Numerical Domain Research Articles

Using H‐Convergence to Calculate the Numerical Errors for 1D Unsaturated Seepage Under Steady‐State Conditions

ABSTRACTUnsaturated zones are important for geotechnical design, geochemical reactions, and microbial reactions. The numerical analysis of unsaturated seepage is complex because it involves highly nonlinear partial differential equations. The permeability can vary by orders of magnitude over short vertical distances. This article defines and uses H‐convergence tests to quantify numerical errors made by uniform meshes with element size (ES) for 1D steady‐state conditions. The quantitative H‐convergence should not be confused with a qualitative mesh sensitivity study. The difference between numerical and mathematical convergences is stated. A detailed affordable method for an H‐convergence test is presented. The true but unknown solution is defined as the asymptote of the numerical solutions for all solution components when ES decreases to zero. The numerical errors versus ES are then assessed with respect to the true solution, and using a log–log plot, which indicates whether a code is correct or incorrect. If a code is correct, its results follow the rules of mathematical convergence in a mathematical convergence domain (MCD) which is smaller than the numerical convergence domain (NCD). If a code is incorrect, it has an NCD but no MCD. Incorrect algorithms of incorrect codes need to be modified and repaired. Existing codes are shown to converge numerically within large NCDs but generate large errors, up to 500%, in the NCDs, a dangerous situation for designers.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Machine learning and domain decomposition methods - a survey

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning and various industrial domains, especially in computational science and engineering. In the present survey, several promising avenues of research will be examined which focus on the combination of machine learning (ML) and domain decomposition methods (DDMs). The aim of this survey is to provide an overview of existing work within this field and to structure it into domain decomposition for machine learning and machine learning-enhanced domain decomposition, including: domain decomposition for classical machine learning, domain decomposition to accelerate the training of physics-aware neural networks, machine learning to enhance the convergence properties or computational efficiency of DDMs, and machine learning as a discretization method in a DDM for the solution of PDEs. In each of these fields, we summarize existing work and key advances within a common framework and, finally, discuss ongoing challenges and opportunities for future research.

A volume‐averaging and stochastic turbulence modeling framework for homogeneous roughness‐induced drag in turbulent flows

AbstractThe goal of this work is the formulation of a model for the parameterization of homogeneous roughness‐induced drag in turbulent flow simulations. We characterize rough surfaces using their surface‐area moments of roughness‐peaks. Additionally, we consider a probabilistic self‐similar power law distribution for the roughness heights, in such a way that an assumed set of discrete roughness elements can behave as a probabilistic fractal set. Upon these assumptions, the surface characterization is complete and yields wall‐normal profiles of porosity, average roughness element diameter, and equivalent pore diameter of the roughness affected porous flow. The profiles are supplied to a (pore‐based) Reynolds number and porosity‐based drag parameterization for staggered cylinder arrays available in the literature, as well as a one‐dimensional map‐based turbulence model, the one‐dimensional turbulence (ODT) model. The purpose of the latter is a way to provide the missing effects of the equivalent dispersion tensor in the porous flow close to the rough surface, as well as that of the nonlinear turbulent transport away from the wall. Simulations are then carried out in order to evaluate the effects of two different rough surfaces on a turbulent channel flow, comparing results with existing direct numerical simulation (DNS) data. Overall, reasonable agreement is obtained, which suggests that, by using the proposed model, it is possible to model the effects of rough surfaces on wall‐bounded turbulent flows, without the explicit need of the full topological representation of the surface on the numerical domain. To that extent, one clear flow control application for the suggested approach would be that of drag‐based surface optimization.

A Numerical Approach to Analyzing Shallow Flows over Rough Surfaces

The hydraulic characteristics (such as velocity profiles, near-bed velocity profile, bed shear stress, and resistance coefficients) of shallow flows over rough surfaces were investigated using numerical simulations. A novel method is presented to simulate shallow flows over rough surfaces in a two-dimensional (2D) numerical domain, where the physical numerical domain represents bed topography. Results reveal that the model can accurately predict spatially averaged velocity profiles, turbulence characteristics, shear stresses, and uniform flow depths. The analysis identified two distinct flow regions based on mean and turbulent flow profiles. Results show that the turbulent shear stress profiles provide a more accurate estimation of the bed shear stresses. Resistance coefficients (friction factor or Manning’s roughness coefficient) vary with Froude number and submergence ratio (depth divided by roughness height).

