Second-order Discretization Research Articles

Numerical Approach of the First Instability Appearance in Inclined Taylor–Couette System

The present study numerically investigates the three-dimensional forced and mixed convection heat transfer in an inclined Taylor–Couette system (η=0.5 and Γ=20) submitted to a radial temperature gradient. The main objective of this study is to determine the effect of the angle inclination duct on the thermal and dynamic fields. Several cases have been dealt with depending on the inclination angle to detect the critical Reynolds number in each case. The model of the conservation equations with their boundary conditions is numerically solved by the finite volume method with a second-order spatiotemporal discretization. The results show that, in forced convection, the effect of the inclination angle is inexistent on the velocity field. However, with the presence of buoyancy effects, which impact flow stability and the transition to turbulence, the inclination influences both velocity and temperature fields. It also shows that selecting the vertical position of the annulus is preferable to obtain hydrodynamic stability in mixed convection. At the same time, from the thermal point of view, it is preferable to select the horizontal position to get dynamic and thermal stability.

Numerical approximation for the MHD equations with variable density based on the Gauge-Uzawa method

In this paper, we consider the numerical approximation of incompressible magnetohydrodynamic (MHD) system with variable density. Firstly, we provide first- and second-order time discretization schemes based on the convective form of the Gauge-Uzawa method. Secondly, we prove that the proposed schemes are unconditionally stable. We also provide error estimates through rigorous theoretical analysis. Then, we construct a fully-discrete first-order scheme with finite elements in space and provide its stability result. Finally, we present some numerical experiments to validate the effectiveness of the proposed schemes. Furthermore, we also present the conserved scheme and its numerical results.

Enhancing the heat transfer in CuO-MWCNT oil hybrid nanofluid flow in a pipe

In this study, three-dimensional steady-state laminar flow simulations were conducted in a horizontal pipe using CuO and multi-walled carbon nanotubes (MWCNT) nanoparticles with engine oil as the base fluid. Various nanoparticle volume fractions were examined under a constant heat flux boundary condition applied to the pipe wall. The main goal was to assess and compare the effects of different nanoparticle volume concentrations, including CuO and MWCNT in ratios of 1:1 and 1:2, on convective heat transfer. A second-order discretisation method was employed for solving the equations, and the SIMPLE algorithm was used for pressure–velocity coupling in the CFD code. The study focused on the impact of nanoparticle volume fraction on the convective heat transfer coefficient and the Nusselt number at a Reynolds number of 750. The findings indicate that increasing the nanoparticle volume fraction enhances both the convective heat transfer coefficient and the Nusselt number, with MWCNT having a more pronounced effect compared to CuO. Specifically, adding 2% CuO increases the heat transfer coefficient by 65%, while a mixture of 1% CuO and 1% MWCNT boosts it by 75%. The thermal boundary layer thickness also grows with higher nanoparticle concentrations, with 1% CuO and 3% CuO increasing the thickness by 1.5% and 3.6%, respectively. A formula for the thermal boundary layer thickness in CuO-oil nanofluids is provided based on volume fraction, and a scale analysis of the average heat transfer coefficient confirms that the simulation results are consistent with this analysis.

A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling

We study the numerical solution of a Cahn-Hilliard/Allen-Cahn system with strong coupling through state and gradient dependent non-diagonal mobility matrices. A fully discrete approximation scheme in space and time is proposed which preserves the underlying gradient flow structure and leads to dissipation of the free-energy on the discrete level. Existence and uniqueness of the discrete solution is established and relative energy estimates are used to prove optimal convergence rates in space and time under minimal smoothness assumptions. Numerical tests are presented for illustration of the theoretical results and to demonstrate the viability of the proposed methods.

Complete quasilinear model for the acceleration-driven lower hybrid drift instability and a computational assessment of its validity.

A complete quasilinear model is derived for the electrostatic acceleration-driven lower hybrid drift instability in a uniform two-species low-beta plasma in which current is perpendicular to the background magnetic field. The model consists of coupled nonlinear velocity space diffusion equationsfor the volume-averaged ion and electron distribution functions. Each species' diffusion coefficient depends on a time-evolving spectral density of the electric-field energy per unit volume and a time-evolving dispersion relation. The dispersion relation is expressed analytically in integral form without the use of asymptotic limits and applies to arbitrary distribution functions, so long as they can be expressed as a function of one velocity coordinate, e.g., f(v_{y}) or f(v_{⊥}). The quasilinear model conserves energy and is complete in that it fully describes the evolution of the distribution functions, including resonant and nonresonant particle-wave interactions, while accounting for distribution-function-dependent mixed-complex frequencies. The quasilinear diffusion model is solved numerically and self-consistently using a Crank-Nicolson temporal discretization and a second-order finite-volume velocity-space discretization. Numerical solutions are compared to nonlinear fourth-order accurate continuum kinetic Vlasov-Poisson simulations. Evolution of electric-field energy, growth rates, distribution functions, and diffusion coefficients are shown to be in agreement with Vlasov simulations. The quasilinear model is shown to predict anomalous transport terms, like resistivity and heating, to within a factor of order unity. Discrepancies between the quasilinear model and Vlasov simulations are assessed and attributed primarily to lack of damping in the quasilinear description and to the use of unperturbed-orbit susceptibilities in the linear theory dispersion relation. The results illuminate the predictive accuracy of the quasilinear model, place approximate bounds on its validity, and provide much needed vetting of quasilinear theory's ability to predict the nonlinear state of a microturbulent plasma.

