Second-order Finite Volume Method Research Articles

Numerical Approximation of a Nonequilibrium Model of Gradient Elution Chromatography Considering Different Functional Relationships between Model Parameters and Solvent Composition.

In this paper, a rigorous theoretical study is conducted to analyze the influence of varying solvent compositions on the retention characteristics of elution profiles within a fixed-bed liquid chromatographic column. In gradient chromatography, the propagation speed of elution profiles is manipulated through a progressive variation in the mobile-phase composition. Consequently, enhanced separation of the mixture components can be achieved together with a reduction in the requisite recycling times for subsequent injections. In other words, both the efficiency and the selectivity of the column can be enhanced. The lumped kinetic model coupled with the convection-diffusion equation for the volume fraction of the solvent is applied to simulate the process. The resulting nonlinear model equations are numerically solved by applying a semidiscrete second-order finite-volume method. The numerical solutions are utilized to quantify the effects of gradient starting and ending times, solvent composition, solvent strength parameters, and gradient slope on the concentration profiles. Additionally, temporal numerical moments are plotted versus the starting and ending times of the gradient, and standard performance criteria are presented for evaluating the process performance. The outcomes of this investigation will contribute to further enhancements in gradient elution chromatography.

A solver for subsonic flow around airfoils based on physics-informed neural networks and mesh transformation

Physics-informed neural networks (PINNs) have recently become a new popular method for solving forward and inverse problems governed by partial differential equations. However, in the flow around airfoils, the fluid is greatly accelerated near the leading edge, resulting in a local sharper transition, which is difficult to capture by PINNs. Therefore, PINNs are still rarely used to solve the flow around airfoils. In this study, we combine physical-informed neural networks with mesh transformation, using a neural network to learn the flow in the uniform computational space instead of physical space. Mesh transformation avoids the network from capturing the local sharper transition and learning flow with internal boundary (wall boundary). We successfully solve inviscid flow and provide an open-source subsonic flow solver for arbitrary airfoils. Our results show that the solver exhibits higher-order attributes, achieving nearly an order of magnitude error reduction over second-order finite volume method (FVM) on very sparse meshes. Limited by the learning ability and optimization difficulties of the neural network, the accuracy of this solver will not improve significantly with mesh refinement. Nevertheless, it achieves comparable accuracy and efficiency to second-order FVM on fine meshes. Finally, we highlight the significant advantage of the solver in solving parametric problems, as it can efficiently obtain solutions in the continuous parameter space about the angle of attack.

A conservative sharp interface method for two-dimensional incompressible two-phase flows with phase change

A conservative sharp interface method is proposed in this work to simulate two-dimensional/axisymmetric incompressible two-phase flows with phase change. In this method, we use the cut cell method to generate unstructured meshes near the interface, of which the cell edges overlap with the interface at each time step. On such mesh, the mass and heat transfer during phase change and all the jump conditions can be incorporated into the calculation of fluxes at the cell edges, to ensure that they are strictly satisfied at the interface in a sharp manner. The governing equations, including the incompressible Navier–Stokes equations, heat equation, and vapor mass fraction equation, are discretized by a second-order finite volume method in the arbitrary Lagrangian–Eulerian framework. To well couple the mass, heat, momentum, and interface evolution, the solution procedure is carefully designed and performed with several techniques. In such a way, the sharp discontinuity of the velocity, stress, temperature gradient, and vapor fraction, caused by the mass/heat transfer during phase change, can be simulated accurately and robustly. The performance of this method is systematically examined by cases of phase change at or below the saturated temperature, including vapor bubble in superheated liquid, film boiling, droplet evaporation at different relative humidity conditions, droplet evaporation under gravity, and droplet evaporation under forced convection. The applicability of the present method for incompressible two-phase flows with phase change is well demonstrated by comparing the numerical results with the benchmark, theoretical or experimental ones.

A Finite Volume Method for a 2D Dam-Break Simulation on a Wet Bed Using a Modified HLLC Scheme

This study proposes a numerical model for depth-averaged Reynolds equations (shallow-water equations) to investigate a dam-break problem, based upon a two-dimensional (2D) second-order upwind cell-centre finite volume method. The transportation terms were modelled using a modified approximate HLLC Riemann solver with the first-order accuracy. The proposed 2D model was assessed and validated through experimental data and analytical solutions for several dam-break cases on a wet and dry bed. The results showed that the error values of the model are lower than those of existing numerical methods at different points. Our findings also revealed that the dimensionless error parameters decrease as the wave propagates downstream. In general, the new model can model the dam-break problem and captures the shock wave superbly.