Numerical modeling and validation of dike-induced water flow dynamics using OpenFOAM

ABSTRACT The basic purpose of the present research is to numerically elucidate the flow around a single dike in four different geometric configurations (CN: no cylinders, C5: 5 cylinders, C10: 10 cylinders, C15: 15 cylinders), under conditions of a constant flow rate and subcritical flow. This involved substituting the impermeable dike with varying numbers of piles to validate the observed experimental flow patterns, particularly the conditions leading to reverse flow formation. OpenFOAM, an open-source algorithm, was utilized for the simulation, incorporating the volume of fluid (VOF) method. Accurately depicting the reverse flow formation with minimal numerical diffusion, a strategy involving multizone meshing and the application of mesh refinement zones was utilized. These refinements were particularly focused on the section between 1.8 m and 2.2 m within the numerical domain. The water surface profile in front section of the dike and the averaged velocity at five different locations, which were then compared with the numerical results. The numerically predicted streamwise velocity showed a percentage error ranging between 0.5% and 4.5%. The numerical results from this study are pivotal in dike design, aiming to mitigate accelerated wear during flood events and to protect both the dike head and the adjacent bank from high energy flow failures.

Experimental implementation of the nonstructural element simulator on shake table (NEST) for an archetype tall building

AbstractNonstructural damage, usually predominates in buildings subjected to low to moderate seismic intensities and contributes significantly to business interruption and economic losses. It promotes a rapidly growing need for experimental research and qualification of various types of nonstructural elements in buildings in the recent decades. To provide a novel option of experimentally simulating the realistic boundary conditions for various nonstructural elements in labs, we developed the Nonstructural Element Simulator on Shake Table (NEST), a passively controlled three‐layer substructural testbed driven by existing shake tables. This paper presents the first experimental implementation of NEST on a challenging 42‐story archetype tall building. The dynamic properties of the substructure were tuned to adapt the archetype building and the required shake table motions were solved as a reverse problem by an open‐loop control algorithm. The test results proved the capability of NEST to replicate the history responses of the target floors in the archetype building under either recorded earthquake ground motion or artificial loading protocols for qualification purposes. In all cases, the synthetic relative error in the floor accelerations and the inter‐story drift within the frequency range of interest was less than 5% in the numerical domain and less than 30% in the physical realization. The seismic responses of a variety of nonstructural elements to the substructural motions show that, in the mid‐story of the archetype tall building, the ceiling sustained minimum damage because of the small floor acceleration while the indoor contents slid significantly and even overturned.

Influence of the Presence of Different Signatures on the Heat Transfer Profile of Laminar Flow Inside a Microchannel

Abstract The process of transferring heat and mass involves a high-pressure decline. Hence microchannels are utilized in extremely efficient heat and mass transfer processes, such as in the systems of the lungs and kidneys. Due to their high surface-to-volume ratio and compact volume, microchannels have demonstrated superior thermal performance. In this work, micro-fins of different topologies and arrangements were used inside microchannels for modulating heat transfer from the surface of the fins, in the presence of flowing media. The influx of media was simulated in sinusoidal form, reproducing simplified form of physiological circulatory blood flow. The study was conducted in numerical domain, while heat transfer and mass transfer equations were solved using pardiso solver. Temperature modulation at the fin surfaces was conducted to examine the transient profile of heat transfer, as function of both variable fluid velocity as well as variable feeding heat sources. In case of different input boundary conditions, the effect of heat distributions on fluid flow with respect to spatial distribution of micro-fins within a microchannel was evaluated. Results revealed the difference in heat transfer profiles in microchannels in presence of different sets of fin configurations. It was found that rectangular fins have the highest heat transfer in fluid at all its spatial configurations; while semi-ellipsoidal-shaped fins had shown the least heat transfer profile at same surface area. At the same time, it was also observed that the rate of heat dissipation was faster and limited in semi-ellipsoidal-shaped fins configuration; while the heat transfer rate was found symmetrical and low in presence of rectangular fins. Hence, this article emphasizes the modulation of temperature and velocity variations within the working fluid by emphasizing the thermo-fluid coupling effects in microchannels.