Improved lattice Boltzmann model for immiscible multicomponent systems with high viscosity gradients at the interface.

We propose alternative discretization schemes for improving the lattice Boltzmann pseudopotential model for incompressible multicomponent systems, with the purpose of modeling the flow of immiscible fluids with a large viscosity ratio. Compared to the original model of Shan-Chen [Phys. Rev. E 47, 1815 (1993)1063-651X10.1103/PhysRevE.47.1815], the present discretization schemes consider: (i) an explicit force term, (ii) a second-order discretization of the stream term, (iii) a moments-based model for the kinetic nonequilibrium distributions, and (iv) a high-order discretization of the spatial derivative terms. To verify the accuracy of the proposed model, the effects of varying the viscosity ratio as well as both fluid's viscosities on spurious currents and capillary number are investigated for the problems dealing with a static bubble, two-component Poiseuille flow, and immiscible fluid-fluid displacement. The resulting algorithm maintains the simplicity of the pseudopotential model while allowing an easy implementation for multicomponent systems. The results of the model herein proposed show improved control of the interface region and interfacial tension, relatively smaller magnitudes of spurious current values with increasing viscosity ratio, and also a significantly wider stability range with respect to the previously best results in the literature.

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.

The impact of wedge and wavy-wall combustor side walls on shock-induced mixing and combustion in a hydrogen-fuelled model scramjet combustor

The mixing and combustion processes are linked and challenging for a supersonic combustion engine. Supersonic air doesn't spend much time in the combustor, making it challenging to blend fuel with supersonic air. The expansion of shear-mixing layer with shock waves and their interactions are the primary characteristics of mixing augmentation. This paper investigates the same methodology for a model scramjet combustion chamber with wedge and wavy sides. The novel scramjet combustion models include wedge and way-wall sidewalls based on the basic scramjet model. By combining the finite volume approach with a second-order upwind discretization method, the Reynolds-averaged Navier-Stokes equations are numerically analyzed. The interplay between wedge and wavy-wall effects and free stream flows is taken into consideration by using the shear stress transport (SST) k−ω turbulence model. The effect of the wedge and wavy wall combustor has been evaluated by analysing the shock production and their interactions with the mixing layer and internal flow structure. The turbulence characteristics are also examined to analyse the degree of flow disturbance for wedge and wavy-wall combustion. The wavy wall combustor model results in a 14.5% increase in turbulence intensity, which raises air-fuel mixing and there by improves mixing and combustion efficiency by 12.59% and 11.23%, compared to the base wedge type combustor.

Quasi-Newton positivity-preserving scheme for the Spalart-Allmaras turbulence model using unstructured grids

A new implicit method for solving the Spalart-Allmaras (SA) turbulence model is proposed using a formal second-order upwind scheme on unstructured grids for steady-state flows. The mean-flow set of equations and the SA model equation are solved in a loosely coupled manner. While the Jacobian of the mean-flow equations is derived from a low-order approximation of the residual (i.e. the defect-correction approach), the SA Jacobian is obtained by modifying a direct linearization of the second-order residual. The modified Jacobian is designed to form an M-matrix without the artificial time derivative commonly used for steady-state problems. Namely, no time step is involved in solving the SA model equation. Thanks to a special design of the M-matrix, the positivity of the SA model working variable and the convergence of the SA model linearized problem (fixed-point iterative problem) is guaranteed. To further enhance the overall stability of the flow solver, the Runge-Kutta implicit algorithm is employed, allowing CFL numbers as high as one to two thousand for the mean-flow equations. A second-order upwind scheme is achieved with a least-square method using a limiter to suppress oscillation in the solution. However, second-order upwind scheme discretization of turbulence models may lack robustness, especially when using unstructured grids. The limiter can hinder the convergence of the non-linear iterations, especially of the turbulence model. A square root transformation is applied to the dependent variable of a baseline SA model to improve the non-linear convergence characteristics. Three variants of the SA model are implemented and tested together with two limiters. Numerical simulations of a broad range of aerodynamic applications are conducted. The transformed variant exhibits the least sensitivity to the limiter type.