A Second-Order Finite Volume Method for Field-Scale Reservoir Simulation

Subsurface reservoirs are large complex systems. Reservoir flow models are defined on complex grids that follow geology with relatively large block sizes to make consistent simulations feasible. Reservoir engineers rely on established reservoir simulation software to model fluid flow. Nevertheless, fluid front position inaccuracies and front smearing on large grids may cause significant errors and make it hard to predict hydrocarbon production efficiency. We investigate higher-order methods that reduce these undesired effects without refining the grid, thus making reservoir simulation more accurate and robust. For this paper, we implemented a second-order finite volume method with linear programming (LP) reconstruction in the open-source industry-grade reservoir simulator OPM Flow (part of the open porous media initiative, OPM). We benchmark it against the first-order method on full-scale cases with standard coarse and refined grids. We prepared open refined-grid models of a synthetic reservoir with an unstructured grid and refined Norne field example. Our results confirm that the LP method predicts front positions as accurately as the first-order method on the refined grid for problems dominated by transport. These include the water alternating gas scenario on the synthetic reservoir and piston-type injection on the Norne field. Moreover, we study the gains from the LP method for CO_{2} injection problems on the Norne field with full multi-phase complexity beyond transport. We observe the relevant difference between the first- and the second-order methods in these cases. However, in some configurations, the reservoir complexity overshadows the gains from the second-order methods.

Transient solidification and melting numerical simulation of lauric acid PCM filled stepped solar still basin used in water desalination process

It is crucial to produce potable water from salt water in arid places using a stepped solar still desalination process. This work presents the temperature distribution and phase transformation of lauric acid PCM in inclined stepped solar still desalination system. A second-order finite volume method is employed for discretizing the two-dimensional geometry, while energy, continuity, and momentum equations are solved and then coupled using the second-order PRESTO algorithm. The solidification and melting CFD models have been used to capture the melting stages of lauric acid PCM. Four steps with an inclination of 30° to ground and two different cases with saline water levels of 10 mm and 40 mm have been reported. The results reveal that the temperature and mass fraction of lauric acid PCM strongly depends on the saline water level at each step, the number of steps, the angle of inclination, and the location of the solar still step. The maximum percentage increase in PCM melting fraction is 50%, and a PCM temperature rise of 4 °C is observed in the case of SWL-max compared to SWL-min. The total enthalpy of lauric acid PCM is increased by 30% for the SWL-max water level compared to the SWL-min water level.

Finite Volume Method for Transient Pipe Flow with an Air Cushion Surge Chamber Considering Unsteady Friction and Experimental Validation

In various water transmission systems such as long-distance water transfer projects and hydropower stations, accurate simulation of water hammer is extremely important for safe and stable operation and the realization of intelligent operations. Previous water hammer calculations usually consider only steady-state friction, underestimating the decay of transient pressure. A second-order Finite Volume Method (FVM) considering the effect of unsteady friction factor is developed to simulate the water hammer and the dynamic behavior of air cushion surge chamber in a water pipeline system, while an experimental pipe system is conducted to validate the proposed numerical model. Two unsteady friction models, Brunone and TVB models, were incorporated into the water hammer equations, which are solved by the MUSCL–Hancock method. One virtual boundary method was proposed to realize the FVM simulation of Air Cushion Surge Chamber. Comparisons with water hammer experimental results show that, while the steady friction model only accurately predicts the first pressure peak, it seriously underestimates pressure attenuation in later stages. Incorporating an unsteady friction factor can better predict the entire pressure attenuation process; in particular, the TVB unsteady friction model more accurately reproduces the pressure peaks and the whole pressure oscillation periods. For water pipeline systems with an air cushion surge chamber, energy attenuation of the elastic pipe water hammer is primarily due to pipe friction and the air cushion. The experimental results with the air cushion surge chamber demonstrate that the proposed FVM model with the TVB unsteady friction model and the air chamber polytropic exponent near 1.0 can well reproduce the experimental pressure oscillations.