On the measurement of the aerodynamic features of composite structures under subsonic airflow using mathematical modeling and machine learning method

This study investigates the aerodynamic features of functionally graded triply periodic minimal surface (FG-TPMS) structures under subsonic airflow using a combination of mathematical modeling and machine learning methods. The modified supersonic piston theory of Krumhaar is used to quantify the aerodynamic stiffness and damping matrices. Using characteristics and six distinct patterns of volume distribution of FG-TPMS materials in terms of their mechanical qualities, Hamilton’s principle, and parabolic shear deformation theory are used to extract the governing equations of the presented composite structure. After that, the differential quadrature method at the discrete places and Lagrange interpolated polynomials and Kronecker product are used to solve the governing equations in the numerical domain. This study highlights the potential of integrating advanced computational techniques with machine learning to address complex engineering problems. The findings provide valuable insights into the aerodynamic optimization of sandwich structures, paving the way for their more efficient and effective design in aerospace engineering. As an important outcome for related industries, the displacement field along with the Z direction increases by increasing the angle of airflow by approximately 9 % while this value for the X, and Y directions are 7 %, and 6 %, respectively. The proposed methodology offers a powerful tool for engineers and researchers seeking to optimize the performance of advanced composite materials under subsonic airflow conditions.

Airflow and heat transfer characteristics for different cargo pallet arrangements in a heat transfer enhanced refrigerated container

Improving the performance of a refrigerated container is crucial in effectively preserving thermal-sensitive cargo. The combined effect of vertical airflow resistance and airflow short-circuiting in air gaps play a vital role in cargo cooling. This study numerically investigates the effects of cargo vertical airflow resistance and air gaps between cargo pallets using two commonly used apple packaging boxes, each stacked in six different cargo pallet arrangements. A heat transfer enhanced refrigerated container design and a numerical model developed from the present authors’ previous studies are used. The numerical model uses the porous medium model to incorporate the refrigeration unit and cargo within the refrigerated container. Additionally, the numerical domain includes the T-bar floor in the refrigerated container. Initially, the airflow characteristics based on vertical airflow resistance and air gaps are discussed. Then, the corresponding effects on cargo cooling are addressed. Results show that the air gaps between the cargo pallets have positive and negative effects on cargo cooling. In cargos with lower vertical airflow resistance, an increase in air gaps leads to reduced cargo cooling and elevated temperature non-uniformity. Conversely, cargos with higher vertical airflow resistance experience increased cooling rates and reduced temperature non-uniformity with the increase in air gaps. However, avoiding airflow short-circuiting through the air gaps between the cargo pallets and improving the vertical airflow inside the cargo pallets can provide better cargo cooling, which outweighs the positive cooling effect of the air gap.

Computational Fluid Dynamics Methodology to Estimate the Drag Coefficient of Balls in Rolling Element Bearings

The emergence of electric vehicles has brought new issues such as the problem of rolling element bearings (REBs) operating at high speeds. Losses due to these components in mechanical transmissions are a key issue and must therefore be taken into account right from the design stage of these systems. Among these losses, the one induced by the motion of rolling elements, known as drag loss, becomes predominant in high-speed REBs. Although an experimental approach is still possible, it is difficult to isolate this loss in order to study it properly. A numerical approach based on CFD is therefore a possible way forward, even if other issues arise. The aim of this article is to study the ability of such an approach to correctly estimate the drag coefficient associated with the motion of rolling elements. The influence of the numerical domain extension, the mesh refinement, the simplification of the ring shape, and the presence of the cage on the values of the drag coefficient is presented. While it seems possible to compromise on the calculation domain and mesh size, it appears that the other parameters must be taken into account as much as possible to obtain realistic results.

