Coarse Grid Research Articles

Diagnosis of Second-Order Turbulent Properties of the Surface Layer for Three-Dimensional Flow Based on the Mellor–Yamada Model

Abstract Accurate representation of heterogeneous surface layer processes is essential for numerical weather prediction (NWP) with sub-kilometer grid spacing. NWP models such as the Weather Research and Forecasting (WRF) Model generally use second-moment turbulent models for parameterizing the planetary boundary layer (PBL). The most common parameterizations follow Mellor–Yamada and account for the vertical turbulent mixing only; that is, standard PBL parameterizations are one-dimensional (1DPBL). The horizontal diffusion of momentum is parameterized based on Smagorinsky’s model for numerical stability. Although the combination of 1DPBL and 2D Smagorinsky parameterizations is successful at coarse grid resolutions (e.g., grid-size dx ∼ 12–2 km), it does not represent well the effect of horizontal turbulence as gridcell size decreases (<1 km). To reconcile the representation of vertical and horizontal turbulent mixing, a full three-dimensional PBL scheme (3DPBL) based on the Mellor–Yamada model was implemented in WRF. The 3DPBL uses the horizontal and vertical turbulent fluxes diagnosed from the flow gradients to handle the turbulent mixing. These gradients cannot be directly calculated near the surface. Therefore, the 3DPBL parameterization is coupled herein to a second-order diagnostic model of the three-dimensional turbulent fluxes in the surface layer. Several adjustments to the original Mellor–Yamada model, including a modified length scale, were introduced to capture flow anisotropy and dependence on stability conditions. The results are compared against data from the Wind Forecast Improvement Project 2 (WFIP2) for different weather regimes and using different grid resolutions to examine stability and scale dependency.

Numerical investigations of the vortex feature-based vorticity confinement models for the assessment in three-dimensional vortex-dominated flows

In order to improve the vortex resolution in aerodynamic wakes, a locally normalized vortex feature-based vorticity confinement method is implemented into the multi-block, structured computational fluid dynamics solver (ROSITA). In this method, the second vorticity confinement (VC2) scheme with two well-known vortex feature detection methods (non-dimensional Q criterion, non-dimensional lambda _2 criterion) is employed to counterbalance the truncation error introduced by the numerical discretization of the convective term. The flow field of two benchmark three-dimensional steady vortex-dominated cases, the NACA0015 wing and the Caradonna–Tung hovering rotor, is simulated with the implemented method. The improvements in aerodynamics prediction, vorticity preservation, computational stability, and efficiency are demonstrated. From the numerical results, the vortex feature-based confinement models significantly improve the computational stability, the aerodynamic loads prediction and vorticity preservation capability, especially for the lambda _2–based VC2 model. In addition, it allows the use of higher confinement parameters on a coarse grid with a relatively higher computational efficiency to obtain better results than those of a finer grid.

Gridding Optimization for Hydraulic Fractured Well in Reservoir Simulation Using Well Test Analysis for Long Term Prediction

Analytic models, complex simulations, and simple models are being used to predict the production performance of hydraulically fractured shale. Analytical models such as decline curve analysis and rate transient analysis are used for a quick evaluation of reservoir performance. However, they have considerable limitations. For instance, decline curve analysis cannot honor the physical phenomena in shale wells that are related to hydraulic fracture, reservoir characteristics, and fluid flow. On the other hand, even though explicit hydraulic fracture modeling is the most comprehensive approach when compared with other traditional techniques, it cannot guarantee to model enough hydraulic fracture effects. Hence, calibration of the model, which commonly is referred to in the oil and gas industry as history matching, becomes a must. However, history matching of an explicit hydraulic fracture model with limited information is time-consuming and cumbersome. Especially history matching of a full field shale gas/oil model with many wells is a daunting task. In this study, we propose a workflow to integrate numerical reservoir simulation and well test analysis. In the workflow, information such as fracture half-length and enhanced effective permeability are obtained from pressure transient analysis and are used to calibrate grid properties in the vicinity of the plane covered by the fracture length and width. Finally, the simulation model is calibrated using pressure and flow rate data, and it is used for the long-term performance prediction of a hydraulic fractured well. The workflow was evaluated by using a synthetic reservoir model whose permeability mimicked that of a shale formation. As a result, the workflow thus enabled the use of coarse grid blocks, which, in turn, reduced the simulation time to just 1.5% of the simulation runtime consumed by a reference fine grid model.

