Order Finite Volume Method Research Articles

Numerical studies of hydrodynamic relativistic astrophysical jet: The modification of vorticity transport equation

In this Letter, we have carried out the two-dimensional numerical simulation of axisymmetric relativistic jet in cylindrical coordinates by employing higher order finite volume method in PLUTO [Mignone et al., “PLUTO: A numerical code for computational astrophysics,” Astrophys. J. Suppl. Ser. 170, 228 (2007)] solver. The modified vorticity transport equation has been proposed for relativistic flow by taking the curl of the momentum equation, which shows significant change in the baroclinic vorticity production term due to relativistic effect. Both mathematical analysis and numerical results show that the vorticity production term due to baroclinic torque is heavily influenced due to the presence of specific enthalpy gradient and square of Lorentz factor gradient in a relativistic fluid flow.

Improved Weighted Essentially Non-Oscillatory Schemes by Mixed Stencils

Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping method. The restriction that conventional ENO or WENO schemes only use r order stencils, is removed. Higher resolution can be achieved by introducing the r + 1 order stencils. This method is verified by three cases, i.e. the interaction of a moving shock with a density wave problem, the interacting blast wave problem and the double mach reflection problem. The numerical results show that the Improved Weighted Essential Non-oscillatory method is a stable, accurate high-resolution finite volume scheme.

A bound-preserving high order scheme for variable density incompressible Navier-Stokes equations

For numerical schemes to the incompressible Navier-Stokes equations with variable density, it is a critical property to preserve the bounds of density. A bound-preserving high order accurate scheme can be constructed by using high order discontinuous Galerkin (DG) methods or finite volume methods with a bound-preserving limiter for the density evolution equation, with any popular numerical method for the momentum evolution. In this paper, we consider a combination of a continuous finite element method for momentum evolution and a bound-preserving DG method for density evolution. Fully explicit and explicit-implicit strong stability preserving Runge-Kutta methods can be used for the time discretization for the sake of bound-preserving. Numerical tests on representative examples are shown to demonstrate the performance of the proposed scheme.

Computational study on drilling mud flow through wellbore annulus by Giesekus viscoelastic model

Cuttings transport from wellbore annulus to the surface via drilling fluids is one of the most important problems in gas and oil industries. In the present paper, the effects of viscoelastic property of drilling fluids on flow through wellbore annulus are studied numerically by use of computational fluid dynamics simulation in OpenFOAM software. This problem is simulated as the flow between two coaxial annulus cylinders and the inner cylinder is rotating through its axes. Here, the Giesekus model is used as the nonlinear constitutive equation. This model brings the nonlinear viscosity, normal stress differences, extensional viscosity and elastic property. The numerical solution is obtained using the second order finite volume method by considering PISO algorithm for pressure correction. The effect of elasticity, Reynolds number, Taylor number and mobility factor on the velocity and stress fields, pressure drop, and important coefficient of drilling mud flow is studied in detail. The results predicted that increasing elastic property of drilling mud lead to an initial sharp drop in the axial pressure gradient as well as Darcy-Weisbach friction coefficient. Increasing the Reynolds number at constant Taylor number, resulted an enhancing in the axial pressure drop of the fluid but Darcy-Weisbach [Formula: see text] friction coefficient mainly reduced.

Cavitation bubble interaction with a rigid spherical particle on a microscale

Cavitation bubble collapse close to a submerged sphere on a microscale is investigated numerically using a finite volume method in order to determine the likelihood of previously suspected mechanical effects to cause bacterial cell damage, such as impact of a high speed water jet, propagation of bubble emitted shock waves, shear loads, and thermal loads. A grid convergence study and validation of the employed axisymmetric numerical model against the Gilmore’s equation is performed for a case of a single microbubble collapse due to a sudden ambient pressure increase. Numerical simulations of bubble-sphere interaction corresponding to different values of nondimensional bubble-sphere standoff distance δ and their size ratio ε are carried out. The obtained results show vastly different bubble collapse dynamics across the considered parameter space, from the development of a fast thin annular jet towards the sphere to an almost spherical bubble collapse. Although some similarities in bubble shape progression to previous studies on larger bubbles exist, it can be noticed that bubble jetting is much less likely to occur on the considered scale due to the cushioning effects of surface tension on the intensity of the collapse. Overall, the results show that the mechanical loads on a spherical particle tend to increase with a sphere-bubble size ratio ε, and decrease with their distance δ. Additionally, the results are discussed with respect to bacteria eradication by hydrodynamic cavitation. Potentially harmful mechanical effects of bubble-sphere interaction on a micro scale are identified, namely the collapse-induced shear loads with peaks of a few megapascals and propagation of bubble emitted shock waves, which could cause spatially highly variable compressive loads with peaks of a few hundred megapascals and gradients of 100 MPa/μm.

