Sub-grid Corrections Research Articles

Subgrid modeling for compound flooding in coastal systems

ABSTRACT Compound flooding, the concurrence of multiple flooding mechanisms such as storm surge, heavy rainfall, and riverine flooding, poses a significant threat to coastal communities. To mitigate the impacts of compound flooding, forecasts must represent the variability of flooding drivers over a wide range of spatial scales while remaining timely. One approach to develop these forecasts is through subgrid corrections, which utilize information at smaller scales to “correct” water levels and current velocities averaged over the model scale. Recent studies have shown that subgrid models can improve both accuracy and efficiency; however, existing models are not able to account for the dynamic interactions of hydrologic and hydrodynamic drivers and their contributions to flooding along the smallest flow pathways when using a coarse resolution. Here, we have developed a solver called CoaSToRM (Coastal Subgrid Topography Research Model) with subgrid corrections to compute compound flooding in coastal systems resulting from fluvial, pluvial, tidal, and wind-driven processes. A key contribution is the model’s ability to enforce all flood drivers and use the subgrid corrections to improve the accuracy of the coarse-resolution simulation. The model is validated for Hurricane Eta 2020 in Tampa Bay, showing improved prediction accuracy with subgrid corrections at 42 locations. Subgrid models with coarse resolutions (R2 = 0.70, 0.73, 0.77 for 3-, 1.5-, 0.75-km grids) outperform standard counterparts (R2 = 0.03, 0.14, 0.26). A 3-km subgrid simulation runs roughly 50 times faster than a 0.75-km subgrid simulation, with similar accuracy.

Numerical extensions to incorporate subgrid corrections in an established storm surge model

ABSTRACT Inundation models represent coastal regions with a grid of computational points, often with varying resolution of flow pathways and barriers. Models based on coarse grid solutions of shallow water equations have been improved recently via the use of subgrid corrections, which account for information (ground surface elevations, roughness characteristics) at smaller scales. In this work, numerical approaches of an established storm surge model are extended to include subgrid corrections. In an attempt to maintain continuity with existing users and results, model extensions were limited to those needed to provide basic subgrid capabilities, and included two major additions. First, a finite volume method is used to incorporate corrections to the mass and momentum equations using high-resolution ground surface elevations. Second, the no-slip condition imposed on the B-grid wet/dry interface in the model is modified to a slip condition to enable flows in channels with widths comparable to cell size. Numerical results demonstrate these numerical extensions can significantly enhance the accuracy of the model’s predictions of coastal flooding, with low additional computational cost.

Frame-invariant sub-grid corrections to the Fictitious Domain Method for the simulation of particulate suspensions in nonlinear flows using OpenFOAM

A particle-scale simulation method for non-Brownian suspensions at low Reynolds numbers is presented, which is well suited to nonlinear flows. It is based on a Fictitious Domain Method supplemented by sub-grid corrections (Gallier et al., 2014 [16]). In the original method, the standard sub-grid corrections involve the underlying suspension linear flow. The conditions required to remove this ambient flow from the sub-grid corrections, while keeping frame indifference, are determined. A sub-grid correction matrix is built for the particle-particle and particle-wall hydrodynamic interactions, such that the mentioned conditions inherently hold. The method is validated against various flows that involve a few particles or a concentrated suspension in linear and nonlinear flows, with various boundary conditions.

Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds

Abstract The simulation of the elastodynamics equations at high frequency suffers from the well-known pollution effect. We present a Petrov–Galerkin multiscale sub-grid correction method that remains pollution-free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the log ⁡ ( k ) \log(k) , 𝑘 being the wave number. Key to this method are polynomial-in-𝑘 bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-𝑘 bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in R 3 \mathbb{R}^{3} . Previous methods relied on variational techniques, Rellich identities, and geometric constraints. In the context of elastodynamics, these suffer from the need to hypothesize a Korn’s inequality on the boundary. The methods in this work are based on boundary integral operators and estimation of Green’s function’s derivatives dependence on 𝑘 and do not require this extra hypothesis. We also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.