New formulations and convergence analysis for reduced tracer mass fragmentation model: an application to depolymerization

In this work, two discrete formulations based on the finite volume approach for a reduced fragmentation model are developed. The important features such as mass conservation and accurate prediction of the zeroth order moments are accomplished by the modification of the selection function. The new schemes can compute the second order moment, which plays a significant role in predicting the area of the particles in real life applications, with high accuracy without taking any specific measures. A thorough convergence analysis of both schemes including Lipschitz condition and consistency is presented and exhibit second order convergence. The accuracy and efficiency of both schemes is demonstrated by applying them to the depolymerization problem which commonly arises in polymer sciences and chemical engineering. It is demonstrated that the new schemes are easy to implement, computationally efficient and able to compute the numerical results with higher precision even on a coarser grid.

Optimal Convergence Analysis of Two-Level Nonconforming Finite Element Iterative Methods for 2D/3D MHD Equations.

Several two-level iterative methods based on nonconforming finite element methods are applied for solving numerically the 2D/3D stationary incompressible MHD equations under different uniqueness conditions. These two-level algorithms are motivated by applying the m iterations on a coarse grid and correction once on a fine grid. A one-level Oseen iterative method on a fine mesh is further studied under a weak uniqueness condition. Moreover, the stability and error estimate are rigorously carried out, which prove that the proposed methods are stable and effective. Finally, some numerical examples corroborate the effectiveness of our theoretical analysis and the proposed methods.

심층학습 기반 초해상화 기법을 이용한 슬로싱 압력장 복원에 관한 연구

Deep-learning-based Super-Resolution (SR) methods were evaluated to reconstruct pressure fields with a high resolution from low-resolution images taken from a coarse grid simulation. In addition to a canonical SRCNN(super-resolution convolutional neural network) model, two modified models from SRCNN, adding an activation function (ReLU or Sigmoid function) to the output layer, were considered in the present study. High resolution images obtained by three models were more vivid and reliable qualitatively, compared with a conventional super-resolution method of bicubic interpolation. A quantitative comparison of statistical similarity showed that SRCNN model with Sigmoid function achieved best performance with less dependency on original resolution of input images.

Generalized multiscale finite element methods for the reduced model of Darcy flow in fractured porous media

In this work, we combine the generalized multiscale finite element method (GMsFEM) with a reduced model based on the discrete fracture model (DFM) to resolve the difficulties of simulating flow in fractured porous media while efficiently and accurately reducing the computational complexity resulting from resolving the fine scale effects of the fractures. The geometrical structure of the fractures is discretely resolved within the model using the DFM. The advantage of using GMsFEM is to represent the fracture effects on a coarse grid via multiscale basis functions constructed using local spectral problems. Solving local problems leads to consideration and usage of small scale information in each coarse grid. Besides, the multiscale basis functions, generated following GMsFEM framework, are parameter independent and constructed once in what we call offline stage. These basis functions can be re-used for solving the problem for any input parameter when it is needed. Combining GMsFEM and DFM has been introduced in other works assuming continuous pressure across the fractures interface. This continuity is obtained when the fractures are much more permeable than that in the matrix domain. In this work, we consider a general case for the permeability in both fracture and matrix domain using the reduced model presented in Martin et al. (2005). The proposed reduction technique has significant impact on enabling engineers and scientist to efficiently, accurately and inexpensively solve the large and complex system resulted from modeling flow in fractured porous media.