Comparison of heat transfer characteristics of a heat exchanger with straight helical tube and a heat exchanger with coiled flow reverser

Vertical shell and coiled tube heat exchangers are an important aspect of numerous industrial processes and are renowned for their high heat transfer efficiency, versatility, compact design, easy manufacturability and maintenance. As a novel technique, a coiled flow reverser was used to enhance the thermal performance of shell and coiled tube heat exchangers. Therefore, the impact of employing a coiled flow reverser on the heat transfer characteristics and pressure drop in a counter-current shell and tube heat exchanger is investigated using both numerical and experimental approaches. The results are compared to a straight helical coiled tube heat exchanger. The numerical simulations were carried out by employing computational fluid dynamics (CFD) techniques, and the SIMPLE scheme was implemented through the second-order upwind discretization method. The laboratory device used for experimentation features a shell with an inner diameter of 15 cm, and the coil pitch and coil diameter of both coiled tubes are set at 3 cm and 12.1 cm, respectively. Results reveal that the overall heat transfer coefficient of the heat exchanger with a coiled flow reverser is 39.5 % to 49.9 % higher than that of the heat exchanger with a straight helical tube. The effectiveness ratio of the heat exchanger with coiled flow reverser to the straight helical tube heat exchanger ranges from 1.254 to 1.379. Optimal results are observed at a cold flow rate of 6 LPM and a hot flow rate of 1 LPM for both heat exchangers. Additionally, the coiled flow reverser enhances heat transfer performance at the cost of increased pressure drop. Numerical results are in good agreement with experimental findings, establishing the reliability of the study.

Control of large amplitude limit cycle of a multi-dimensional nonlinear dynamic system of a composite cantilever beam

For the first time, a control strategy based on Fuzzy Sliding Mode Control is implemented in the control of a large amplitude limit cycle of a composite cantilever beam in a multi-dimensional nonlinear form. In the dynamic model establishment of the investigated structure, the higher-order shearing effect is applied, as well as the second-order discretization. Numerical simulation demonstrates that a multi-dimensional nonlinear dynamic system of the investigated structure is demanded for accurate estimation of large amplitude limit cycle responses. Therefore, a control strategy is employed to effectively suppress such responses of the beam in multi-dimensional nonlinear form.

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

DUGKS-GPU: An efficient parallel GPU code for 3D turbulent flow simulations using Discrete Unified Gas Kinetic Scheme

This paper presents a parallel implementation of the Discrete Unified Gas Kinetic Scheme (DUGKS) on the GPU system using the CUDA Fortran and CUDA C++ programming languages. Firstly, we conducted an extensive revision of our original CPU-based code, resulting in a threefold decrease in memory usage. This new implementation is also paired with a novel approach to compute cell face flux using trilinear interpolation. It is shown analytically that the interpolation-based approach to flux calculation is more accurate compared to the one used in the original DUGKS. The initial simulation results using this new approach suggest that trilinear interpolation can reduce numerical errors on a coarse mesh. For example, in the case of the decaying Taylor-Green vortex flow at a 1283 mesh resolution, the relative numerical error in the energy dissipation rate at t⁎=2, using the spectral simulation result as the benchmark, is approximately 30% lower than that of the original implementation. The improved GPU DUGKS method is applied to laminar and turbulent flows in periodic and wall-bounded boundary configurations. A performance comparison of the GPU implementation is also presented and compared to the previous CPU implementation. A maximum speedup of 7.64x was achieved on a desktop-level GPU compared to a 32-core CPU. The strong scaling test, conducted on an eight-GPU node, demonstrated the efficient utilization of available multiple GPU resources by the code. Program summaryProgram Title: DUGKS-GPUCPC Library link to program files:https://doi.org/10.17632/yykv5s9g2n.1Developer's repository link:https://github.com/kairzhan/DUGKS-GPUCode Ocean capsule:https://codeocean.com/capsule/3b1e4f74-cdd3-4781-8923-8514d5923dfb/Licensing provisions: GNU General Public License 3Programming language: CUDA C++, CUDA FortranSupplementary material:Nature of problem: DUGKS-GPU is an advanced numerical code designed to accelerate the simulation of turbulent fluid flows through the application of the kinetic method known as the Discrete Unified Gas Kinetic Scheme (DUGKS). This computational tool comprises a CUDA Fortran version, tailored for a single GPU, as well as a CUDA C++ version developed for multi-GPU platforms.Solution method: The code employs a second-order finite-volume discretization of the discrete velocity Boltzmann equation with the BGK collision model. It can be used to simulate small to medium-scale problems on modern multi-GPU platforms with CUDA architecture.