A three-dimensional high-order flux reconstruction lattice boltzmann flux solver for incompressible laminar and turbulent flows

A three-dimensional (3D) high-order flux reconstruction lattice Boltzmann flux solver (FR-LBFS) is developed in this paper for accurate and efficient simulations of incompressible laminar and turbulent flows. The original lattice Boltzmann flux solver (LBFS) is a second-order finite volume method (FVM) which combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. But it is not convenient to construct high-order FVM, and the compactness is always a bottleneck. The present work separates LBFS from FVM and combines LBFS with high-order FR scheme. Thus, the FR-LBFS inherits the superiority of LBFS and FR and shares the following attractive features: (i) a fully-explicit arbitrary order compact method, (ii) weakly compressible (WC) model for unsteady incompressible flows and (iii) evaluating inviscid and viscous fluxes simultaneously and no particular technique requirement, such as local discontinuous Galerkin method (LDG), for viscous term discretization. To validate the current method, various 3D incompressible laminar isothermal and thermal numerical experiments are presented first. Good agreements are achieved between the current results and the previous experimental and numerical data whilst the present high-order solver adopt coarse meshes. Furthermore, the ability of the proposed method to perform accurate and stable computations of incompressible decaying homogeneous isotropic turbulent flows and turbulent channel flows with different mesh resolutions and accuracy orders are investigated. The results show that the present 3D FR-LBFS is a promising tool for under-resolved or implicit large eddy simulation (ILES) of incompressible turbulent flows.

Second-order accurate implicit finite volume method for two-dimensional modeling of PFAS transport in unsaturated porous media with variable surface tension

Per- and polyfluoroalkyl substances (PFASs) have become emerging contaminants of critical concern. Comprehensive understanding of the transport and fate of PFAS in the vadose-zone, a type of water-unsaturated porous media, is key to determination of the risks of the PFAS contamination in the subsurface and to the development of the effective remediation strategies. Accurate modeling of the PFAS transport in the unsaturated porous media is still a challenge due to the variable surface tension induced by the adsorption of PFAS to the air-water interfaces. In an effort to address this challenge, we propose a multidimensional modeling framework for the transient PFAS transport in the unsaturated porous media based on the second order accuracy finite volume method. In the modeling, the adsorption of PFAS to the solid surfaces and to the air-water interfaces is described by the two-domain sorption kinetics model, i.e., both the instantaneous and the rate limited PFAS adsorptions are taken into account. The diffusive and convective terms in the governing equations for the PFAS transport and the water flow are discretized by the central difference and the quadratic upstream interpolation for convective kinetics schemes, respectively. We investigate the effects of the convergent criteria, coupling method, and variation of the surface tension on the average and the local PFAS concentration and water content in the computational domain. We find that the convergent criteria should be chosen carefully so as to get the accurate results. The differences between the different coupling methods are affected by the boundary conditions. The variation in the surface tension due to the variation of the PFAS concentration cannot be neglected. The dimensionless parameters relevant to the properties of porous media as well as the relation between the surface tension and the PFAS concentration play an important role in transport of PFAS in the unsaturated porous media. The effects of the Péclet number, Damköhler values, and fraction of instantaneous sorption are not significant in the ranges studied in this work. These findings provide a better understanding of transport of PFAS in the vadose zone.

A three-dimensional solver for simulating detonation on curvilinear adaptive meshes

A generic solver in a parallel Cartesian adaptive mesh refinement framework is extended to simulate detonations on three-dimensional structured curvilinear meshes. A second-order accurate finite volume method is used with grid-aligned Riemann solvers for thermally perfect gas mixtures. Detailed, multi-step chemical kinetic mechanisms are employed and numerically incorporated with a splitting approach. The adaptive mesh refinement technique is applied to a mapped mesh using modified prolongation and restriction operators. The flux along the coarse-fine interface is considered in a correction procedure to ensure the conservation of the solver. The numerical accuracy, conservation and robustness of the simulations are verified and validated with suitable benchmark tests. The new solver is then used to simulate detonation problems in non-Cartesian geometries. A simulation is conducted of the three-dimensional detonation propagation in a 90-degree pipe bend. A detonation in a round tube is also simulated in a Galilean frame of reference. Both a rectangular mode and a spinning mode are observed in the simulations. In addition, the fundamental problem of detonation wave/boundary layer interaction is studied. The results show that the new solver can simulate high-speed reactive flows efficiently by the combined use of a curvilinear mapping with mesh adaptation.

