Finite Volume Formulation Research Articles

A finite volume formulation for one-dimensional Stefan problems

This paper presents a numerical formulation that combines the finite-volume method with a time-dependent boundary-fitted coordinate system to tackle the moving boundary problems for the one-dimensional (1-D) heat equation. Following the formulation process, an algorithm was created with the help of the linear algebra package Eigen to generate the data for the numerical solution. Four benchmark cases of heat conduction were investigated by this approach and compared to the analytical solution: a fixed boundary case, the liquid phase of a melting process, the solid phase of a melting process, and an ablating process. The results demonstrate the capability of the presented numerical formulation in efficiently solving the general 1-D heat problems, especially the one-phase Stefan problems, as well as the consistency of the numerical and analytical solutions. Future works can strive to extend the formulations to Stefan problems of two phases or higher dimensions. The work can then be extended to real-life applications, such as the spacecraft re-entry ablation process, and the process of iceberg melting, demonstrating its critical value in solving practical problems.

A non-neutral 1D fluid model of hall thruster discharges: full electron inertia and anode sheath reversal

A non-neutral model (NNM) of the axial plasma discharge in a Hall thruster, including full electron inertia, is presented. In the finite-volume formulation, two types of sheath boundary conditions previously used in the literature are tested and proven to behave practically identically in this model. Both normal and reversed (i.e. electron repelling and attracting, respectively) anode sheaths are admitted. This model is compared with the quasineutral model developed in a previous work, which includes only azimuthal electron inertia and normal anode sheaths. Both models agree excellently within the parametric region where steady-state solutions with a normal anode sheath exist. The NNM shows the absence of steady-state solutions with a reversed anode sheath. Nonetheless, a reversed sheath can appear during the transient to a steady-state solution with a normal sheath and the periodic transition from a normal to a reversed sheath can be observed in the presence of breathing-mode oscillations. In other cases, the reversed sheath leads to the discharge shut-off. Full electron inertia is always important in the presence of a reversed sheath. The parametric threshold of the wall accommodation parameter from a stationary solution to a breathing mode one differs slightly between the non-neutral and the quasi-neutral model.

Large-eddy simulation of wall-bounded incompressible turbulent flows based on multi-moment finite volume formulation

The multi-moment based numerical model is proposed for performing large-eddy simulation (LES) of wall-bounded incompressible turbulent flows on the unstructured grids. In this model, velocity is defined as volume integral average (VIA) and point values (PVs) which are updated as prognostic variables simultaneously in time. Using VIA and PV in each cell, this method constructs higher-order polynomial on a compact stencil and more importantly, provides a new approach to design subgrid-scale (SGS) models for LES. Various SGS models are developed in this framework where the eddy-viscosity is computed separately at both VIA and PVs using the least-square method on different interpolation stencils. Compared with conventional finite volume schemes, the proposed method effectively reduces numerical dissipation and largely suppresses the sensitivity of numerical solution on SGS models and resolves more elaborated flow structures on the same mesh resolution. It also reduces solution dependence on the shape and quality of grid elements, as demonstrated by comparing numerical results on the different types of grids. The proposed method can attain high-fidelity simulation without loss of numerical efficiency and robustness, which is highly appealing for complex turbulent problems with high Reynolds numbers and complicated geometries.

Optimized compact finite volume formulation with dispersion relation preserving characteristics in structured space discretizations

Computationally extensive fluid flow simulations usually require numerical schemes with a high degree of numerical precision, where high wavenumber and frequency capturing capability is desired in solutions with high physical fidelity. Numerical dispersion and diffusion errors must be reduced to a minimum in order to caputre low amplitude and high frequency flow aspects, such as flow induced sound waves and minor instabilities. Such schemes also must be capable of dealing with flow discontinuities, such as shockwaves and expansion waves in cases where the flow velocity stands near or above sound velocity. In this work the Dispersion Relation Preserving (DRP) optimization is formally extended to the compact Finite Volume (FV) framework where the error is minimized along a given spectral interval. This is done by equaling the resulting algebraic compact spectral optimized Finite Difference (FD) operators with the Finite Volume formulation with the desired numeric characteristics. Numerical tests are carried over linear and nonlinear governing equations models for one and three-dimensional cases, where the travelling Gaussian wave, the transient expansion wave and the smooth to nonsmooth step wave flows are considered. The proposed compact DRP FV schemes surpasses its counterparts in some numerical tests while maintaining comparable capabilities with other more basic schemes in the remaining numerical tests. Excellent broadband content capturing capability is observed, while possessing reduced numerical instabilities generation.