Analysis on a High Accuracy Fully Implicit Solution for Strong Nonlinear Diffusion Problem - Convergence, Stability, and Uniqueness

Some fundamental properties are analyzed for a fully implicit finite difference (FIFD) solution of conservative strong nonlinear diffusion problem. The scheme is constructed by combining a second-order backward difference temporal discretization and a central finite difference spatial discretization, and therefore highly nonlinear. Theoretical analysis is carried out under a coercive condition according with the diffusion feature of the strong nonlinear diffusion model. Benefiting from the boundedness estimates of the FIFD solution itself and its first- and second-order spatial difference quotients, some novel argument techniques are developed to overcome the difficulties coming from the nonlinear approximation for the strong nonlinear conservative diffusion operator. Consequently, it is proved rigorously that the FIFD scheme is unconditionally stable, its solution is unique and convergent to the exact solution of the original problem with second-order space-time accuracy. Numerical examples are provided to confirm its advantages on precision and efficiency over its first-order time accurate counterpoint.

3D numerical simulation of frequency-domain elastic wave equation with a compact second-order scheme

Finite-difference frequency-domain (FDFD) modeling has gained popularity in recent years. However, its main drawback lies in the need to solve large sparse linear systems, which is computationally expensive, especially for 3D elastic wave simulations. The existing elastic wave optimal schemes improve the simulation accuracy at the cost of using large-sized stencils or high-order finite-difference (FD) operators, and most of them only consider equal grid spacing and solid media. To address these issues, we have proposed a compact second-order scheme for simulating 3D elastic wave propagation. Starting from the second-order FD discretization of the first-order velocity-stress expression, the compact discrete scheme of the second-order elastic wave equation is obtained by using a parsimonious staggered-grid strategy to eliminate the stress variables. In this way, the scheme preserves the ability to handle high-contrast velocity and liquid-solid media, and has a more compact and sparse impedance matrix than existing schemes, which helps save computational costs. We also use an antilumped mass strategy to further improve the modeling accuracy. Dispersion analysis indicates that our compact second-order scheme has better accuracy than the classic second-order scheme, and even better accuracy than the average derivative 27-point scheme at large Poisson's ratio. Compared with the existing 3D elastic wave schemes, the compact second-order scheme has advantages in computational cost for the same grid. Finally, we employ these schemes to implement 3D numerical simulations, validating the effectiveness of our proposed scheme.

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.

A modification of explicit time integrator scheme for unsteady power-law nanofluid flow over the moving sheets

This paper introduces an exponential time integrator scheme for solving partial differential equations in time, specifically addressing the scalar time-dependent convection-diffusion equation. The proposed second-order accurate scheme is demonstrated to be stable. It is applied to analyze the heat and mass transfer mixed convective flow of power-law nanofluid over flat and oscillatory sheets. The governing equations are transformed into a dimensionless set of partial differential equations, with the continuity equation discretized using a first-order scheme. The proposed time integrator scheme is employed in the time direction, complemented by second-order central discretization in the space direction for the momentum, energy, and nanoparticle volume fraction equations. Quantitative results indicate intriguing trends, indicating that an increase in the Prandtl number and thermophoresis parameter leads to a decrease in the local Nusselt number. This modified time integrator is a valuable tool for exploring the dynamics of unsteady power-law nanofluid flow over moving sheets across various scenarios. Its versatility extends to the examination of unstable fluid flows. This work improves engineering and technological design and operation in nanofluid dynamics. Improving numerical simulations’ precision and computational efficiency deepens our comprehension of fundamental physics, yielding helpful information for enhancing systems that rely on nanofluids.

A Steady-State-Preserving Numerical Scheme for One-Dimensional Blood Flow Model

In this work, an entropy-stable and well-balanced numerical scheme for a one-dimensional blood flow model is presented. Such a scheme was obtained from an explicit entropy-conservative flux along with a second-order discretisation of the source term by using centred finite differences. We prove that the scheme is entropy-stable and preserves steady-state solutions. In addition, some numerical examples are included to test the performance of the proposed scheme.

State-Feedback Control Design for Polynomial Discrete-Time Systems obtained via Second-Order Runge-Kutta Discretization

This paper addresses the state-feedback control problem for the class of state-polynomial discrete-time systems. The continuous-time polynomial nonlinear model is discretized by the second-order Runge-Kutta method. The Lyapunov theory and the exponential stability were employed to derive the conditions. The sum of squares formulation was used to check the constraints. Two approaches are presented, the first makes use of the Lyapunov function to recover the gain matrices. While the second formulation allows the design of rational state feedback control gains. We evaluated the impact of the step size used in the discretization process in the results. Numerical experiments were used to illustrate the potential of the proposed technique.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.