Two Methods to Improve the Efficiency of Supersonic Flow Simulation on Unstructured Grids

The paper presents two methods to improve the efficiency of supersonic flow simulation using arbitrarily shaped unstructured grids. The first method promotes increasing the numerical solution convergence rate and is based on the geometric multigrid method for initialization of the flow field. The method is used to obtain the initial field of distributed physical quantity values, which maximally corresponds to the converged solution. For this purpose, the problem simulation is performed on a series of coarse grids beginning from the coarsest one in this series. Upon completion of simulations, the solution obtained is interpolated to a finer grid and used for initialization of simulations on this grid. The second method allows increasing the numerical solution accuracy and is based on statically adapting the computational grid to the flow specifics. The static adaptation algorithm provides automatic refinement of the computational grid in the region of specific features of flow, such as shock waves typical for supersonic flows. This algorithm provides a better description of the shock-wave front owing to the local grid refinement, with the local refinement region being automatically selected. Results of using these methods are demonstrated for the two supersonic aerodynamics problems: the simulation of the bow shock strength at a given distance under axially symmetric body Seeb-ALR and a mock-up aircraft Lockheed Martin 1021. It is shown that in both cases, the numerical solution convergence rate is increased owing to the use of the geometric multigrid method for initialization and a higher quality and a higher accuracy of solution is gained owing to the local grid refinement (using static adaptation means) near the shock-wave front.

Multiscale coupling of FFT-based simulations with the LDC approach

FFT based solvers have proven to be efficient computational methods for the numerical simulations of composite materials with complex microstructures and sophisticated material behavior. Nevertheless, these solvers require a regular discretization of the investigated domain and periodic boundary conditions.Local multi-grid methods appear as an appropriate framework to perform simulations involving local refinement with FFT solvers. A coarse (global) discretization is improved by one or several local refined discretizations defined in subdomains overlapping the global grid. The Local Defect Correction (LDC) method is adapted to couple the resolution on the global coarse grid problem with the resolution of the local grids problems. Numerical tests demonstrate a significant reduction of CPU time and memory consumption with a relatively low loss in accuracy in comparison to the computation on one fully refined global grid.This approach also provides a baseline for extensions to the coupling of FFT codes with other solvers in a multi-scale and multi-physics framework.

Application of High-Order WENO Scheme in the CFD/FW–H Method to Predict Helicopter Rotor Blade–Vortex Interaction Tonal Noise

The accurate prediction of helicopter rotor blade–vortex interaction (BVI) noise is challenging. This paper presents an implementation of the seventh-order improved weighted essentially non-oscillatory (WENO-Z) scheme for predicting rotor BVI noise using a high-resolution numerical method based on the Reynolds-averaged Navier–Stokes and the Ffowcs Williams–Hawkings equations. The seventh-order improved WENO-Z scheme is utilized to minimize the inherent numerical dissipation of the reconstruction method in the monotone upstream-centered scheme for conservation laws (MUSCL), thereby improving the rotor wake resolution and the BVI noise-prediction accuracy. The three-layer dummy cell method is used to ensure that the flux at the boundary maintains seventh-order accuracy. The effectiveness of the flow solver and the acoustic solver is validated using the Helishape-7A rotor and the UH-1H rotor, respectively. The flow field and BVI noise characteristics of the OLS rotor obtained from the fifth- and seventh-order WENO-Z schemes are compared with that of the third-order MUSCL for coarse and fine background grids. The wake resolution, noise-prediction accuracy, and computational cost of the three schemes are compared. The results show that the high-order WENO scheme provides higher accuracy for flow field simulation and BVI noise prediction than the MUSCL, but the computational cost of the WENO scheme increases substantially as the grid resolution increases. However, the WENO scheme can predict BVI using a coarser grid than the MUSCL. The computational cost of the WENO scheme is relatively low under the same flow field simulation resolution.

Transport and modeling of subgrid-scale turbulent kinetic energy in channel flows

To develop a more convenient subgrid-scale (SGS) model that performs well even in coarse grid cases, we investigate the transport and modeling of SGS turbulent kinetic energy (hereafter SGS energy) in turbulent channel flows based on the stabilized mixed model (SMM). In this paper, we try to increase the convenience of the SMM by replacing the modeled transport equation for the SGS energy with an algebraic model. The SMM quantitatively adequately predicts the total turbulent kinetic energy of the direct numerical simulation (DNS) even in coarse grid cases. For both the filtered DNS (fDNS) and large-eddy simulation (LES), the statistically averaged production term balances with the dissipation in the region away from the wall in the SGS energy transport equation. In contrast, we reveal that the correlation coefficient between the production and dissipation terms is high for the modeled transport equation in LES, whereas that for the fDNS is low. Based on the high correlation or local equilibrium between the production and dissipation observed in the LES, we demonstrate the reduction of the SMM into a zero-equation SMM (ZE-SMM). We construct a new damping function based on the grid-scale Kolmogorov length to reproduce the near-wall behavior of the algebraic model for the SGS energy. The ZE-SMM provides quantitatively the same performance as the original SMM that employs the SGS energy transport model. This result suggests that the local equilibrium model for the SGS energy provides the equivalent performance as the transport model in wall-bounded turbulent flows even in coarse grid cases.