Evaluating the efficiency and effectiveness of non-Newtonian fluid flow in a double-pipe heat exchanger with porous medium via numerical simulation

In the present work, the convection heat transfer and pressure drop of the turbulent flow based on a non-Newtonian fluid were numerically investigated in a double-pipe heat exchanger with various porous rings by considering the inner tube under constant heat flux. The solution of 0.1% and 2% mass carboxy methyl cellulose in deionized (DI) water water was used as the base non-Newtonian fluid. The fluid flow and heat transfer in this study were modeled by using the RNG K-ε turbulent viscosity with the enhanced wall treatment. The non-Newtonian power-law method was used in ANSYS Fluent software to resolve the flow within the numerical solution domain in the double pipe. Furthermore, the thermal performance of the heat exchanger, and the effects of the geometry, number, thickness, and height of the porous rings at the various Reynolds numbers were studied. According to the results, using non-Newtonian fluid had a significant effect on the average Nusselt number and pressure drop which were increased by up to 40.1% and 10 times, respectively. The annular tube possessing six porous rings at Re = 4,000 and height = 2 mm, thickness = 6 mm, applying non-Newtonian fluid can improve the efficiency of the system by up to 11.25%. It also found that by increasing the porous thickness, the Nusselt number increased by up to 56.3% at its maximum. The results have demonstrated a satisfactory level of agreement with experimental and correlation data by comparing them with existing data in similar literature.

Towards limited-domain full waveform ambient noise inversion

SUMMARY Ambient noise tomography is a well-established tomographic imaging technique but the effect that spatially variable noise sources have on the measurements remains challenging to account for. Full waveform ambient noise inversion has emerged recently as a promising solution but is computationally challenging since even distant noise sources can have an influence on the interstation correlation functions and therefore requires a prohibitively large numerical domain, beyond that of the tomographic region of interest. We investigate a new strategy that allows us to reduce the simulation domain while still being able to account for distant contributions. To allow nearby numerical sources to account for distant true sources, we introduce correlated sources and generate a time-dependent effective source distribution at the boundary of a small region of interest that excites the correlation wavefield of a larger domain. In a series of 2-D numerical simulations, we demonstrate that the proposed methodology with correlated sources is able to successfully represent a far-field source that is simultaneously present with nearby sources and the methodology also successfully results in a robustly estimated noise source distribution. Furthermore, we show how beamforming results can be used as prior information regarding the azimuthal variation of the ambient noise sources in helping determine the far-field noise distribution. These experiments provide insight into how to reduce the computational cost needed to perform full waveform ambient noise inversion, which is key to turning it into a viable tomographic technique. In addition, the presented experiments may help reduce source-induced bias in time-dependent monitoring applications.

A Numerical Domain Decomposition Method for Solving Elliptic Equations on Manifolds

A Numerical Domain Decomposition Method for Solving Elliptic Equations on Manifolds

Study on Main Factors Controlling Development Performance of Heterogeneous Composite Flooding in Post-Polymer Flooding Reservoir

Heterogeneous composite flooding has performed well with regard to enhanced oil recovery after polymer flooding in recent years. In order to significantly increase oil recovery, the development parameters should be designed differently for each well. However, it is difficult to rapidly allocate development parameters through the lowering of computational costs. Therefore, the authors of this paper carried out research to clarify the main controlling factors of parameter allocation. Firstly, the numerical simulation domain was separated into several regions, with injection wells and production wells at the center of each region. The statistical parameters of each region were calculated. Then, the water injection rate, liquid production rate, and chemical agent concentration were allocated based on the proportion of statistical parameters in each region. A large number of development schemes were designed by combining different injection and production allocations that were calculated based on each statistical parameter. Finally, the development performance of each scheme was simulated and analyzed. The statistical parameters corresponding to the best performance scheme were regarded as the main controlling factors of heterogeneous composite flooding after polymer flooding. These results showed that the main controlling factors for the allocation of the water injection rate were pore volume and permeability variation coefficient. The main controlling factors for liquid production rate were the remaining oil saturation, formation coefficient, and reservoir pressure. The main controlling factors for chemical agent concentration were pressure and permeability variation coefficient. These findings concerning the main factors controlling development parameter allocation were validated by practical application in several well groups of an actual reservoir model. This study provides references for improving heterogeneous composite flooding performance for post-polymer flooding reservoirs in the future.