Modelling and numerical simulation of the heat transfer and natural ventilation in storage halls

An algorithm for modelling of the heat transfer in storage halls at different combinations of open doors and smoke vents is developed. The models allow numerical simulations of velocity, pressure and temperature fields in the halls in cases of transient and steady state conditions, taking into account the indoor buoyancy, heat sources, heat transfer through the building envelopes and outdoor wind pressures. The models are solved numerically via finite volume method in order to estimate the necessity of additional heating at a design stage of a courier logistic storage hall, situated in the temperate climate zone, at the hardest winter conditions of the region. Recommendations about the organization of the heating and the cargo input are done based on the analyses of the fluid flow and thermal fields at the modelled scenarios.

A Posteriori Error Analysis of Any Order Finite Volume Methods for Elliptic Problems

A Posteriori Error Analysis of Any Order Finite Volume Methods for Elliptic Problems

A high order least square-based finite difference-finite volume method with lattice Boltzmann flux solver for simulation of incompressible flows on unstructured grids

This paper presents a high order least square-based finite difference-finite volume (LSFD-FV) method together with lattice Boltzmann flux solver for accurate simulation of incompressible flows on unstructured grids. Within each control cell, a high order polynomial, which is based on Taylor series expansion, is applied to approximate the solution function. The derivatives in the Taylor series expansion are approximated by the functional values at the centers of neighboring cells using the mesh-free least square-based finite difference (LSFD) method developed by Ding et al. (2004) [36]. In the high order finite volume method, the recently developed lattice Boltzmann flux solver (LBFS) is applied to evaluate the inviscid and viscous fluxes physically and simultaneously at the cell interface by local reconstruction of lattice Boltzmann solution. Compared with traditional k-exact high-order finite volume method, the present method is more accurate and efficient. Various benchmark examples are tested to validate the high-order accuracy, high computational efficiency and flexibility of the proposed method on unstructured grids.

Compact high order finite volume method on unstructured grids IV: Explicit multi-step reconstruction schemes on compact stencil

In the present paper, a multi-step reconstruction procedure is proposed for high order finite volume schemes on unstructured grids using compact stencil. The procedure is a recursive algorithm that can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique and the continuation technique. The partial inversion can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The continuation technique is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. This multi-step procedure is operationally compact since in each step only the relations defined on a compact stencil are used. In the present paper, the third and fourth order finite volume schemes based on two-step quadratic and three-step cubic reconstructions are studied.

Entropy stable essentially nonoscillatory methods based on RBF reconstruction

To solve hyperbolic conservation laws we propose to use high-order essentially nonoscillatory methods based on radial basis functions. We introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Thus, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. The main contribution is the construction of an algorithm and a smoothness indicator that ensures an interpolation function which fulfills the sign-property on general one dimensional grids.

Morphological Approach for Nonlinear Flow and Heat Transfer in Complex Bio-Composite Material

ABSTRACTThe bio-based materials have experienced in recent years a relevant evolution in construction and for many building applications. Despite the efforts to develop new solutions to achieve the objectives of numerical characterizations, a few researches in this area have been carried out using a heterogeneous wood-cement composite. This study is based on a constructive approach using the finite volume method in order to generate a numerical model capable of reproducing the equivalent transfer properties of a wood-cement composite. The challenge is to understand, design, and control this multiphase inhomogeneous structure and predict transfers and flows within the community in order to achieve an optimal functionality. Experimental measurements and microscopic characterization using X-ray microtomography were performed to compare and demonstrate the method validation. The comparative results demonstrate a less than 10% deviation between the measurements and the simulations of the equivalent thermal properties. Thus, thermal conductivity and permeability, under different medium configurations, were characterized. Therefore, it was observed that these properties depend strongly on the porosity and the pore distribution in the continuous phase. Moreover, these two parameters are highly dependent on cement percentages in the structure.