Characterization of low levels of turbulence generated by grids in the settling chamber of a laminar wind tunnel

Wind tunnel investigations of how Natural Laminar Flow (NLF) airfoils respond to atmospheric turbulence require the generation of turbulence, whose relevant characteristics resemble those in the atmosphere. The lower, convective part of the atmospheric boundary layer is characterized by low to medium levels of turbulence. The current study focuses on the small scales of this turbulence. Detailed hot-wire measurements have been performed to characterize the properties of the turbulence generated by grids mounted in the settling chamber of the Laminar Wind Tunnel (LWT). In the test section, the very low base turbulence level of Tuu ≅ 0.02% (10 ≤ f ≤ 5000 Hz) is incrementally increased by the grids up to Tuu ≅ 0.5%. The turbulence spectrum in the u-direction shows the typical suppression of larger scales due to the contraction between grids and test section. Still, the generated turbulence provides a good mapping of the spectrum measured in flight for most of the frequency range 500 ≤ f ≤ 3000 Hz, where Tollmien-Schlichting (TS)-amplification occurs for typical NLF airfoils. The spectra in v and w-direction exhibit distinct inertial subranges with slopes being less steep compared to the − 5/3 slope of the Kolmogorov spectrum. The normalized spectra in u-direction collapse together well for all grids, whereas in v- and w-directions the inertial- and dissipative subranges are more clearly distinguished for the coarser grids. It is demonstrated that the dissipation rate ε is a suitable parameter for comparing the wind tunnel turbulence with the atmospheric turbulence in the frequency range of interest. By employing the grids, turbulence in the range 4.4 × 10–7 ≤ ε ≤ 0.40 m2/s3 at free-stream velocity U∞ = 40 m/s can be generated in the LWT, which covers representative dissipation rates of free flight NLF applications. In the x-direction, the spectra of the v and w-components develop progressively more pronounced inertial- and dissipative subranges, and the energy below f ≈ 400 Hz decreases. In contrast, the spectral energy of the u-component increases across the whole frequency range, when moving downstream. This behavior can be explained by the combination of energy transport along the Kolmogorov cascade and the incipient return to an isotropic state.Graphic

On the efficiency of 5(4) RK-embedded pairs with high order compact scheme and Robin boundary condition for options valuation

When solving the American options with or without dividends, numerical methods often obtain lower convergence rates if further treatment is not implemented even using high-order schemes. In this article, we present a fast and explicit fourth-order compact scheme for solving the free boundary options. In particular, the early exercise features with the asset option and option sensitivity are computed based on a coupled of nonlinear PDEs with fixed boundaries for which a high order analytical approximation is obtained. Furthermore, we implement a new treatment at the left boundary by introducing a third-order Robin boundary condition. Rather than computing the optimal exercise boundary from the analytical approximation, we simply obtain it from the asset option based on the linear relationship at the left boundary. As such, a high order convergence rate can be achieved. We validate by examples that the improvement at the left boundary yields a fourth-order convergence rate without further implementation of mesh refinement, Rannacher time-stepping, and/or smoothing of the initial condition. Furthermore, we extensively compare the performance of our present method with several 5(4) Runge–Kutta pairs and observe that Dormand and Prince and Bogacki and Shampine 5(4) pairs are faster and provide more accurate numerical solutions. Based on numerical results and comparison with other existing methods, we can validate that the present method is very fast and provides more accurate solutions with very coarse grids.

