Second-order Spatial Discretization Research Articles

A generalized finite difference method for solving elliptic interface problems with non-homogeneous jump conditions on surfaces

A meshless generalized finite difference method is presented to solve elliptic interface problems with non-homogeneous jump conditions on surfaces. The innovation is that based on a splitting treatment of the interface problem, the generalized finite difference method combined with the local tangential lifting approach is developed for the second-order spatial discretization with low complexity and small stencil. The generalized finite difference method considered is based on Taylor expansion and weighted moving least squares approximation. Under the local tangential lifting framework, the numerical scheme formed by the generalized finite difference method on the tangent plane is regarded as that for the original surface problem. The biggest advantage for combining two methods is that the local tangential lifting framework can reduce the complexity of the discrete process of generalized finite difference method on the surface and resulting relatively small stencil. Several numerical examples are presented to validate the convergence of the proposed method. Numerical results suggest that the proposed approach is effective to the surface interface problem with non-homogeneous jump conditions.

On the conservative phase-field method with the N-component incompressible flows

This paper presents a conservative Allen–Cahn model coupled with the incompressible Navier–Stokes equation for tracking the interface with the N-component immiscible fluids system. The proposed conservative phase-field model can track the interface with large deformation in divergence-free velocity fields. The erroneous estimation of the normal vector is a challenging numerical issue for the interface capturing due to the appearance of spurious oscillations. The improved phase-field-based method combines the nonlinear preprocessing operation guided by the level-set method with local artificial viscosity stabilization to improve the computation of the discrete normal vector. The interfaces between different immiscible components are replaced by the transition region with finite thickness in the continuous phase field. The surface tension effects are represented with the continuous surface tension force in the system, which is not limited by the number of components. The third-order Runge-Kutta time discretization and second-order spatial discretization are applied for the multi-component system. To eliminate the spurious oscillations caused by discontinuous and steep gradient for capturing the shocks and sharp interfaces, we apply the third-order weighted essentially non-oscillatory method for the advection term. Several quantitative results of numerical tests, such as error estimation with the proposed method, comparative tests with different methods, and convergence rate for classical benchmark test, have been performed to illustrate that our method works well for the interface tracing issue with high numerical accuracy. In addition, various representative qualitative tests have been presented to demonstrate the applicability of our method.

Comparing the performance of RANS turbulence models between different cavity flow benchmarks

To evaluate the performance of RANS turbulence models, this study compares four different cavity flow benchmarks using the prevailing two-equation turbulence models for indoor airflows, namely the standard and RNG k-ε and the standard and SST k-ω models. A cavity flow consists of one air inlet and one outlet slot. The inlet slot is positioned on the upper left corner of the cavity, whereas the outlet slot is located in the lower right. This cavity flow is representative of mixing ventilation. These four cavity benchmarks differ by their geometry (i.e., the aspect ratio of the room), flow regime and whether the flow is isothermal or not. Measurements of the air velocity and temperature in these benchmarks are used to evaluate the accuracy of the RANS turbulence models. Many existing studies have investigated the airflow and heat transfer over these benchmarks. However, the numerical methods and other relevant CFD parameters are not always described in detail, reducing the transparency and reproducibility of these works. To compare the influence of the RANS turbulence model on the four cavity flows, a same CFD setup is adopted here for all benchmarks. This setup is based on the best practice in RANS, namely a steady second-order spatial discretization on a wall-resolved structured mesh and with a grid convergence analysis. The results show that k-ε models, particularly the standard k-ε model, are best suited in a fully turbulent flow regime without strong pressure gradients. On the opposite, the SST k-ω model performs best in the transitional regime while the k-ε models only give moderate to poor results.

Single recalcitrant bubble simulation using a hybrid lattice boltzmann finite difference model

In this paper, 2D simulations of a single recalcitrant bubble using a hybrid finite difference lattice Boltzmann method are presented. To solve the flow field, the conservative phase field lattice Boltzmann model of Geier et al. (2015) is employed with a multi-relaxation-time (MRT) scheme which makes it capable of handling multiphase flows with high density and viscosity contrasts. A finite difference approach is applied to obtain the temperature field with a second-order spatial discretization (central) which marches in time with the 2nd order Runge-Kutta marching technique. The well-known continuum surface tension force (CSF) is implemented to define the thermocapillary force which depends on the temperature field. Before delving into the main aim of this work, we first assess the accuracy of the present model by verifying the Laplace law for a static bubble with four different values of surface tension. To further validate the model, thermocapillary migration of a deformable drop, and thermocapillary flow with two superimposed planar fluids are studied as our second and third benchmarks. Comparing the analytical solutions for both velocity and temperature fields with the simulation ones, good agreements are found. We then propose a simple theory with some simplification for the study of the migration of a single recalcitrant bubble, and after that weperform a series of 2D simulations to analyze the effects of density ratios, viscosity ratios, entrance average velocities of the flow, and the values of second derivative of surface tension with respect to the temperature.