The effect of conical shell and converging/diverging tube on the charging performance of shell and tube latent heat thermal energy storage system

Latent heat storage is a promising method to overcome the problem of the intermittent nature of renewable sources. However, the main challenge in utilizing phase change materials is their poor thermal conductivity which leads to the slow charging and discharging of latent heat thermal energy storage tanks. Geometric optimization is a passive method to enhance the charging and discharging rate. In this paper, the geometric optimization of a shell and tube storage tank is studied numerically. Different configurations, including conical shell and tube, cylindrical shell and diffuser, cylindrical shell and nozzle, and a combination of conical shell and diffuser with different tilting angles is investigated. The height of all cases is assumed to be a constant number. The melting process is modeled by the enthalpy-porosity method, and the Boussinesq approximation is employed to consider the effect of natural convection. The governing equations are solved by a spatially and temporally second-order finite volume method. It is found that the main determining parameter is the cross-sectional area ratio of the phase change material at the top to its value at the bottom of the tank. By increasing this ratio, the effect of conductive heat transfer is attenuated, but natural convection is augmented. Since the charging rate is affected by these two heat transfer mechanisms, an optimum value of around 10 is obtained for this shape factor which results in an increased storage rate of 25 % compared to the case of cylindrical shell and tube storage tank. The optimum shape factor can be obtained either by a conical shell and tube or by combining a conical shell and a diffuser.

Second-order accurate finite volume method for G-equation on polyhedral meshes

Second-order accurate finite volume method for G-equation on polyhedral meshes

Finite Volume Method for Modeling the Load-Rejection Process of a Hydropower Plant with an Air Cushion Surge Chamber

The pipe systems of hydropower plants are complex and feature special pipe types and various devices. When the Method of Characteristics (MOC) is used, interpolation or wave velocity adjustment is required, which may introduce calculation errors. The second-order Finite Volume Method (FVM) was presented to simulate water hammer and the load-rejection process of a hydropower plant with an air cushion surge chamber, which has rarely been considered before. First, the governing equations were discretized by FVM and the flux was calculated by a Riemann solver. A MINMOD slope limiter was introduced to avoid false oscillation caused by data reconstruction. The virtual boundary strategy was proposed to simply and effectively handle the complicated boundary problems between the pipe and the various devices, and to unify the internal pipeline and boundary calculations. FVM results were compared with MOC results, exact solutions, and measured values, and the sensitivity analysis was conducted. When the Courant number was equal to 1, the results of FVM and MOC were consistent with the exact solution. When the Courant number was less than 1, compared with MOC, the second-order FVM results were more accurate with less numerical dissipation. As the Courant number gradually decreased, the second-order FVM simulations were more stable. For the given numerical accuracy, second-order FVM had higher computational efficiency. The simulations of load rejection showed that compared with the MOC results, the second-order FVM calculations were closer to the measured values. For hydropower plants with complex pipe systems, wave velocity or the Courant number should be adjusted during MOC calculation, resulting in calculation error, and the error value is related to the parameters of the air cushion surge chamber (initial water depth, air cushion height, etc.). The second-order FVM can more accurately, stably, and efficiently simulate the load-rejection process of hydropower plants compared with MOC.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Lid-Driven Square Cavity Flow: A Benchmark Solution With an 8192 × 8192 Grid

Abstract The incompressible steady-state fluid flow inside a lid-driven square cavity was simulated using the mass conservation and Navier–Stokes equations. This system of equations is solved for Reynolds numbers of up to 10,000 to the accuracy of the computational machine round-off error. The computational model used was the second-order accurate finite volume (FV) method. A stable solution is obtained using the iterative multigrid methodology with 8192 × 8192 volumes, while degree-10 interpolation and Richardson extrapolation were used to reduce the discretization error. The solution vector comprised five entries of velocities, pressure, and location. For comparison purposes, 65 different variables of interest were chosen, such as velocity profile, its extremum values and location, and extremum values and location of the stream function. The discretization error for each variable of interest was estimated using two types of estimators and their apparent order of accuracy. The variations of the 11 selected variables are shown across 38 Reynolds number values between 0.0001 and 10,000. In this study, we provide a more accurate determination of the Reynolds number value at which the upper secondary vortex appears. The results of this study were compared with those of several other studies in the literature. The current solution methodology was observed to produce the most accurate solution till date for a wide range of Reynolds numbers.