A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel

The main aim of this paper is to solve the two-dimensional nonlinear fractional partial integro-differential equation (PIDE) with a weakly singular kernel by using the time two-grid finite difference (FD) algorithm. The second-order backward difference formula (BDF) and L1 scheme are used in time. The time two-grid algorithm is constructed to improve the solving efficiency of nonlinear systems. The Newton iteration is used to solve nonlinear discrete system on the coarse grid, and then we apply Lagrangian linear interpolation to attain the function value used in constructing the difference scheme on the fine grid. The second-order finite difference method (FDM) is used in space. The unconditional stability and convergence are attained for the two-grid fully discrete system. Numerical experiments show that the used CPU time for the presented two-grid numerical algorithm is lower than the general finite difference method for solving the nonlinear system.

Local and parallel finite element methods based on two-grid discretizations for a non-stationary coupled Stokes-Darcy model

In this paper, some local and parallel finite element methods based on two-grid discretizations are proposed and investigated for a non-stationary coupled Stokes-Darcy model. Based on two-grid discretizations, a semi-discrete scheme is presented. With backward Euler scheme for the temporal discretization and two-grid discretizations for the spatial discretization, some fully discrete schemes are proposed. The crucial idea is to adopt a decoupling scheme based on interface approximation via temporal extrapolation to approximate the mixed model by utilizing a coarse grid on the whole domain, then solve some residual equations with a finer grid on some overlapped subdomains by some local and parallel procedures at each time step. The interface coupling term on the subdomains with fine grid is approximated by the coarse-grid approximations. To reach a global continuous approximation, a new parallel algorithm based on the partition of unity is devised. Some local a priori estimate that is crucial for the theoretical analysis is obtained. Finally, some numerical experiments are conducted to support our theoretical results and demonstrate the computational effectiveness.

Continental United States may lose 1.8 petagrams of soil organic carbon under climate change by 2100

AbstractAimsHigh‐resolution information on soils’ vulnerability to climate‐induced soil organic carbon (SOC) loss can enable environmental scientists, land managers, and policy makers to develop targeted mitigation strategies. This study aims to estimate baseline and decadal changes in continental US surface SOC stocks under future emission scenarios.LocationContinental United States.Time period2014–2100.MethodsWe used recent SOC field observations (n = 6,213 sites), environmental factors (n = 32), and an ensemble machine learning (ML) approach to estimate baseline SOC stocks in surface soils across the continental United States at 100‐m spatial resolution, and decadal changes under the projected climate scenarios of Coupled Model Intercomparison Project Phase Six (CMIP6) earth system models (ESMs).ResultsBaseline SOC projections from ML approaches captured more than 50% of variability in SOC observations, whereas ESMs represented only 6–16% of observed SOC variability. ML estimates showed a mean total loss of 1.8 Pg C from US surface soils under the high‐emission scenario by 2100, whereas ESMs showed no significant change in SOC stocks with wide variation among ESMs. Both ML and ESM predictions agree on the direction of SOC change (net emissions or sequestration) across 46–51% of continental US land area. These differences are attributable to the high‐resolution site‐specific data used in the ML models compared to the relatively coarse grid represented in CMIP6 ESMs.Main conclusionsOur high‐resolution estimates of baseline SOC stocks, identification of key environmental controllers, and projection of SOC changes from US land cover types under future climate scenarios suggest the need for high‐resolution simulations of SOC in ESMs to represent the heterogeneity of SOC. We found that the SOC change is sensitive to key soil related factors (e.g. soil drainage and soil order) that have not been historically considered as input parameters in ESMs, because currently more than 95% variability in the SOC of CMIP6 ESMs is controlled by net primary productivity, temperature, and precipitation. Using additional environmental factors to estimate the baseline SOC stocks and predict the future trajectory of SOC change can provide more accurate results.

Analysis of high Reynolds free surface flows

ABSTRACT In this paper, we will combine an upwind radial basis function-finite element with direct velocity–pressure formulation to study the two-dimensional Navier-Stokes equations with free surface flows. We will examine this formulation in an improved mixed-order finite element and localized radial basis function method. A particle tracking method and the arbitrary Lagrangian-Eulerian scheme will then be applied to simulate the two-dimensional high Reynolds free surface flows. An upwind improved finite element formulation based on a localized radial basis function differential quadrature (LRBFDQ) method is used to deal with high Reynolds number convection dominated flows. This study successfully obtained very high Reynolds number free surface flows, up to Re = 500 000. Finally, we will demonstrate and discuss the capability and feasibility of the proposed model by simulating two complex free surface flow problems: (1) a highly nonlinear free oscillation flow and (2) a large amplitude sloshing problem. Using even very coarse grids in all computing scenarios, we have achieved good results in accuracy and efficiency.