Influence of different subgrid-scale models in low-order LES of supersonic jet flows

The present work is concerned with a study of large eddy simulations (LES) of unsteady turbulent jet flows. In particular, the present analysis is focused on the effects of the subgrid-scale modeling used when a second-order spatial discretization methodology is employed for the numerical simulations. The present effort addresses perfectly expanded supersonic jets, because the authors want to emphasize the effects of the jet mixing phenomena. The LES formulation is discretized using the finite difference approach, after the equations are rewritten in a generalized coordinate system. Both space and time discretizations are second-order accurate and an explicit time march is adopted. Special care is dedicated to the discretization of the energy equation to appropriately model the set of filtered equations appearing in the LES formulation. The classical Smagorinsky, the dynamic Smagorinsky and the Vreman models are the subgrid-scale closures selected for the present work. The computational results are compared to data in the literature to validate the present simulation tool. Results indicate that the characteristics of numerical discretization can be as important as the effects of the subgrid-scale models for such low-order spatial discretization schemes. A detailed analysis is presented for the performance of each subgrid closure in the numerical context here considered.

Application of high-order numerical schemes and Newton-Krylov method to two-phase drift-flux model

This paper concerns the application and solver robustness of the Newton-Krylov method in solving two-phase flow drift-flux model problems using high-order numerical schemes. In our previous studies, the Newton-Krylov method has been proven as a promising solver for two-phase flow drift-flux model problems. However, these studies were limited to use first-order numerical schemes only. Moreover, the previous approach to treating the drift-flux closure correlations was later revealed to cause deteriorated solver convergence performance, when the mesh was highly refined, and also when higher-order numerical schemes were employed. In this study, a second-order spatial discretization scheme that has been tested with two-fluid two-phase flow model was extended to solve drift-flux model problems. In order to improve solver robustness, and therefore efficiency, a new approach was proposed to treating the mean drift velocity of the gas phase as a primary nonlinear variable to the equation system. With this new approach, significant improvement in solver robustness was achieved. With highly refined mesh, the proposed treatment along with the Newton-Krylov solver were extensively tested with two-phase flow problems that cover a wide range of thermal-hydraulics conditions. Satisfactory convergence performances were observed for all test cases. Numerical verification was then performed in the form of mesh convergence studies, from which expected orders of accuracy were obtained for both the first-order and the second-order spatial discretization schemes. At last, the drift-flux model, along with numerical methods presented, were validated with three sets of flow boiling experiments that cover different flow channel geometries (round tube, rectangular tube, and rod bundle), and a wide range of test conditions (pressure, mass flux, wall heat flux, inlet subcooling and outlet void fraction).

New analytical solutions to the two-phase water faucet problem

The one-dimensional water faucet problem is one of the classical benchmark problems originally proposed by Ransom to study the two-fluid two-phase flow model. With simplifications, such as massless gas phase and no wall and interfacial frictions, analytical solutions had been previously obtained for the transient liquid velocity and void fraction distribution. The water faucet problem and its analytical solutions have been widely used for code assessment, benchmark, and numerical verification. In this work, we present a new set of analytical solutions to the water faucet problem at the steady-state condition, with the gas-phase density’s effect on pressure distribution considered. This new set of analytical solutions is used in a rigorous numerical verification process from which the anticipated second-order spatial accuracy is achieved for a second-order spatial discretization scheme. On the contrary, the same anticipated order of accuracy could not be obtained using the Ransom solutions as the reference. In addition, extended Ransom transient solutions for the gas-phase velocity and pressure are derived with the assumption of decoupled liquid and gas pressures. Numerical benchmark on the extended Ransom solutions is also presented.

Design Optimization of an Oil-Air Catch Can Separation System

The Positive Crankcase Ventilation (PCV) system in a car engine is designed to lower the pressure in the crankcase, which otherwise could lead to oil leaks and seal damage. The rotation of crankshaft in the crankcase causes the churn up of oil which conducts to occurrence of oil droplets which in turn may end in the PCV exhaust air intended to be re-injected in the engine admission. The oil catch can (OCC) is a device designed to trap these oil droplets, allowing the air to escape from the crankcase with the lowest content of oil as possible and thus, reducing the generation and emission of extra pollutants during the combustion of the air-fuel mixture. The main purpose of this paper is to optimize the design of a typical OCC used in many commercial cars by varying the length of its inner tube and the relative position of the outlet from radial to tangential fitting to the can body. For this purpose, CFD parametric analysis is performed to compute a one-way coupled Lagrangian-Eulerian two-phase flow simulation of the engine oil droplets driven by the air flow stream running through the device. The study was performed using the finite volume method with second-order spatial discretization scheme on governing equations in the Solid Works-EFD CFD platform. The turbulence was modelled using the k-? model with wall functions. Numerical results have proved that maximum efficiency is obtained for the longest inner tube and the tangential position of the outlet; however, it is recommended further investigation to assess the potential erosion on the bottom of the can under such a design configuration.

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.

High order conservative Lagrangian schemes with Lax–Wendroff type time discretization for the compressible Euler equations

In this paper, we explore the Lax–Wendroff (LW) type time discretization as an alternative procedure to the high order Runge–Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in [3,5]. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy–Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes [3] and third-order spatial discretization using curvilinear meshes [5]. Comparing with the Runge–Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.

High accuracy stable numerical solution of 1D microscale heat transport equation

AbstractWe investigate the use of a fourth‐order compact finite difference scheme for solving a one‐dimensional heat transport equation at the microscale.The fourth‐order compact scheme is used with a Crank–Nicholson type integrator by introducing an intermediate function for the heat transport equation. The new scheme is proved to be unconditionally stable with respect to initial values. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second‐order spatial discretization. It is shown that the new scheme is computationally more efficient and more accurate than the second‐order scheme. Copyright © 2001 John Wiley & Sons, Ltd.