Subgrid corrections in finite-element modeling of storm-driven coastal flooding

Coastal flooding models are used to predict the timing and magnitude of inundation during storms, both for real-time forecasting and long-term design. However, there is a need for faster flooding predictions that also represent flow pathways and barriers at the scales of critical infrastructure. This need can be addressed via subgrid corrections, which use information at smaller scales to ‘correct’ the flow variables (water levels, current velocities) averaged over the mesh scale. Recent studies have shown a decrease in run time by 1 to 2 orders of magnitude, with the ability to decrease further if the model time step is also increased.In this study, subgrid corrections are added to a widely used, finite-element-based, shallow water model to better understand how they can improve the accuracy and efficiency of inundation predictions. The performance of the model, with and without subgrid corrections, is evaluated on scenarios of tidal flooding in a synthetic domain and a small bay in Massachusetts, as well as a scenario with a real atmospheric forcing and storm surge in southwest Louisiana. In these tests we observed that the subgrid corrections can increase model speed by 10 to 50 times, while still representing flow through channels below the mesh scale to inland locations.

Verification and validation of the DDPM-EMMS model for numerical simulations of bubbling, turbulent and circulating fluidized beds

Compared with the traditional Eulerian-Eulerian two-fluid modeling (TFM) approach, dense discrete phase model (DDPM) is potentially promising for industrial-scale applications of fluidized bed reactors, due to its capability of tracking the trajectories of particle parcels and resolving the particle size distributions (PSD). However, DDPM is yet at its early stage of mature implementation. Hence, verification and validation of the DDPM approach is still required for future applications of different gas-solid fluidization regimes.In the present work, verification and validation of the DDPM is systematically investigated for the first time, with regards to bubbling, turbulent and circulating fluidized beds. The sensitivity of the drag force, a key modeling parameter in gas-solid fluidized bed reactors is investigated with two different drag models i.e., Gidaspow and EMMS/bubbling. Numerical results of bubbling, turbulent and circulating fluidization regimes are presented in terms of discrete particle distributions, axial and radial solid concentrations and radial solid velocity profiles distributions. The results indicate that the sub-grid correction based on the EMMS approach is essential for the DDPM to correctly account for the effects of dissipative structures in all three fluidization regimes. While, the DDPM with the conventional Gidaspow drag failed to do so. Further, validations of the numerical results with the experimental data also prove that the DDPM approach with EMMS/bubbling drag model can resolve the time-averaged axial and radial solid concentrations and radial solid velocity profiles better than the conventional Gidaspow drag model.

Subgrid theory for storm surge modeling

Averaging techniques are used to generate upscaled forms of the shallow water equations for storm surge including subgrid corrections. These systems are structurally similar to the standard shallow water equations but have additional terms related to integral properties of the fine-scale bathymetry, topography, and flow. As the system only operates with coarse-scale variables (such as averaged fluid velocity) relating to flow, these fine-scale integrals require closures to relate them to the coarsened variables. Closures with different levels of complexity are identified and tested for accuracy against high resolution solutions of the standard shallow water equations. Results show that, for coarse grids in complex geometries, inclusion of subgrid closure terms greatly improves model accuracy when compared to standard solutions, and will thereby enable new classes of storm surge models.

An effective three-marker drag model via sub-grid modeling for turbulent fluidization

The effect of meso-scale structures on hydrodynamic predictions is not considered in the classically uniform drag models when the coarse grid is used. To address this issue, this study tries to develop an effective three-marker drag correlation via straightforward sub-grid modeling, which accounts for a parabolic spatial concentration distribution within a computational grid. The reliability and accuracy of the developed model is then assessed in detail. How the uniform drag inputs affect the derived sub-grid correction is quantified for the first time. Besides, a comprehensive comparison between several typical sub-grid models and present work is implemented. Results reveal a systematic dependence of our drag modification on the concentration gradient as an additional marker. Coarse-grid hydrodynamic validation shows that the developed model yields a fairly improved agreement with experiments under various operating conditions in a 3D turbulent fluidized bed. Furthermore, results demonstrate that the present model using different uniform drag inputs still can exhibit satisfactory performance. The developed model is able to resolve the heterogeneous flow behavior both cheaply and adequately, which is potentially applied for industrial reactor design and optimization.

Numerical Homogenization of Heterogeneous Fractional Laplacians

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form; however, the Caffarelli--Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a subgrid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is our main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ di...

Filtered sub-grid constitutive models for fluidized gas-particle flows constructed from 3-D simulations

The accuracy of fluidized-bed CFD predictions using the two-fluid model can be improved significantly, even when using coarse grids, by replacing the microscopic kinetic-theory-based closures with coarse-grained constitutive models. These coarse-grained constitutive relationships, called filtered models, account for the unresolved gas-particle structures (clusters and bubbles) via sub-grid corrections. Following the previous 2-D approaches of Igci et al. [AIChE J., 54(6), 1431–1448, 2008] and Milioli et al. [AIChE J., 59(9), 3265–3275, 2013], new closures for the filtered inter-phase drag and stresses in the gas and particle phases are constructed from highly-resolved 3-D simulations of gas-particle flows. These new closure relations are then validated through the bubbling-fluidized-bed challenge problem presented by National Energy Technology Laboratory and Particulate Solids Research Inc.

