Temporal Discretization Schemes Research Articles

Inconsistencies in unstructured geometric volume-of-fluid methods for two-phase flows with high density ratios

Geometric flux-based Volume-of-Fluid (VOF) methods (Marić et al., 2020) are widely considered consistent in handling two-phase flows with high density ratios. However, although the conservation of mass and momentum is consistent for two-phase incompressible single-field Navier–Stokes equations without phase-change (Liu et al., 2023), discretization may easily introduce inconsistencies that result in very large errors or catastrophic failure. We apply the consistency conditions derived for the ρLENT unstructured Level Set/Front Tracking method (Liu et al., 2023) to flux-based geometric VOF methods (Marić et al., 2020), and implement our discretization into the plicRDF-isoAdvector geometrical VOF method (Roenby et al., 2016). We find that computing the mass flux by scaling the geometrically computed fluxed phase-specific volume can ensure equivalence between the mass conservation equation and the phase indicator (volume conservation) if consistent discretization schemes are chosen for the temporal and convective term. Based on the analysis of discretization errors, we suggest a consistent combination of the temporal discretization scheme and the interpolation scheme for the momentum convection term. We confirm the consistency by solving an auxiliary mass conservation equation with a geometrical calculation of the face-centered density (Liu et al., 2023). We prove the equivalence between these two approaches mathematically and verify and validate their numerical stability for density ratios within [1, 106] and viscosity ratios within [102, 105].

A mixed finite element spatial discretization scheme and a higher‐order accurate temporal discretization scheme for a strongly discontinuous electromagnetic interface condition in ideal sliding electrical contact problems

AbstractSliding electrical contact involves multiple conductors sliding in contact at different speeds, with current flowing through the contact surfaces. The Lagrangian method is commonly used to describe the electromagnetic field in order to overcome the trouble of convective dominance, especially in high‐speed sliding electrical contact problems. However, to maintain correct field continuity, magnetic vector potential and scalar potential taken as variables cannot be continuous simultaneously at the ideal sliding electrical contact interface. This involves a strongly discontinuous condition for variables. Further, commonly used spatial–temporal discretization algorithms are invalid, for example, the classic finite element (CFEM) framework does not allow discontinuous variables, and the backward Euler method in time domain introduces a significant interface error source associated with the velocity of relative motion. To accurately handle strongly discontinuous conditions in numerical calculations, a mixed nodal finite element scheme and a higher‐order accurate temporal discretization scheme are introduced. In this scheme, classical finite element method is performed in each subdomain, and the derivative terms at the boundary are added as new variables. The effectiveness and accuracy of the above methods are verified by comparing them with a standard solution in a two‐dimensional railgun model and analyzing the current density distribution in a three‐dimensional railgun model.

Meshless analysis of fractional diffusion-wave equations by generalized finite difference method

In this paper, a meshless generalized finite difference method (GFDM) is proposed to solve the time fractional diffusion-wave (TFDW) equations. A second-order temporal discretization scheme is developed to tackle the Caputo fractional derivative, and then spatial discretization formulas are derived by the GFDM. Theoretical accuracy and convergence of the GFDM for TFDW equations are analyzed. Numerical results verify the theoretical results and the efficiency of the method.

Numerical investigation of mixing and heat transfer in a 7.9 m JP-8 pool fire

The response of objects engulfed in, and adjacent to, large-scale pool fires is of interest in accident and safety assessments. In this study, Fuego, a low-Mach turbulent reacting flow code was used to study conjugate heat transfer in a 7.9 m diameter JP-8 pool fire. Simulations were designed to replicate past experimental measurements (Blanchat et al., 2006) of incident heat flux to three cylindrical calorimeters in and around the pool fire. Two turbulent combustion models were compared directly - the eddy dissipation concept and a more recently developed unsteady flamelet model. First and second order spatial and temporal discretization schemes were also compared to assess the performance of low-dissipation numerical operators. Heat flux predictions to the transportation size calorimeter outside the fire were within experimental uncertainties. Inside the fire, experimental measurements were higher than predicted values and may have been a consequence of soot deposition and augmented participating media radiation from soot and fuel vapor. Simulation predictions improved in cases where turbulent kinetic energy and mixing were more resolved. This work, and others referenced herein, suggest that spatial resolution on the order of 0.5–1.0 cm may be required to fully resolve fluid instabilities, vortex production between fire plumes and crosswind, soot production, and fuel-air mixing. This presents a substantial computational challenge for safety assessments of engulfed objects in fully turbulent pool fires.

Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation

In this paper, a fast element-free Galerkin (EFG) method is proposed for solving the multi-term time-fractional diffusion equation (TFDE). Through the use of the multi-term L2−1σ formula to discrete the multi-term Caputo time-fractional derivative, a fast second-order scheme is presented for the temporal discretization, and then the EFG method is used to perform the spatial discretization. The stability of the temporal discretization scheme is discussed. By verifying the fractional Grönwall inequality, the global error of the proposed EFG method for the multi-term TFDE is analyzed in theory. Numerical results validate the theoretical results and the effectiveness of the method.

Numerical Analysis and Computation of a Subgrid-Sparse-Grad-Div Stabilization Method for Incompressible Flow Problems

In this article, a subgrid-sparse-grad-div method for incompressible flow problem was proposed, which is a combination of the subgrid stabilization method and the recently proposed sparse-grad-div method. The method maintains the advantage of both methods: (i) It is robust for solving incompressible flow problem with dominance of the convection, especially when the viscosity is too small. (ii) It can keep mass conservation. Therefore, the method is very efficient for solving incompressible flow. Moreover, based on the Crank–Nicolson extrapolated scheme for temporal discretization, and mixed finite element in spatial discretization, we derive the unconditional stability and optimal convergence of the method. Finally, numerical experiments are proposed to validate the theoretical predictions and demonstrate the efficiency of the method on a test problem for incompressible flow.

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

A robust weak Galerkin finite element method for two parameter singularly perturbed parabolic problems on nonuniform meshes

In this study, we proposed a weak Galerkin finite element method (WG-FEM) for solving two-parameter singularly perturbed parabolic problems (TP-SPPPs) of convection–diffusion–reaction type on nonuniform mesh. The WG-FEM approach incorporates an interpolation operator IN and achieves optimal order convergence in an energy-like norm for continuous and fully discrete schemes. In the fully-discrete analysis, we combined WG-FEM for spatial space on a Shishkin-type mesh with the Crank–Nicolson scheme for temporal discretization on an equidistant mesh. Since the energy norm is insufficient to capture the behavior of the boundary layer functions accurately, we have derived the optimal order of convergence in a strong balanced norm. This norm exhibits an order of O(N−1lnN)k in spatial convergence and second-order convergence in time when using an L2-projection QN as an intermediate operator. We conducted numerical examples to validate the proposed method’s theoretical findings. The results of these experiments confirmed the theoretical conclusions and demonstrated the robustness of the proposed method.

Numerical Study of the Effect of Primary Nozzle Geometry on Supersonic Gas-Solid Jet of Bypass Injected Dry Powder Fire Extinguishing Device

A two-way coupled model between polydisperse particle phases with compressible gases and a density-based coupling implicit solution method, combining the third-order MUSCL with QUICK spatial discretization scheme and the second-order temporal discretization scheme, are constructed based on the discrete-phase model (DPM) and the stochastic wander model (DRWM) in the Eulerian–Lagrangian framework in conjunction with a unitary particulate source (PSIC) approach and the SST k-ω turbulence model. The accuracy of the numerical prediction method is verified using previous supersonic nozzle gas-solid two-phase flow experiments. Numerical simulation of a two-phase jet of dry powder extinguishing agent gas with pilot-type supersonic nozzle was performed to analyze the influence of geometrical parameters, such as the length ratio rL and the area ratio rA of the main nozzle on the two-phase flow field, as well as on the jet performance indexes, such as the particle mean velocity vp,a, velocity inhomogeneity Φvp, particle dispersion Ψp, particle mean acceleration ap,a, etc. By analyzing the parameters, we indicate the requirements for the combination of jet performance metrics for different flame types such as penetrating, spreading, and dispersing.

Development of a Fully-Implicit Second-Order system Thermal-Hydraulics analysis code for reactor modeling and simulations

There has been continuous advancement in numerical techniques and software engineering in the field of system thermal-hydraulics over the past decade. The current state-of-the-art involves the use of fully-implicit high-resolution schemes in conjunction with the Jacobian-Free Newton-Krylov (JFNK) method for both single-phase and two-phase flows. However, the application of these advanced schemes in system thermal-hydraulics codes is not a common practice. Given the opportunity and the need to develop a new modern system thermal–hydraulic code from scratch, there is no doubt that these recent advancements should be leveraged. In this work, we introduce a new modern object-oriented system thermal-hydraulics code named RETA. RETA is characterized by several key features, including: second-order temporal and spatial discretization schemes for both incompressible and compressible flows, as well as single-phase and two-phase flows; fully-implicit solution schemes utilizing both the Newton and PJFNK methods; and an object-oriented design of major fluid components and physical models, which allows for easy extension. A series of verification and demonstration studies are conducted to verify the implementation, demonstrate the accuracy improvement, and evaluate the performance of the new code. It is concluded that results from the second-order scheme are generally more reliable. Without the concern of a code stability issue, the second-order scheme should be the preferred and default choice for a code user or an analyst.