Effect of Geometry Design on Mixing Performance of Newtonian Fluids using Helical Overlapped Mixer Elements in Kenics Static Mixer

The laminar flow pattern and mixing behavior of incompressible Newtonian fluids in different modified mixer ‎configurations were numerically investigated using Computational Fluid Dynamics (CFD) simulations in the range of ‎Re=0.15-100. The governing equations were solved by ANSYS Fluent 14 using the second-order finite volume method ‎‎(FVM) and the SIMPLE algorithm scheme. The computational model is assessed by comparing the predicted pressure ‎drop results to empirical correlations in the literature. The effects of incorporated helical overlapped mixer elements and ‎the diameter aspect ratio (C) on the mixing efficiency for different mixer geometries were examined and evaluated by ‎characteristics measures of Intensity of Segregation (IOS), pressure drop, extensional efficiency, and G-factor. The ‎performance of new modified mixers is evaluated via comparison with the standard industrial Kenics static mixer. The ‎static mixers with modified internal geometry achieved fast mixing and better mixing quality than the Kenics mixer. ‎Besides, an increase in diameter aspect ratio C benefited from a decrease in pressure drop within the static. The ‎modified mixer: C=1.5 was found to have the highest mixing efficiency, concerning short mixing length with marginally ‎higher pressure drop than the other mixers. In contrast, the mixer: C=2 is the most efficient based on low pressure drop ‎and energy requirement with slightly greater mixing length‎‎.

Performance evaluation of standard second-order finite volume method for DNS solution of turbulent channel flow

Performance evaluation of standard second-order finite volume method for DNS solution of turbulent channel flow

A solver for simulating shock-induced combustion on curvilinear adaptive meshes

A generic solver in a structured Cartesian adaptive mesh refinement framework is extended to simulate unsteady shock-induced combustion problems on a structured curvilinear mesh. A second-order accurate finite volume method is used with a grid-aligned Riemann solver for inviscid thermally perfect gas mixtures. To solve these reactive problems, detailed chemical kinetic mechanisms are employed with a splitting approach. The prolongation and restriction operators are modified to implement the adaptive mesh refinement algorithm on a mapped mesh. The developed solver is verified with several benchmark tests and is then used to simulate unsteady shock-induced combustion. The results show that the computed stand-off distance of waves and oscillation frequencies of mass fraction of products observed at the stagnation point are in good agreement with the results from experiments.

Monotonicity correction for second order element finite volume methods of anisotropic diffusion problems

We apply the monotonicity correction to second-order element finite volume methods, which include quadratic element on triangular meshes and biquadratic element on quadrilateral meshes, for anisotropic diffusion problems, and obtain a second-order monotone finite volume scheme. The main ideas of monotonicity correction are as follows: When we formulate the second-order element finite volume schemes, we need to calculate line integrals on the boundary line segment of the dual element, and regard these line integrals as numerical fluxes, which are in the form of multi-point flux. Among these numerical fluxes, we can separate a two-point flux structure on both sides of the line segment. Therefore, we can apply nonlinear monotone correction to these numerical fluxes, then we obtain a corrected second order element finite volume schemes whose stiffness matrix is monotone matrix, so the first high order unconditional positivity-preserving finite volume schemes are obtained by us. Moreover, we analyze the truncation error of the corrected numerical fluxes. Furthermore, for the time dependent problem, we also apply monotone correction to time derivative term. Numerical experiments show that the corrected second order element finite volume schemes have monotonicity, and maintain the convergence order of the original schemes, which are second order in H1 norm and third order in L2 norm.

Power performance enhancement of vertical axis wind turbines by a novel gurney flap design

PurposeNowadays flaps and winglets are one of the main mechanisms to increase airfoil efficiency. This study aims to investigate the power performance of vertical axis wind turbines (VAWT) that are equipped with diverse gurney flaps. This study could play a crucial role in the design of the VAWT in the future.Design/methodology/approachIn this paper, the two-dimensional computational fluid dynamics simulation is used. The second-order finite volume method is used for the discretization of the governing equations.FindingsThe results show that the gurney flap enhances the power coefficient at the low range of tip speed ratio (TSR). When an angled and standard gurney flap case has the same aerodynamic performance, an angled gurney flap case has a lower hinge moment on the junction of airfoil and gurney flap which shows the structural excellence of this case. In all gurney flap cases, the power coefficient increases by an average of 20% at the TSR range of 0.6 to 1.8. The gurney flap cases do not perform well at the high TSR range and the results show a lower amount of power coefficient compare to the clean airfoil.Originality/valueThe angled gurney flap which has the structural advantage and is deployed to the pressure side of the airfoil improves the efficiency of VAWT at the low and medium range of TSR. This study recommends using a controllable gurney flap which could be deployed at a certain amount of TSR.