CFD modeling and simulation of industrial scale olefin polymerization fluidized bed reactors

A two-fluid model for the numerical simulation of industrial scale olefin polymerization fluidized bed reactors is presented. However, a fully resolved simulation of industrial scale reactor is still nearly unfeasible. We, therefore, use sub-grid models (Schneiderbauer and Pirker, 2014) for the interphase drag and the solids stresses to account for the effect of the small unresolved structures on large resolved scales when using coarse grids. The sub-grid correction for the drag force is modified to consider the wide particle size distribution of high density polyethylene (HDPE). Furthermore, the sub-grid modification for the solids stresses is adapted to include the rheological properties of the polymer. On the one hand, the presented model is validated in the case of the coarse grid simulation of lab-scale bubbling fluidized bed by comparing bed expansion, bubble size and bubble rise velocities with experimental data. On the other hand, the model is applied to the coarse grid simulation of an industrial scale fluidized bed – moving bed reactor assembly. The numerical results demonstrate that our model reveals fairly good agreement with experimental data of average bed voidage, bubble diameters and bubble rise velocities. Finally, the impact of a barrier gas injection is studied, which is aimed to separate the fluidization gas from the gas in the moving bed reactor.

Statistical properties of an ideal subgrid-scale correction for Lagrangian particle tracking in turbulent channel flow

One issue associated with the use of Large-Eddy Simulation (LES) to investigate the dispersion of small inertial particles in turbulent flows is the accuracy with which particle statistics and concentration can be reproduced. The motion of particles in LES fields may differ significantly from that observed in experiments or direct numerical simulation (DNS) because the force acting on the particles is not accurately estimated, due to the availability of the only filtered fluid velocity, and because errors accumulate in time leading to a progressive divergence of the trajectories. This may lead to different degrees of inaccuracy in the prediction of statistics and concentration. We identify herein an ideal subgrid correction of the a-priori LES fluid velocity seen by the particles in turbulent channel flow. This correction is computed by imposing that the trajectories of individual particles moving in filtered DNS fields exactly coincide with the particle trajectories in a DNS. In this way the errors introduced by filtering into the particle motion equations can be singled out and analyzed separately from those due to the progressive divergence of the trajectories. The subgrid correction term, and therefore the filtering error, is characterized in the present paper in terms of statistical moments. The effects of the particle inertia and of the filter type and width on the properties of the correction term are investigated.

A note on variational multiscale methods for high-contrast heterogeneous porous media flows with rough source terms

In this short note, we discuss variational multiscale methods for solving porous media flows in high-contrast heterogeneous media with rough source terms. Our objective is to separate, as much as possible, subgrid effects induced by the media properties from those due to heterogeneous source terms. For this reason, enriched coarse spaces designed for high-contrast multiscale problems are used to represent the effects of heterogeneities of the media. Furthermore, rough source terms are captured via auxiliary correction equations that appear in the formulation of variational multiscale methods [23]. These auxiliary equations are localized and one can use additive or multiplicative constructions for the subgrid corrections as discussed in the current paper. Our preliminary numerical results show that one can capture the effects due to both spatial heterogeneities in the coefficients (such as permeability field) and source terms (e.g., due to singular well terms) in one iteration. We test the cases for both smooth source terms and rough source terms and show that with the multiplicative correction, the numerical approximations are more accurate compared to the additive correction.

Coarse-Grid Simulation of Gas-Particle Flows in Vertical Risers

Continuum model equations for unsteady gas-particle flows in devices such as fluidized beds and circulating fluidized bed risers contain unstable modes whose length scale is of the order of 10 particle diameters. Yet, because of limited computational resources, these flows are routinely simulated by solving the discretized version of continuum models over coarse spatial grids. These simulations resolve the large-scale flow structures but not the finer scale structures. In most industrial applications involving large devices, it is impractical to resolve all the fine-scale structures, and therefore the effects of the unresolved structures must be addressed through suitable subgrid models. Using gas-particle flows in a wide and very tall vertical channel as an example, we have demonstrated in this study that the results obtained in coarse-grid integration of the microscopic equations for gas-particle flows change appreciably if subgrid corrections to account for the effects of unresolved structures are incl...