Total Computational Cost Research Articles

A multi-scale IMEX second order Runge-Kutta method for 3D hydrodynamic ocean models

Understanding complex physical phenomena often involves dealing with partial differential equations (PDEs) where different phenomena exhibit distinct timescales. Fast terms, associated with short characteristic times, coexist with slower ones requiring relatively longer time steps for resolution. The challenge becomes more manageable when, despite the varying characteristic times of fast and slow terms, the computational cost associated with faster terms is significantly lower than that of slower terms. Additionally, slower terms can also exhibit two distinct longer characteristic times, adding complexity to the system and resulting in a total of three characteristic timescales. In this paper, an innovative split second-order IMEX (IMplicit-EXplicit) temporal scheme is introduced to address this temporal complexity. It is used to solve the primitive equation ocean model. Extremely short times are handled explicitly with small time steps, while longer timescales are managed explicitly and semi-implicitly using larger time steps. The decision to solve a portion of the slower terms semi-implicitly is due to the fact that it does not significantly increase the total computational cost, allowing for greater flexibility in the time step without imposing a substantial burden on the overall computational efficiency. This strategy enables efficient management of the various temporal scales present in the equations, thereby optimizing computational resources. The proposed scheme is applied to solve 3D hydrodynamics equations encompassing three time scale: fast terms representing wave phenomena, slow terms describing horizontal aspects and stiff terms for vertical ones. Furthermore, the scheme is designed to respect crucial physical properties, namely global and local conservation. The obtained results on different test cases demonstrate the robustness and efficiency of the IMEX approach in simulating these complex systems.

A Comprehensive Quantitative Lifecycle Cost and Environmental Impact Analysis Model for Computing Infrastructure

This paper presents a comprehensive quantitative model for quantitative assessment of the lifecycle costs and environmental impacts of computing infrastructure, with a focus on internet data centers (IDCs) and high-performance computing (HPC) facilities. The key innovation lies in the integration of interdisciplinary cost evaluation and carbon emission methods for the establishment of this quantitative model. This framework, which outlines key cost components and carbon emission factors, enables the calculation of total costs, electricity expenses, and greenhouse gas emissions throughout the lifecycle of infrastructure. With IDCs as a case study, the research clarifies the intricate cost structure associated with equipment procurement, energy usage, land acquisition, and operational expenses. This paper provides an in-depth understanding of the cost structure and environmental impact of computing infrastructure in support of sustainable decision-making in its development.•Based on established cost estimation methods, such as Lifecycle Cost Analysis (LCCA) and the Analogous Estimating Method, this study examines costs across construction and operation phases.•The Emission Factor Method is used to quantify environmental impact, emphasizing the significance of regional energy mix and power usage effectiveness (PUE).

Analisa Pengendalian Persediaan Bahan Baku pada CV. Es Mitra Jaya Surabaya

Manual data collection of raw materials causes inventory problems such as data collection errors, duplicate data, and warehouse loading processes that consume a lot of time. This research was conducted on a milktea boba frenchise company that is very maintaining quality. Raw materials are provided and arranged by the central company, the time of marking goods can only be ordered according to a predetermined schedule. By implementing the Qr-Code Scan system using the Scan-IT to Office application, the warehouse goods data collection process becomes more effective. The results of system implementation were evaluated using the PIECES method with an average satisfaction of 3.58 (SATISFIED) from the six domains tested. The results of the total cost calculation that have been carried out prove that the decrease in inventory costs using Material Requirement Planning (MRP) with the Economic Order Quantity (EOQ) method has decreased. The calculation carried out by the company shows the total inventory cost of Rp.32,346,426.25. After calculations using the MRP method, it decreased by Rp.1,460,780.23.

Exploring cost changes with time-driven activity-based costing after service delivery redesign in Dutch maternity care.