Investigating the effects of numerical algorithms on global magnetohydrodynamics simulations of solar wind in the inner heliosphere

This paper explores the effects of numerical algorithms on global magnetohydrodynamics simulations of solar wind (SW) in the inner heliosphere. To do so, we use sunRunner3D, a 3-D magnetohydrodynamics model that employs the boundary conditions generated by CORHEL and the PLUTO code to compute the plasma properties of the SW with the ideal magnetohydrodynamics approximation up to 1.1 AU in the inner heliosphere. Mainly, we define three different combinations of numerical algorithms based on their diffusion level. This diffusion level is related to the way of solving the magnetohydrodynamics equations using the finite volume formulation, and, therefore, we set in terms of the divergence-free condition methods, Riemann solvers, variable reconstruction schemes, limiters, and time-steeping algorithms. According to the simulation results, we demonstrate that sunRunner3D reproduces global features of Corotating Interaction Regions observed by Earth-based spacecraft for a set of Carrington rotations that cover a period that lays in the late declining phase of solar cycle 24, independently of the numerical algorithms. Moreover, statistical analyses between models and in-situ measurements show reasonable agreement with the observations, and remarkably, the high diffusive method matches better with in-situ data than low diffusive methods.

Improving Global Barotropic Tides With Sub‐Grid Scale Topography

AbstractIn recent years, efforts have been made to include tides in both operational ocean models as well as climate and earth system models. The accuracy of the barotropic tides is often limited by the model topography, which is in turn limited by model horizontal resolution. In this work, we explore the reduction of barotropic tidal errors in an ocean general circulation model (Modular Ocean Model version 6; MOM6) using sub‐grid scale topography representation. We follow the methodology from Adcroft (2013, https://doi.org/10.1016/j.ocemod.2013.03.002), which utilizes statistics from finer resolution topographic data sets to represent sub‐grid scale features with a light computational cost in a structured finite volume formulation. The geometric effect from sub‐grid scale topography can be introduced to the model with only a few parameters at each grid cell. The porous barriers, which are implemented at the walls of the grid cells, are used to modify transport between grid cells. Our results show that the globally averaged tidal error in lower‐resolution simulations is significantly reduced with the use of porous barriers. We argue this method is a potentially useful tool to improve simulations of tides (and other flows) in low‐resolution simulations.

An Adaptive PML Finite Volume Algorithm for the Scattering by Periodic Gratings

In this paper, we studied the finite volume formulations for solving the diffraction grating problem that is truncated by the perfectly matched layer (PML) technique. Based on a reliable the a-posteriori error estimate, an adaptive PML finite volume method is discussed for the numerical approximation of the diffraction grating problem. The PML parameters are obtained numerically by sharp a-posteriori error estimates of the PML finite volume method such as the thickness of the layer and the medium property. It is worth mentioning in the a-posteriori error estimates that we derive the error representation formula and use a [Formula: see text]-orthogonality property of the residual similar to the Galerkin orthogonality used in the finite element method. Furthermore, the lower bound is established to demonstrate the efficiency of the a-posteriori error estimates. Numerical experiments are given to illustrate the accuracy and robustness of our adaptive algorithm.

Face‐centred finite volume methods for Stokes flows with variable viscosity

SummarySix face‐centred finite volume formulations are derived and compared for the simulation of Stokes flows with spatially varying viscosity. The main difference between the methods derived is the mixed variable used in the mixed formulation and the use of a weak or strong form in each element using integration by parts. A brief discussion about the properties of the different methods is provided, including comments on the computational cost and the symmetry of the resulting global system of equations. Finally, numerical examples in two and three dimensions are used to compare the accuracy of all the formulations presented. The examples include a problem where the methods are employed to simulate a steep variation of the viscosity, showing the ability to perform these simulations without using a mesh conforming to a material interface. The performance of different element types and different choices of the stabilisation is also discussed.

Finite volume form factors in integrable theories

We develop a new method to calculate finite size corrections for form factors in two-dimensional integrable quantum field theories. We extract these corrections from the excited state expectation value of bilocal operators in the limit when the operators are far apart. We elaborate the finite size effects explicitly up to the 3rd Lüscher order and conjecture the structure of the general form. We also fully recover the explicitly known massive fermion finite volume form factors.