A Conforming Virtual Element Method for Parabolic Integro-Differential Equations

Abstract This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L 2 L^{2} projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.

Fully discrete scheme for a time-dependent Ginzburg-Landau equation in macromolecular microsphere composite hydrogels

In this article, we propose and analyze a fully discrete scheme for the time-dependent Ginzburg-Landau equation, which describes the phase field evolution of macromolecular microsphere composite hydrogels. This fully discrete scheme is a combination of the mass lumped finite element approximation for spatial discretization and the first order backward Euler scheme for temporal discretization. Besides, the convex-concave decomposition is adopted technically in the treatment of energy functional, where the concave part is treated explicitly and the convex part is treated implicitly. Moreover, we obtain the unique solvability and unconditional energy stability of the scheme. Finally, several numerical experiments are presented to illustrate the convergence order and effectiveness of the presented full-discrete scheme.

Energy-stable finite element method for a class of nonlinear fourth-order parabolic equations

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L∞(L2)- and L∞(H1)-norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results.

CFD Simulation of Low Speed Turbulent Flow Problems Using Unsteady Rans and Large Eddy Simulation Approach

The present paper focuses on the recent development of an implicit pressure-based finite volume algorithm for numerical solution of Navier Stokes equation in an inertial frame of reference for prediction of unsteady incompressible flow problems. The algorithm uses boundary-conforming, multiblock structured grid with moving boundaries, collocated variable arrangement with momentum equations resolved along cartesian directions, second order accurate spatial and temporal discretisation schemes for the convective fluxes and a pressurevelocity solution strategy. Effect of turbulence is simulated using one of the two different approaches. In the Unsteady Reynolds Averaged (URANS) approach coupled to appropriate eddy viscosity based turbulence models, the Navier Stokes (NS) equations, averaged over the whole range of turbulent length scales of the flow, are solved numerically. On the other hand, in the Large Eddy Simulation (LES) approach, the model filtered 3D NS equations are directly solved for the flow variables to resolve the large scale turbulent motions whereas the transport processes at the fine subgrid scale level only are simulated using simple algebraic turbulence model. The capabilities and limitations of both the cost-effective URANS approach and the relatively expensive but rich in physics LES approach have been demonstrated for few application problems of engineering interest.

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

A fully decoupled numerical method for Cahn–Hilliard–Navier–Stokes–Darcy equations based on auxiliary variable approaches

A fully decoupled, linearized, and unconditionally stable finite element method is developed to solve the Cahn–Hilliard–Navier–Stokes–Darcy model in the coupled free fluid region and porous medium region. By introducing two auxiliary energy variables, we derive the equivalent system that is consistent with the original system. The energy dissipation law of the proposed equivalent model is proven. To lay a solid foundation, we first present a coupled linearized time-stepping method for the reformulated system, and prove its unconditionally energy stability. In order to further improve the computational efficiency, special treatment for the interface conditions and the artificial compression approach are utilized to decouple the two subdomains and the Navier–Stokes equation. Therefore, with the discretization techniques of two existing auxiliary variable approaches, a fully decoupled and linearized numerical scheme can be developed, under the framework of a semi-implicit semi-explicit scheme for temporal discretization and Galerkin finite element method for spatial discretization. The grad-div stabilization is also employed to further improve the stability of auxiliary variable algorithm. The full discretization obeys the desired energy dissipation law without any temporal restriction. Moreover, the implementation process is discussed, including the adaptive mesh strategy to accurately capture the diffuse interface. Ample numerical experiments are performed to validate the typical features of developed numerical schemes, such as the accuracy, energy stability without restriction for time step size, and adaptive mesh refinement in space. Furthermore, we apply the proposed numerical method to simulate the shape relaxation and the Buoyancy-driven flows, which demonstrate the applicability of the proposed method.

Effect of cube spacings on the three-dimensional flow structure over an array of wall-mounted cube