The implementation of Value-Based Healthcare (VBHC) has spread across international healthcare systems, aiming to improve decision-making by combining information about patient outcomes and costs of care. Time-Driven Activity-Based Costing (TDABC) is introduced as a pragmatic yet accurate method to calculate costs of care pathways. It is often applied to demonstrate value-improving opportunities, such as interventions aimed at service delivery redesign. It is imperative for healthcare managers to know whether these interventions yield the expected outcome of improving patient value, for which TDABC is also suitable. However, its application becomes more complex and labour intensive if the intervention extends beyond activity-level changes in existing care pathways, to the implementation of entirely new care pathways. The complexity arises from the potential influence of such interventions on the costs of related care pathways. To fully comprehend the impact of such interventions on organizational costs, it is important to include these factors in the cost calculation. Given the substantial effort required for this analysis, this may explain the limited number of prior TDABC studies with similar objectives. This methodological development paper addresses this gap by offering a pragmatic enrichment of the TDABC methodology. This enrichment is twofold. First, it provides guidance on calculating a change in costs without the need for a total cost calculation. Second, to secure granularity, a more detailed level of cost-allocation is proposed. The aim is to encourage further application of TDABC to conduct financial evaluations of promising interventions in the domain of VBHC and service delivery redesign.

Research on Autonomous Vehicle Obstacle Avoidance Path Planning with Consideration of Social Ethics

Self-driving car research can effectively reduce the occurrence of traffic accidents, but when encountering sudden people or obstacles cutting into the lane, how to reduce the damage hazard to traffic participants and make ethical decisions is the key point that the development of self-driving technology must break through. When faced with sudden traffic participants, self-driving vehicles need to make ethical decisions between ramming into the traffic participants or other obstacles. Therefore, in this paper, we propose a model decision planning method based on multi-objective evaluation function path evaluation of local path planning. This method addresses the ethical model disagreement problem of self-driving vehicles encountering traffic participants and other obstacles. The aim is to ensure the safety of the lives of the traffic participants and achieve the vehicle’s reasonable ethical decision planning. Firstly, when anticipating traffic participants and other obstacles, the vehicle’s planning intention decisions are obtained through fuzzy algorithms. Different sets of curves for various positions are generated based on dynamic programming algorithms. These curves are then fitted using B-spline curves, incorporating obstacle collision costs, and classifying obstacles into different types with varying cost weights. Secondly, factors such as path length and average path curvature are considered for path total cost calculations. Finally, a local path that avoids traffic participants is obtained. This path is then tracked using a pure pursuit algorithm. The proposed algorithm’s effectiveness is verified through simulation experiments and comparative analyses conducted on the MATLAB platform. In conclusion, this research promotes a safer and more sustainable transport system in line with the principles of sustainable development by addressing the challenges associated with safety and ethical decision making in self-driving cars.

EQONOMIC ORDER QUANTITY (EOQ) METHOD FOR INVENTORY CONTROL OF SOYBEAN RAW MATERIALS IN TOFU INDUSTRY FAUSI BONDOWOSO

tofu industry is an industry engaged in the food sector, namely processing agricultural products in the form of soybeans that sell food needs in the form of tofu. The process of making tofu in the Fausi tofu industry with a large amount every day makes the supply of soybean raw materials must be maintained optimally. Recording for the purchase of soybean raw materials which is still done manually by writing in a daily book makes it difficult for the owner or employees involved to review stock inventory. Descriptive analysis method with a quantitative approach is done by identifying existing data, comparing the results of the data obtained with the existing theory, producing an overview of the problem to be studied and drawing conclusions. The results of the analysis of soybean raw material inventory control in the fausi tofu industry, namely the Economic Order Quantity method can be more optimal than the manual method. Reorder point (ROP) which determines the point at which raw material inventory must be added before running out of inventory in 2023 is at least 1449 Kg. And the calculation of Total Inventory Cost in the Fausi Tofu Industry is more effective using the EOQ method than the manual method because it can reduce total inventory cost by Rp 6,710,485 so that the costs incurred are lower.

