10 km grid cells within a 50 km grid indicated that aggregation of forest types and air temperature from fine to coarse grid cells contributed most to the spatial aggregation error. The aggregation error for the OSU climate was similar to the GFDL-Q climate and both results were similar to the aggregation error of the baseline climate in magnitude, sign, and spatial pattern. While aggregation error was similar across the baseline, GFDL-Q and OSU scenarios, NPP response to the GFDL-Q and OSU climates increased 13 to 30% above the baseline NPP. Within each climate scenario, the estimated NPP response to climate change differed by less than 1% between the coarse and fine resolutions. Except for transition regions and regions with substantial variability in air temperature, our simulations indicate that the use of 0.5° resolution provides an acceptable level of aggregation error at the 3 scales of analysis in this study. Improvements could be made by focusing computational intensity in heterogeneous regions and avoid computational intensity in regions that are relatively homogeneous with respect to vegetation and air temperature.

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.

Most natural landscapes are characterized by multiscale (often multifractal) topography with well-known scale-invariance properties. For example, the spectral density of landscape elevation fields is often found to have a power-law scaling behaviour (with a -2 slope on a log-log scale) over a wide span of spatial scales, typically ranging from tens of kilometres down to a few metres. Even though the effect of topography on the atmospheric boundary layer (ABL) has been the subject of numerous studies, few have focussed on multiscale topography. In this study, large-eddy simulation (LES) is used to investigate boundary-layer flow over multiscale topography, and guide the development of parametrizations needed to represent the effects of subgrid-scale (SGS) topography in numerical models of ABL flow. Particular emphasis is placed on the formulation of an effective roughness used to account for the increased aerodynamic roughness associated with SGS topography. The LES code uses the scale-dependent Lagrangian dynamic SGS model for the turbulent stresses and a terrain-following coordinate transformation to explicitly resolve the effects of the topography at scales larger than the LES resolution. The terrain used in the simulations is generated using a restricted solid-on-solid landscape evolution model, and it is characterized by a -2 slope of the elevation power spectrum. Results from simulations performed using elevation fields band-pass filtered at different spatial resolutions indicate a clear linear relation between the square of the effective roughness and the variance of elevation.