Compact High Order Finite Volume Research Articles

A MOOD-like compact high order finite volume scheme with adaptive mesh refinement

In this paper, a novel Finite Volume (FV) scheme for obtaining high order approximations of solutions of multi-dimensional hyperbolic systems of conservation laws within an Adaptive Mesh Refinement framework is proposed. It is based on a point-wise polynomial reconstruction that avoids the recalculation of reconstruction stencils and matrices whenever a mesh is refined or coarsened. It also couples both the limiting of the FV scheme and the refinement procedure, taking advantage of the Multi-dimensional Optimal Order Detection (MOOD) detection criteria. The resulting computational procedure is employed to simulate test cases of increasing difficulty using two models of Partial Differential Equations: the Euler system and the radiative M1 model, thus demonstrating its efficiency.

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.

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.

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 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).

Compact high order finite volume method on unstructured grids III: Variational reconstruction

This paper presents a variational reconstruction for the high order finite volume method in solving the two-dimensional Navier–Stokes equations on arbitrary unstructured grids. In the variational reconstruction, an interfacial jump integration is defined to measure the jumps of the reconstruction polynomial and its spatial derivatives on each cell interface. The system of linear equations to determine the reconstruction polynomials is derived by minimizing the total interfacial jump integration in the computational domain using the variational method. On each control volume, the derived equations are implicit relations between the coefficients of the reconstruction polynomials defined on a compact stencil involving only the current cell and its direct face-neighbors. The reconstruction and time integration coupled iteration method proposed in our previous paper is used to achieve high computational efficiency. A problem-independent shock detector and the WBAP limiter are used to suppress non-physical oscillations in the simulation of flow with discontinuities. The advantages of the finite volume method using the variational reconstruction over the compact least-squares finite volume method proposed in our previous papers are higher accuracy, higher computational efficiency, more flexible boundary treatment and non-singularity of the reconstruction matrix. A number of numerical test cases are solved to verify the accuracy, efficiency and shock-capturing capability of the finite volume method using the variational reconstruction.

Entropy generation due to three-dimensional double-diffusive convection of power-law fluids in heterogeneous porous media

The article reports a numerical study of entropy generation due to the double-diffusive natural convection in a 3D heterogeneous porous cubic saturated with power-law fluids and submitted to horizontal thermal and concentration gradients. The governing equations based on a generalized non-Darcy model are solved by using the compact high order finite volume method. This approach is devised to detect the effects of the heterogeneity level, the buoyancy ratio and the porous thermal Rayleigh number on the fluid flow and its associated irreversibility characteristics. The results show that the mean Nusselt and Sherwood number reduce, while the total entropy generation increases, as the level of heterogeneity increases by the exponential distribution of the permeability; the mean Nusselt, Sherwood number and total entropy generation increases as the porous thermal Rayleigh number increases or the buoyancy ratio increases (decreases) for the thermal dominated flow (solutal dominated flow) by the heat and mass transfer performance. Apart from that, our numerical tests of the approach on the shear-thinning, the Newtonian and shear-thickening fluids show that the impacts of different power-law indexes on the entropy generation due to fluid friction, heat and mass transfer are mainly manifested in rheological properties, which elucidate that the shear-thinning fluids is more effective than shear-thickening fluids. The studies may help us establish a physically reasonable methodology to systemically assess fluid flow and energy consumption in heterogeneous porous media in the real world.

Three-dimensional numerical investigation on thermosolutal convection of power-law fluids in anisotropic porous media

The present study simulates the 3D unsteady double-diffusive natural convection subject to opposing thermal and solutal buoyancy forces (N<0) in a porous cubic by a generalized non-Darcy model, in which the effects of the crucial parameters such as the porous thermal Rayleigh numbers, buoyancy ratio and anisotropy ratio on the flow structure, heat and mass transfer of power-law fluids are investigated independently. The top and bottom walls are given different temperatures and concentrations, while the other walls are adiabatic and impermeable. A compact high order finite volume method is adopted to describe the flow structure and the resulting heat and mass transfer characteristic of non-Newtonian fluids in the anisotropy porous cubic. Our simulations show that the flow structure develops from conduction-dominated to convection-dominated as buoyancy ratio or anisotropy ratio or porous thermal Rayleigh number increases. The average Nusselt and Sherwood numbers keep constants during the conduction-dominated stage, then increase along the transition route. On the other hand, the impacts of different power-law indexes on the convection are mainly manifested in rheological properties, which elucidate that the shear-thinning fluids is more effective in heat and mass transfer enhancement than shear-thickening fluids. The studies may help us establish a physically reasonable methodology to systemically assess double-diffusive convection of non-Newtonian power-law fluids in anisotropy porous media in the real world.

Compact high order finite volume method on unstructured grids I: Basic formulations and one-dimensional schemes

The large reconstruction stencil has been the major bottleneck problem in developing high order finite volume schemes on unstructured grids. This paper presents a compact reconstruction procedure for arbitrarily high order finite volume method on unstructured grids to overcome this shortcoming. In this procedure, a set of constitutive relations are constructed by requiring the reconstruction polynomial and its derivatives on the control volume of interest to conserve their averages on face-neighboring cells. These relations result in an over-determined linear equation system, which, in the sense of least-squares, can be reduced to a block-tridiagonal system in the one-dimensional case. The one-dimensional formulations of the reconstruction are discussed in detail and a Fourier analysis is presented to study the dispersion/dissipation and stability properties. The WBAP limiter based on the secondary reconstruction is used to suppress the non-physical oscillations near discontinuities while achieve high order accuracy in smooth regions of the solution. Numerical results demonstrate the method's high order accuracy, robustness and shock capturing capability. A new reconstruction procedure for high order finite volume scheme is proposed.This procedure can achieve arbitrary order of accuracy using a compact stencil consisting of only face neighboring cells.The spectral properties of the proposed finite volume schemes are analyzed.Some test cases are solved numerically to show the superior performances of the proposed schemes.

Compact high order finite volume method on unstructured grids II: Extension to two-dimensional Euler equations

Abstract In this paper, the compact least-squares finite volume method on unstructured grids proposed in our previous paper is extended to multi-dimensional systems, namely the two-dimensional Euler equations. The key element of this scheme is the compact least-squares reconstruction in which a set of constitutive relations are constructed by requiring the reconstruction polynomial and its spatial derivatives on the control volume of interest to conserve their averages on the face-neighboring cells. These relations result in an over-determined linear equation system. A large sparse system of linear equations is resulted by using the least-squares technique. An efficient solution strategy is of crucial importance for the application of the proposed scheme in multi-dimensional problems since both direct and iterative solvers for this system are computationally very expensive. In the present paper, it is found that in the cases of steady flow simulation and unsteady flow simulation using dual time stepping technique, the present reconstruction method can be coupled with temporal discretization scheme to achieve high computational efficiency. The WBAP limiter and a problem-independent shock detector are used in the simulation of flow with discontinuities. Numerical results demonstrate the high order accuracy, high computational efficiency and capability of handling both complex physics and geometries of the proposed schemes.