Numerical study on combined buoyancy–Marangoni convection heat and mass transfer of power-law nanofluids in a cubic cavity filled with a heterogeneous porous medium

In this paper, the combined buoyancy–Marangoni convection of non-Newtonian power-law nanofluids in a 3D heterogeneous porous cubic cavity is investigated in detail with the compact high order finite volume method. Special attentions are given to detect the effects of heterogeneity level, Marangoni number, thermal Rayleigh number, buoyancy ratio and nanoparticle volume fraction on the fluid flow as well as on rates of heat and mass transfer. It is observed that as a result of the exponential distribution of the permeability, the heat and mass transfer rates reduce as the level of heterogeneity enhances. The effect of the surface tension on the heat and mass transfer intensity becomes insignificant when the buoyancy force is strengthened. Further, the average Nusselt and Sherwood numbers increase as the Marangoni number and thermal Rayleigh number increase due to the combined effects of buoyancy and surface tension. The augmentation of the buoyancy ratio causes convection heat and mass transfer to increase for the thermal dominated flow while decrease in buoyancy ratio also augments it for the solutal dominated flow. The heat transfer (mass transfer) rate is found to increase (decrease) with increasing the nanoparticle volume fraction. Moreover, for all above studied parameters, an intensification of the flow and an increase in average Nusselt and Sherwood numbers occur with the decrease in power-law index.

Numerical artifacts in the Generalized Porous Medium Equation: Why harmonic averaging itself is not to blame

The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable k(p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient k(p) for small p, and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.

Analyses and implications of higher order finite volume methods on first-order macroscopic traffic flow models

Selection of suitable numerical scheme for solving macroscopic traffic flow models should be based on the numerical errors in various traffic parameters. Although such errors associated with state variables such as density are known to some extent, their implications to the derived parameters such as travel time and queue-length are not understood well. This paper proposes a general basis for selecting the spatial order of accuracy of a finite volume scheme for solving first-order traffic flow models. Four Measures of Accuracy that are especially relevant for traffic flow have been introduced. We illustrate that, although higher order schemes outperform the lower order ones in the quality of simulation in terms of resolution as well as computational cost, their superior performance is underestimated with the use of conventional error measures of state predictions. Localized parameters such as size and duration of the queue are more sensitive towards the order of the scheme, than domain aggregated parameters such as duration of congestion and travel time.

A higher-efficient non-hydrostatic finite volume model for strong three-dimensional free surface flows and sediment transport

In order to accurately simulate strong three-dimensional (3-D) free surface flows and sediment transport, the fully 3-D non-hydrostatic pressure models are developed based on the incompressible Navier-Stokes equations and convection–diffusion equation of sediment concentration with the mixing triangle and quadrilateral grids. The governing equations are discretized with the unstructured finite volume method in order to provide conservation properties of mass and momentum, and flexibility with practical application. It is shown that it is first-order accurate on nonuniform plane two-dimensional (2-D) grids and second-order accurate on uniform plane grids. A third-order approximation of the vertical velocity at the top-layer is applied. In such a way, free surface zero stress boundary condition is satisfied maturely, and very few vertical layers are needed to give an accurate solution even for complex discontinuous flow and short wave simulation. The model is applied to four examples to simulate strong 3-D free surface flows and sediment transport where non-hydrostatic pressures have a considerable effect on the velocity field. The newly developed model is verified against analytical solutions with an excellent agreement.