Towards a robust CFD modelling approach for reliable hydrodynamics and mass transfer predictions in aerobic stirred fermenters

A strategy based on the Reynolds averaged two-fluid model is proved to be effective for the simulation of a turbulent gas-liquid stirred tank in the whole range of possible flow regimes, namely complete recirculation, loading, and flooding. The modelling approach is validated through the simulation of a laboratory scale tank stirred with a Rushton turbine, with superficial gas velocities ranging from 1.5 mm/s to 4.4 mm/s and specific power consumption from 39 W/m3 to 837 W/m3, since such a system in this range of operative conditions is well-characterized and several well-established correlations are available for validation purposes. The gas segregation due to the formation of aerated cavities was accounted for in the determination of the specific interfacial area available for the mass transfer and the numerical predictions of the kLa better agree with the available correlations, especially in the loading regime when clinging and ‘3–3’ cavities develop, with respect to the traditional specific interfacial area formulation that does not consider the phase segregation. Finally, starting from grid independent results, the effect of computational grid coarsening is quantified, obtaining guidelines for the affordable simulation of bioreactors of large scale.

On the injectivity estimation in foam EOR

In this work, we study injectivity issues caused by the use of the Peaceman equation in the numerical simulation of chemical enhanced oil recovery (EOR) processes aimed at reducing fluid mobility, such as foam injection, on coarse grids. Employing analytical solutions, we demonstrate that the Peaceman equation, commonly applied to mathematical modeling of the injectivity in commercial simulators, leads to errors of more than two orders of magnitude in the injection well pressure drop when the foam flow effects near the well are considered for assuming homogeneous mobility in a coarse-well block. To circumvent this issue, we investigate numerical treatments focused on the grid of well blocks through local grid partitioning strategies (radial and Cartesian) to improve the injection well bottom-hole pressure (BHP) estimation. This methodology does not change the input data nor the injectivity model characteristics of the commercial simulator. It does not significantly affect the computational cost of the simulation, since the grid treatment occurs only in the blocks containing the wells. Thus, the radial and Cartesian grid partitioning for the well block are compared using the STARS simulator. Our results show the clear capability of the methodology to reduce the well BHP overestimation, mitigating the errors caused by the Peaceman equation. Indeed, in some simulated scenarios, the BHP overestimation was reduced a hundredfold after applying the partitioning technique. Furthermore, we discuss the choice of simulation parameters leading to more accurate and numerically stable results.

On turbulent flow and aerodynamic noise of generic side-view mirror with cell-centred finite difference method

This paper investigates the turbulent flow and aerodynamic noise of a half-cylinder body mounted on a flat plate at using high-order cell-centred finite difference method with delayed detached-eddy simulation (DDES) and large-eddy simulation (LES). Transient flow patterns from the two simulations are found to be very different in consideration of the small-scale structures. The profiles of mean velocity, resolved turbulent kinetic energy and resolved Reynolds shear stress are found to be similar among all the simulations, indicating mean quantities are relatively insensitive to turbulence modelling and grid resolution. The power spectra density of the pressure fluctuations show that LES is more capable of resolving energies in high-frequency range than DDES. After computing the normalised wavenumber-frequency spectra of fluctuating pressure on the window, we further carried out the wavenumber-frequency decomposition to separate the acoustic and the hydrodynamic components from the pressure fluctuations. The energy distribution shows that the acoustic energy has a much slower decaying rate in the high-frequency range than the hydrodynamic energy. In addition, the space-averaged sound pressure levels of pressure fluctuations on the window indicate that the present simulation with a high-order method is able to improve the accuracy in predicting pressure spectra. Finally, we carry out proper orthogonal decomposition to extract the dominating features of the decomposed acoustic and hydrodynamic components of pressure fluctuation. Patterns of multi-scale turbulence in hydrodynamic modes and propagating wavefronts of cylinder shape in acoustic modes are identified. The present research indicates that a relatively coarse grid is still capable of resolving fluctuating quantities of energy-containing structures, and LES is suggested against DDES when near-wall aerodynamic noise is the main concern.