The turbulent flow over an array of cubes mounted on one of the walls of a channel has been investigated using direct numerical simulation for cube spacing that ranges between 2.0 and 4.0. The Reynolds number based on the cube size and the average streamwise velocity is chosen to be 4000. The Navier–Stokes equations have been discretized using second-order spatial and temporal discretization schemes. The present investigation focuses on the flow structures and comprehensive characterization of the separated zones surrounding the cubes, as well as the associated wall-shear stress. A vortex shedding has been observed for the cube spacings of 3.0 and 4.0 without any evidence of vortex shedding for the lowest pitch of 2.0. For the two cases having pitches of 3.0 and 4.0, the presence of the unsteady separation bubbles at the cube's top and side surfaces results in a decrease in wall-shear stress. The quadrant analysis for the region close to the top surface of the cube is performed with the help of the joint probability density function, which reveals dual peaks within the recirculation bubble at the top surface of the cube for higher cube spacings. By conducting an invariant analysis of the Reynolds stress tensor for different cube spacings, we have explored the characteristic of Reynolds stress anisotropy due to the total fluctuations. The production of negative turbulent kinetic energy (TKE) is observed in different regions within the flow domain, among which the horseshoe vortex region for each cube spacing reveals its dominant presence. The physical mechanism responsible for the production of the negative TKE has also been attempted by decomposing the production term into two parts, namely, normal and shear components.

An unconditionally energy stable finite element scheme for a nonlinear fluid–fluid interaction model

Abstract In this paper, we design a decoupled scheme for solving a fluid–fluid interaction problem, which includes two Navier–Stokes equations coupled by some nonlinear interface conditions. Compared with two decoupled schemes proposed by Connors et al. (2012, Decoupled time stepping methods for fluid–fluid interaction. SIAM J. Numer. Anal., 50, 1297–1319) for the fluid–fluid interaction problem, we deal with these nonlinear interface conditions by applying explicit scheme. The presented fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization and explicit treatment for the interface conditions and the nonlinear terms. Moreover, the unconditional energy stability is established and error estimate for the fully discrete scheme is also showed. Finally, some numerical experiments are provided to verify the theoretical results, which illustrate the accuracy and efficiency of the presented scheme.

Transient analyses of wave propagations in nonhomogeneous media employing the novel finite element method with the appropriate enrichment function

An enriched finite element approach with appropriate enrichment function is proposed for the transient analyses of wave propagations in nonhomogeneous media. In present method, the traditional linear nodal interpolation functions for low order finite elements are enriched by the augmented enrichment function. For the purpose of improving the computation accuracy and reducing the numerical dispersion error, the polynomial bases and the trigonometric functions are utilized to form the enrichment functions. In addition, a simple and direct approach is introduced with the aim to tackle the possible linear dependence of the system equations. The numerical results show that the accuracy of the obtained numerical results can be markedly improved by the proposed method in comparison with the standard finite elements. What's more, it can be found that the calculated solutions from the proposed method with the appropriate temporal discretization scheme are able to monotonically converge to the exact solution in transient wave analysis. In consequence, it is possible to consistently increase the solution accuracy by directly decreasing temporal discretization step size. This valuable and important property enable the present approach to be particularly suitable to tackle the wave propagations in nonhomogeneous media.

Numerical Study of Wax Deposition from Multiphase Flow in Oil Pipelines with Heat and Mass Transfer

Wax deposition in field‐scale crude oil pipelines poses a significant challenge to the oil and gas industry, leading to reduced flow rates, increased pressure drops, and potential blockages. Understanding the mechanisms governing wax deposition is crucial for developing effective mitigation strategies. This study investigates the impact of multiphase flow conditions, including water‐in‐oil emulsion, wax precipitation kinetics, shear dispersion, and molecular diffusion, on wax deposition in field‐scale crude oil pipelines. A numerical model is developed that employs second‐order semi‐implicit temporal discretization schemes, such as Crank–Nicolson and Adams–Bashforth methods, in conjunction with a bivariate spectral collocation scheme using Chebyshev–Gauss–Lobatto grid points. The impact of various flow parameters, including Reynolds number (Re), mass Grashof number (Gr), Schmidt number (Sc), and Weber number (We), on the flow variables, wall shear stress, and heat and mass fluxes are investigated. The numerical simulations demonstrate that flow parameters significantly influence the flow behavior, wall shear stress, wall heat flux, and wall mass flux in waxy crude oil pipelines. Specifically, the aggregation of wax crystals in the pipeline decreases by at most 2.5% with increasing Reynolds number from 2.2361 to 3.1361. Conversely, it increases by at most 3.4% with increasing mass Grashof number from 5 to 11 and by at most 4.8% with increasing Weber number from 1.0 to 2.5. Furthermore, the Nusselt number increases from 1.9907 to 4.9834 with increasing Reynolds number from 2.2361 to 5.2361 and from 1.9907 to 2.0225 with increasing mass Grashof number from 5 to 20. It also increases from 1.9907 to 2.0434 with increasing Weber number from 1.0 to 2.5. The insights gained from this study can be applied to optimize pipeline design, operational parameters, and wax deposition mitigation strategies, leading to enhanced pipeline performance and reduced operational costs. The numerical model developed in this work serves as a valuable tool for simulating and predicting wax deposition behavior under various operating conditions.