Analysis of entropy generation in combined buoyancy-Marangoni convection of power-law nanofluids in 3D heterogeneous porous media

In this paper, entropy generation due to the combined buoyancy-Marangoni convection of non-Newtonian power-law nanofluids in a 3D heterogeneous porous cubic cavity is investigated in detail with the compact high order finite volume method. The fluid in the cavity is a CMC-based nanofluids containing CuO nanoparticles, and the numerical study is conducted for a wide range of heterogeneity level ranging from 0 to 1.5, Marangoni number from 0 to 1000, thermal Rayleigh number from 104 to 106, buoyancy ratio from −2.0 to −0.1, nanoparticle volume fraction from 0 to 0.1 and power-law index from 0.76 to 1.0 to identify the irreversibility characteristics. The results show that the total entropy generation is considerably affected by the heterogeneity in permeability; it increases when the level of heterogeneity increases. Marangoni number becomes a more effective parameter on total entropy generation for lower values of thermal Rayleigh numbers. For thermal dominated flow, the increase in the buoyancy ratio provokes various irreversibilities to enhance; on the other hand, for solutal dominated flow, the rise of the buoyancy ratio declines the entropy generation. Moreover, it is also shown that the enhancement of the nanoparticle volume fraction reduces the total entropy generation. Apart from that, our numerical tests show the flow field and entropy generation are influenced appreciably by the presence of the power-law index.

A family of HLL-type solvers for the generalized Riemann problem

The generalized Riemann problem (GRP) is the initial value problem for a conservation law with piecewise smooth, but discontinuous initial data. We provide a new method for solving the GRP approximately, that can be used as a building block for high order finite volume or discontinuous Galerkin methods. Our new GRP solvers use the approximate states and wave speeds obtained through a HLL-type Riemann solver and use this information to build an approximation of the state in the GRP of any order. What is new about this approach compared to most previous solvers is that we no longer need to solve a classical Riemann problem exactly. We give a detailed explanation of this strategy for HLL and HLLC solvers for the Euler equations, as well as for the HLLD solver for MHD equations. We demonstrate the performance of the solvers from this new family of GRP solvers for a broad range of test problems.

A thermal non-equilibrium model for 3D double diffusive convection of power-law fluids with chemical reaction in the porous medium

The article reports a numerical study of the double-diffusive natural convection of power-law fluids in a porous cubic cavity with chemical reaction under the local thermal non-equilibrium (LTNE) state and submitted to horizontal temperature and concentration gradients. The complete governing equations, in which the generalized non-Darcy model is employed as momentum equation and a two-temperature model that represents temperature fields of the fluid and solid phases separately is used for energy equation, are solved by the compact high order finite volume method. Special attentions are given to detecting the effects of the porosity modified conductivity ratio (γ), the inter-phase heat transfer coefficient (H), the chemical reaction parameter (λ), the Dufour parameter (Du), the Soret parameter (Sr) and the Lewis number (Le) on the fluid flow as well as on rates of heat and mass transfer. Our simulations show that the heat transfer rate of solid increases, while the heat transfer rate of fluid and mass transfer rate keep constants, as the LTNE parameters (i.e., γ and H) increase. For any relatively large value of γ and H, the almost thermal equilibrium state is achieved due to the similar thermal distribution of fluid and solid phases. On the contrary, the chemical reaction parameter has significant influence on mass transfer than on heat transfer under the LTNE conditions. In addition, the heat and mass transfer enhance with the increment of the Soret and Dufour parameters. Augmentation of Lewis number enhances mass transfer but reduces heat transfer. Apart from that, our numerical tests show the flow field is influenced appreciably by the presence of the power-law index (n), Darcy number (Da) and porous thermal Rayleigh number (Ra).

A Simple Bound-Preserving Sweeping Technique for Conservative Numerical Approximations

In this paper, we propose a simple bound-preserving sweeping procedure for conservative numerical approximations. Conservative schemes are of importance in many applications, yet for high order methods, the numerical solutions do not necessarily satisfy maximum principle. This paper constructs a simple sweeping algorithm to enforce the bound of the solutions. It has a very general framework acting as a postprocessing step accommodating many point-based or cell average-based discretizations. The method is proven to preserve the bounds of the numerical solution while conserving a prescribed quantity designated as a weighted average of values from all points. The technique is demonstrated to work well with a spectral method, high order finite difference and finite volume methods for scalar conservation laws and incompressible flows. Extensive numerical tests in 1D and 2D are provided to verify the accuracy of the sweeping procedure.

Mathematical modeling and its analysis for instability of the immune system induced by chemotaxis.

In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.