Analysis of Production Cost Calculation Using Job Order Costing Method at PT Secma Energy Cell

Production Cost Calculation is very important for companies as support in financial reporting. Recording production costs encourages business managers to create accurate and fast administrative documentation, so that companies can predict selling price and profit and loss in accordance with expectations. This study aims to examine whether PT Secma Energy Cell, situated in Gresik Regency, East Java, to the principles of cost accounting in implementing the Job Order Costing method for determining production costs. Utilizing a descriptive qualitative approach with a case study methodology, the study aims to evaluate the application of cost calculations for plastic bags and rolls of varying sizes. Data gathering techniques encompass observation, interviews, and documentation of total and variable cost calculations. The study spanned around six months, from December 2022 to May 2023, whereas the data recorded dates back to October 2021. Findings revealed that PT Secma Energy Cell employs a total of 62 individuals, with 34 engaged directly in production activities, while the remainder are non-production staff. Over the course of one month, the company utilized 115,400 kilograms of raw materials, amounting to IDR 2,311,471,500.00, the production of ready-to-sell goods for 1 month was 42,775,005 kg, plastic roll production was 105,020.00 kg, and plastic bag production was 147,795 kg. Up to this point, the company has established the product price per kilogram with an approximately markup of 70%, so that each order received has been calculated in advance regarding the selling price of each product. The analysis revealed that using the Job Order Costing method, the production cost amounted to IDR16,559.48 per kilogram for plastic bags and IDR20,118.81 for plastic rolls. This indicates that PT Secma Energy Cell effectively implements the job order costing method by accurately accounting for expenses associated with raw materials, direct labor, and factory overhead costs. This is intended to enhance the accuracy of the set selling price and potentially boost the company’s future profitability.

Spatially dependent node regularity in meshless approximation of partial differential equations

In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatially varying computational node regularity. Rather than covering the entire domain with scattered nodes, only regions with geometric details are covered with scattered nodes, while the rest of the domain is discretized with regular nodes. A simpler approximation can be used in regions covered by regular nodes, effectively reducing the required stencil size and computational cost compared to the approximation on scattered nodes where a set of polyharmonic splines is added to ensure convergent behaviour.This paper is an extended version of conference paper entitled “Spatially-varying meshless approximation method for enhanced computational efficiency” (Jančič et al., 2023) presented at “International Conference on Computational Science (ICCS) 2023”. The paper is extended with discussion on development and implementation of a hybrid regular-scattered node positioning algorithm (HyNP). The performance of the proposed HyNP algorithm is analysed in terms of separation distance and maximal empty sphere radius. Furthermore, it is demonstrated that HyNP nodes can be used for solving problems from fluid flow and linear elasticity, both in 2D and 3D, using meshless methods.The extension also provides additional analyses of computational efficiency and accuracy of the numerical solution obtained on the spatially-variable regularity of discretization nodes. In particular, different levels of refinement aggressiveness and scattered layer widths are considered to exploit the computational efficiency gains offered by such solution procedure.

A gene-inspired metaheuristic for scheduling workflow tasks in mobile edge computing-supported cyber–physical systems

This paper proposes a cost-minimization scheduling algorithm for the joint optimization of task offloading and resource allocation for workflow applications in a mobile edge computing (MEC)-supported cyber–physical system. We model this scheduling problem as an integer program that minimizes the total communication and computation cost under resources and deadline constraints upon workflow applications. To cope with this complicated optimization problem, we develop an efficient heuristic method to dispatch the workflow tasks onto MEC servers with sufficient resources. By using the task dispatching heuristic to evaluate the quality of each candidate solution, we construct a novel gene-inspired metaheuristic algorithm (GIMA) that incorporates an offspring-production operator and a conditional insertion scheme into the improvement strategy to explore high-quality solutions. Specifically, the offspring-production operator can enhance the scheduling quality by generating improved offspring solutions based on existing solutions. The conditional insertion scheme is further incorporated to reduce the computational overhead involved in solution exploration. We perform extensive simulations to justify the performance of the proposed GIMA in solving the scheduling problem studied in this work. Experimental results on real-world and synthetic workflow applications show that GIMA outperforms other metaheuristics in terms of cost reduction and computational efficiency.