Impact of the numerical domain on turbulent flow statistics: scalings and considerations for canopy flows

Large eddy simulations (LES) are widely used to study the effects of surface morphology on turbulence statistics, exchange processes and turbulence topology in urban canopies. However, as LES are only approximations of reality, special attention is needed for the computational model set-up to ensure an accurate representation of the physical processes of interest. This paper shows that the choice of the numerical domain can significantly affect the accuracy of turbulent flow statistics, potentially causing a mismatch between numerical studies and experimental data. The study examines the influence of cross-stream aspect ratio (YAR), streamwise aspect ratio (XAR) and scale separation (SS) on first- and second-order flow statistics and turbulence topology. It is found that domains with a low YAR underestimate the velocity variance, while those with a low XAR overestimate the variance value. The study proposes a new approach based on the Buckingham Pi theorem to evaluate the effect of SS, as the existing method has major limitations for canopy flows. The results suggest that domains with small SS underpredict the variance value. To minimise the artificial impact of the numerical domain on turbulent flow statistics, the study recommends guidelines for future research, including a YAR of 3 or more, an XAR of 6 or more and an SS of 12 or more. Error tables are presented to allow researchers to select smaller domains than recommended, depending on their research interests in specific parts of the flow.

Linear and nonlinear receptivity mechanisms in boundary layers subject to free-stream turbulence

Large-eddy simulations of a flat-plate boundary layer, without a leading edge, subject to multiple levels of incoming free-stream turbulence are considered in the present work. Within an input–output model, where nonlinear terms of the incompressible Navier–Stokes equations are treated as an external forcing, we manage to separate inputs related to perturbations coming through the intake of the numerical domain, whose evolution represents a linear mechanism, and the volumetric nonlinear forcing due to triadic interactions. With these, we perform the full reconstruction of the statistics of the flow, as measured in the simulations, to quantify pairs of wavenumbers and frequencies more affected by either linear or nonlinear receptivity mechanisms. Inside the boundary layer, different wavenumbers at near-zero frequency reveal streaky structures. Those that are amplified predominantly via linear interactions with the incoming vorticity occur upstream and display transient growth, while those generated by the nonlinear forcing are the most energetic and appear in more downstream positions. The latter feature vortices growing proportionally to the laminar boundary layer thickness, along with a velocity profile that agrees with the optimal amplification obtained by linear transient growth theory. The numerical approach presented is general and could potentially be extended to any simulation for which receptivity to incoming perturbations needs to be assessed.

A physical picture for the acoustic resonant drag instability

ABSTRACT Mixtures of gas and dust are pervasive in the Universe, from active galactic nuclei (AGNs) and molecular clouds to protoplanetary discs. When the two species drift relative to each other, a large class of instabilities can arise, called ‘resonant drag instabilities’ (RDIs). The most famous RDI is the streaming instability, which plays an important role in planet formation. On the other hand, acoustic RDIs, the simplest kind, feature in the winds of cool stars, AGNs, or starburst regions. Unfortunately, owing to the complicated dynamics of two coupled fluids (gas and dust), the underlying physics of most RDIs is mysterious. In this paper, we develop a clear physical picture of how the acoustic RDI arises and support this explanation with transparent mathematics. We find that the acoustic RDI is built on two coupled mechanisms. In the first, the converging flows of a sound wave concentrate dust. In the second, a drifting dust clump excites sound waves. These processes feed into each other at resonance, thereby closing an unstable feedback loop. This physical picture helps decide where and when RDIs are most likely to happen, and what can suppress them. Additionally, we find that the acoustic RDI remains strong far from resonance. This second result suggests that one can simulate RDIs without having to fine-tune the dimensions of the numerical domain.