Nonconvex weighted variational metal artifacts removal via convergent primal-dual algorithms

Direct reconstruction through filtered back projection engenders metal artifacts in polychromatic computed tomography images, attributed to highly attenuating implants, which further poses great challenges for subsequent image analysis. Inpainting the metal trace directly in the Radon domain for the extant variational method leads to strong edge diffusion and potential inherent artifacts. With normalization based on pre-segmentation, the inpainted outcome can be notably ameliorated. However, its reconstructive fidelity is heavily contingent on the precision of the pre-segmentation, and highly accurate segmentation of images with metal artifacts is non-trivial in actuality. In this paper, we propose a nonconvex weighted variational approach for metal artifact reduction. Specifically, in lieu of employing a binary function with zeros in the metal trace, an adaptive weight function is designed in the Radon domain, with zeros in the overlapping regions of multiple disjoint metals as well as areas of highly attenuated projections, and the inverse square root of the measured projection in other regions. A nonconvex L1−αL2 regularization term is incorporated to further enhance edge contrast, alongside a box-constraint in the image domain. Efficient first-order primal-dual algorithms, proven to be globally convergent and of low computational cost owing to the closed-form solution of all subproblems, are devised to resolve such a constrained nonconvex model. Both simulated and real experiments are conducted with comparisons to other variational algorithms, validating the superiority of the presented method. Especially in comparison to Reweighted JSR, our proposed algorithm can curtail the total computational cost to at most one-third, and for the case of inaccurate pre-segmentation, the recovery outcomes by the proposed algorithms are notably enhanced.

Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method

We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods are a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.

Switching Sudan’s Power Generation Units’ Fuel to Natural Gas Enhances the Efficiency, Durability and Lower the Cost

Sudan ranks among the top 20 Sub-Saharan countries facing electricity shortages, affecting approximately 20 million people as of 2020. Political and economic instability led to an increase in the electricity deficit, reaching over 62% in 2021, an 8% rise from 2019. Thermal power stations in Sudan, which contribute 37% of the electricity supply, encounter operational, maintenance, and fuel procurement challenges. They rely on various fuels, including liquefied petroleum gas (LPG), light diesel oil (LDO), and heavy fuel oil (HFO), with fuel procurement difficulties amid declining oil production. This study explores the potential of using natural gas instead of the conventional crude oil-based fuels listed above. Natural gas is known for its cost-effectiveness and lower greenhouse gas emissions, offering a cleaner alternative. The study involves the calculation of total gas volumes, flared gas, and operational costs for diesel and natural gas, with an evaluation of their compatibility with the operational requirements of thermal power generation units. The study reveals that upgrading existing thermal stations to a combined cycle system and transitioning to natural gas can reduce costs by 52% compared to diesel operations. The study also demonstrates that the annual gas requirement for operations is approximately 245 billion cubic feet, which is higher than the local production, indicating the need for importing liquified natural gas (LNG).

Aeroelastic reduced-order modeling for efficient static aeroelastic analysis considering geometric nonlinearity

The computational fluid dynamic/computational structural dynamic (CFD/CSD) coupling method plays an important role in static aeroelastic problems accounting for structural geometric nonlinearity. However, a huge computational burden comes from the CFD simulation, which also dominates the computational cost of the CFD/CSD coupling method. Therefore, this paper presents a reduced-order framework for steady aerodynamics to replace the CFD simulation, which is then coupled with the CSD solver for efficient static aeroelastic analysis considering geometric nonlinearity. Results first show that the proposed approach is capable of accurately predicting the surface pressure distribution of a high-aspect-ratio wing in its rigid undeformed shape at different Mach numbers and angles of attack, with at most a 2.5% relative error compared to CFD simulation. In addition, the geometrically nonlinear structure, a seminal benchmark without a tip slender body, is simulated by a three-dimensional open-source finite element solver coupled with either the CFD solver or the proposed reduced-order models (ROMs). The results demonstrate that the proposed ROM/CSD coupling approach can reflect the deformation and surface pressure distribution at static aeroelastic equilibrium with high accuracy and efficiency, thereby reducing the total computational cost at least five times while maintaining reasonable accuracy.

Machine learning tabulation of thermochemistry for turbulent dimethyl ether (DME) flames

The present work applies a machine learning tabulation methodology for the thermochemistry of dimethyl ether (DME), a typical biodiesel fuel, to accelerate the real-time computation of a large-size DME mechanism in turbulent non-premixed combustion. This approach is known as the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) method (Ding et al., 2021). The essence of the HFRD-MMLP method lies in the generation of training data using the HFRD approach, which enhances the capacity of generalisation for the reactive composition space encountered in practical turbulent combustion by using the random process to expand the training dataset from laminar flamelets. The MMLP artificial neural networks (ANNs) are then trained to predict different composition states, aiming to improve the accuracy of ANN predictions. To validate the effectiveness of the ANNs, they are initially tested on 1-D laminar flame simulations with varying strain rates. Subsequently, a test is conducted on Sandia non-premixed turbulent DME series flames D and F, with an increasing jet Reynolds Number. The results regarding species mole fractions and temperature show overall excellent agreement with the direct integration method. The HFRD-MMLP method achieves speed-up factors of over 16 and 10 for the reaction step, and total computational costs, respectively, in the LES-PDF simulation with the DME mechanism in this work, surpassing the speed-up factors of approximately 12–14, and below 5 for the reaction step, and total time costs, respectively, in previous works on GRI-1.2 mechanism for CH4 (Ding et al., 2021) and CH4/H2 (Ding et al., 2022) combustion. This indicates that the HRFD-MMLP method is highly effective for mechanisms of large size in reducing the computational cost by avoiding a significant number of highly stiff ordinary differential equations (ODEs) integration. This approach can be applied in real-time calculations of reaction source terms in methods including Direct Numerical Simulation (DNS), Probability Density Function (PDF) methods, unsteady flamelet, Conditional Moment Closure (CMC), Multiple Mapping Closure (MMC), Linear Eddy Model (LEM), Thickened Flame Model, the Partially Stirred Reactor (PaSR) method (as in OpenFOAM) and laminar flame computation.

Task offloading and trajectory scheduling for UAV-enabled MEC networks: An MADRL algorithm with prioritized experience replay

As a new network architecture, the air-ground cooperative network is a support for future 6G network to achieve ubiquitous connectivities. To effectively relieve the computational pressure of massive data in 6G wireless networks, Unmanned Aerial Vehicles (UAVs) equipped with Mobile Edge Computing (MEC) servers have become an emerging technology that provides computing resources for Mobile Devices (MDs). Due to limited on-board energy and computational capabilities, this paper investigates a multi-UAV collaborative assisted MEC architecture. The optimization problem of minimizing the total computational cost is constructed by jointly optimizing the UAVs trajectories and MDs offloading strategies scheduling. The coupling between the optimization variables and the non-convexity of the problem can make it difficult to solve directly. To address the above concerns, the non-convex optimization problem is converted into a Markov decision process. The UAVs-assisted Offloading Strategy based on Reinforcement Learning (UOS-RL) algorithm is proposed to address the convergence difficulties caused by the high-dimensional continuous action space. Furthermore, the experience data generated by agents interacting with the environment is highly differentiated due to the highly dynamic variation of the environment. Hence, a Priority Experience Replay (PER) mechanism is proposed to improve the training efficiency of the UOS-RL algorithm based on the priority of the experience data. Simulation results show that the proposed PER-UOS-RL algorithm outperforms the existing works in terms of the computational cost.