Analogy between a moving line source illuminating a metallic wire and Compton scattering experiment

AbstractThis paper presents an electromagnetic analysis of a moving infinitely long line source illuminating an infinitely long metallic wire at rest. The study uses the full‐wave numerical finite difference time domain (FDTD) method, which is based on the spatial and temporal discretization of Maxwell's equations. Movement is computed within the FDTD technique by varying the position of the line source at each time loop. An analogy is proposed between the wavelength spectrum of the simulated electromagnetic field and the Compton scattering experiment. A good agreement is obtained between theoretical analysis, numerical results, and Compton experimental data.

(Keynote) Thermal Performance Optimisation at Cell Level, to Achieve Better, Longer Lasting Battery Packs

Temperature is a major issue in the operation of lithium-ion batteries, with both high and low temperatures causing problems. Cold batteries are inefficient, while hot batteries are at risk of thermal runaway and fire, making it crucial to maintain a safe and optimal temperature range. Temperature gradients can negatively impact battery performance, reducing accessible energy and increasing the degradation rate. This is a critical issue for electric vehicles (EVs), which can experience reduced driving range and deteriorating performance over time.Despite these challenges, current lithium-ion cell designs prioritize maximizing specific energy, which can negatively impact thermal performance. Thermal management systems are used to extract heat from poorly designed cells, but they are a significant parasitic load, demanding on average 8 kW in typical operation. Further, they can introduce uneven distribution of current throughout the battery pack, leading to diminished performance and accelerated degradation. The limited performance of these systems contributes towards range anxiety, which is a major barrier to the widespread adoption of electric vehicles in the UK, Europe, and the USA.To address these challenges, the authors propose the use of a new metric, the cell cooling coefficient (CCC). The CCC is a constant for a given cell and thermal management method (e.g. surface cooling or tab cooling), and is measured in W.K-1 (the reciprocal of thermal resistance). The CCC define a cell's ability to reject heat energy. The CCC is defined as the temperature gradient across a cell in idealized operating conditions, normalized against the heat generation rate in the same conditions. This allows for the comparison of geometrically dissimilar cells and can be used to optimize the thermal performance of lithium-ion cells.Experimental results presented in the paper demonstrate that increasing the thickness of the tabs in a typical lithium-ion pouch cell by 34% can yield a tab cooling performance enhancement of 20%, with a specific energy reduction of just 0.7%. Numerical methods predict that for a large format automotive pouch cell, increasing the tab thickness from 0.2 mm (negative) and 0.45 mm (positive) to 2 mm will lead to a 90% increase in the CCC for tab cooling (see in the attached figure). This yields a 51% reduction in temperature gradient across the cell in any instance of operation. As with all geometric optimisation, our models highlight a law of diminishing returns and we discuss the optimal tab thickness for various operations and applications.The authors also consider the use of the CCC metric for edge cooling in pouch cells and cylindrical cells. They find that tabs used to connect jelly rolls to the external can of cylindrical cells create a considerable thermal bottleneck. Redesign of this tab is well documented through Tesla’s proposed tab-less 4680.The use of thermal models at cell-level alongside existing battery models is critical for the development of future lithium-ion cells. The authors demonstrate that optimisation of thermal performance is entirely achievable within the numerical domain, reducing the requirement for expensive and timely prototyping phases during any product development process.By designing cells to have better thermal performance, it is possible to reduce the temperature gradient required between the hottest and coolest points in the thermal management system. This, in turn, reduces the power required by the thermal management system, allowing for more power to be directed towards extending the driving range of electric vehicles. Reduced temperature gradients at the battery pack level will also extend battery pack lifetime, reducing the lifetime cost to end-users and lessening the strain on raw material supply chains and battery recycling technology.The use of the CCC metric offers a promising approach to optimizing the thermal performance of lithium-ion cells. By addressing thermal management challenges, it is possible to reduce the parasitic load of thermal management systems, improve the lifetime and cost-effectiveness of battery packs, and lessen range anxiety for EV users. Figure 1