Influence Maximization in social networks using discretized Harris’ Hawks Optimization algorithm

Influence Maximization (IM) is the task of determining k optimal influential nodes in a social network to maximize the influence spread using a propagation model. IM is a prominent problem for viral marketing and helps significantly in social media advertising. Previous Greedy and Reverse Influence Sampling-based IM approaches are ineffective in real-world social networks due to their significant computational cost and execution time. Further, even heuristic approaches applied to IM generally yield minimal performance gain relative to the decreased time complexity. This presents a challenge in developing cost-effective algorithms with low execution time that can handle diverse social networks. In this paper, we propose the discretization of the nature-inspired Harris’ Hawks Optimization meta-heuristic algorithm using community structures for optimal selection of seed nodes for influence spread. In addition to Harris’ Hawks’ intelligence, we employ a neighbor scout strategy algorithm to avoid blindness and enhance the searching ability of the hawks. Further, we use a candidate nodes-based random population initialization approach, and these candidate nodes aid in accelerating the convergence process for the entire populace, reducing the total computational cost. We evaluate the efficacy of our proposed DHHO approach on eight social networks using the Independent Cascade model for information diffusion. We observe that DHHO is comparable or better than competing meta-heuristic approaches for Influence Maximization across five metrics, and performs noticeably better than competing heuristic approaches.

A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.

A Multimodel Edge Computing Offloading Framework for Deep-Learning Application Based on Bayesian Optimization

With the rapid development of Internet of Things (IoT), data generated by IoT devices are also increasing exponentially. The edge computing has alleviated the problems of limited network and transmission delay when processing tasks of IoT devices in traditional cloud computing. And with the popularity of deep learning, more and more terminal devices are embedded with AI (Artificial Intelligence) processors for higher processing capability at the edge. However, the problems of Deep Learning task offloading in heterogeneous edge computing environment have not been fully investigated. In this paper, a multi-model edge computing offloading framework is proposed, using NVIDIA Jetson edge devices (Jetson TX2, Jetson Xavier NX, and Jetson Nano) and GeForce RTX GPU servers (RTX3080 and RTX2080) to simulate the edge computing environment, and make binary computational offloading decisions for face detection tasks. We also introduce a Bayesian Optimization algorithm, namely MTPE (Modified Tree-structured Parzen Estimator), to reduce the total cost of edge computation within a time slot including response time and energy consumption, and ensure the accuracy requirements of face detection. In addition, we employ the Lyapunov model to obtain the harvesting energy between time slots to keep the energy queue stable. Experiments reveal that MTPE algorithm can achieve the globally optimal solution in fewer iterations. The total cost of multi-model edge computing framework is reduced by an average of 17.94% compared to a single-model framework. In contrast to the Double Deep Q-Network (DDQN), our proposed algorithm can decrease the computational consumption by 23.01% for obtaining the offloading decision.

Achieving high efficiency in reduced order modeling for large scale polycrystal plasticity simulations

Reduced order models for the nonlinear response of heterogeneous microstructures typically require a construction (or training) stage to build the reduced order basis. In this manuscript, an efficient model construction strategy for the eigenstrain homogenization method (EHM) is presented. The proposed strategy relies on a parallel, element-by-element, conjugate gradient solver. Near linear scaling has been achieved with respect to the number of degrees of freedom used to resolve the microstructure. Linear scaling with respect to the number of pre-analyses required to construct the reduced order model (ROM) follows from the EHM formulation. Furthermore, a parallel implementation for fast evaluation of the constructed ROM has been developed using shared memory parallelization. It has been shown that for large microstructures with ≈10,000 grains, the total computational cost of evaluating the nonlinear response of a polycrystal could be reduced by approximately an order of magnitude using 32 cores with respect to serial ROM simulation. The present methodology has been verified using an additively manufactured polycrystalline microstructure of a nickel-based superalloy, Inconel 625. The capability of the developed framework to construct a ROM for such large microstructures, as well as the ability of the ROM to predict average and local quantities of interest has been demonstrated.

RETRACTED – Reynolds-number scaling and convergence time scale in two-dimensional Rayleigh–Bénard convection

An equation for the evolution of mean kinetic energy, $E$ , in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height $L$ is derived. Assuming classical Nusselt-number scaling, $Nu \sim Ra^{1/3}$ , and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$ , we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that $\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$ , where $\tilde {\tau }$ is the non-dimensional convergence time scale. For the 3-D system, we make the estimate $\tilde {\tau } \gtrsim Ra^{1/6}$ for $Pr = 1$ . It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.