A robust unstructured finite volume scheme for urban flash flood simulation: Duhok city as a case study

ABSTRACT A robust well-balanced finite volume scheme is proposed for simulating the flash floods due to rainfall-runoff process. The solver’s implementation on unstructured triangular meshes is based on a Godunov-type second-order upwind finite volume formulation, and Roe’s average scheme is used to determine the fluxes. The bed slope, friction and rainfall source terms are discretized using an upwind approach. The suggested algorithm uses the slope limiter and Hancock’s predictor-corrector method in order to secure numerical stability. The proposed model overwhelms the drawbacks of many commercial codes that use the finite difference, finite elements and finite volume in their scheme, especially in dealing with rapid change in topography. The prepared scheme solves the two-dimensional shallow water equations over a real topography of Hishkaro basin located in Duhok city in Kurdistan of Iraq where the variation of bed elevation reaches 717 m. The scheme is validated against real observed data in Duhok city provided by the Sewage directory. The data of five observation points over the study area were used to validate the code. The results showed that the max water level reaches 3.67, 2.12, 1.20, 0.67 and 0.3 m at observed points 3, 2, 4, 1 and 5, respectively.

Influence of External Magnetic Field on 3D Thermocapillary Convective Flow in Various Thin Annular Pools Filled with Silicon Melt

Thermocapillary convection flows can have an impact on the homogeneity of floating zone semiconductor crystals. An external magnetic field can also help to reduce this non-homogeneity. The goal of this research is to minimize thermocapillary convection in various thin annular pools filled with silicon melt. A three-dimensional (3D) numerical technique is proposed that employs an implicit finite volume formulation. The steady-state thermocapillary flow in six thin annular pools (R=0.3, 0.4, 0.5, 0.6, 0.7, and 0.8) subjected to an externally induced magnetic field was observed. Under magnetic field influence, the effects of increasing annular gap, R on the hydrothermal wave number and azimuthal pattern are obtained. The results reveal that hydrothermal waves m=14, m=11, m=8, m=6, m=4, and m=3 are observed in steady flow for R=0.3; 0.4; 0.5; 0.6; 0.7, and R=0.8, respectively. The maximum temperature occurs in the intermediate zone between the inner and outer walls when there is no magnetic field. Under a strong enough magnetic field, isothermal lines change form and become concentric circles. As the amplitude of the magnetic field (Ha) grows, the azimuthal velocity and temperature at the free surface reduce, and the asymmetric 3D flow becomes axisymmetric steady when Ha surpasses a threshold value.

Numerical Simulation of a Sonic Jet Injected into a Supersonic Crossflow With K.(l) Turbulence Model

Numerical simulation of normal injection of a sonic jet into supersonic crossflows were performed with a k-ul turbulence model. Favre-averaged Navier-Stokes equations along with k-6p equations were solved in a coupled manner by calculating inviscidfluxes with Roe's flux-dffirence splitting, central dffirencing of derivatives in the viscous terms, and by performing Runge-Kutta time integration in a finite volume formuLation. Experiments of sLot injection at several freestream Mach numbers and injection pressure ratios were simulated. It was observed that simulations agreed well at low pressure ratios but departed with increasing pressure ratio as had been observed in other simulations. A qualitative difference in the structure of the jet was observed when explicit compressibility fficts were included. In turn, the agreement with experiment improves significantly. It was also observed that imposing a lower ReynoLds number in the caLculations than the value determined from reported experimental conditions resulted in solutions expected of transitional separation and vvhich then agreed more closely with experiment.

A multi-moment finite volume formulation for the interaction between free surface flow and moving bodies with THINC method

A numerical model is developed for fluid-rigid body interaction involving free surfaces on moving unstructured grids using the Arbitrary-Lagrangian–Eulerian (ALE) formulation. The multi-moment finite volume method is used for the spatial discretization of Navier–Stokes equations in which volume integrated average (VIA) and point values (PVs) are defined as prognostic variables and updated by integral and differential form respectively. Given VIA and PV in each cell, a higher-order reconstruction function can be built in a compact stencil which improves the numerical accuracy and robustness on the unstructured grids. More importantly, it facilitates imposing a more accurate Dirichlet–Neumann condition for the partitioned interface between the fluid and solid region. The geometrical conservation law (GCL) is also formulated to govern the spatial volume element under an arbitrary mapping. In order to capture the free surfaces of multiphase flows, the volume-of-fluid (VOF) equation is solved by THINC/QQ (THINC method with quadratic surface representation and Gaussian quadrature) scheme in varying mesh, which achieves geometrical faithful interface representation, particularly for curved surfaces in a complex and moving domain. The present model is validated by some benchmark tests in two dimensions for fluid–structure interaction problems with free surfaces. The numerical results show closer agreement with the experimental data than the reference solution of other numerical models. It convinces that the present model provides accurate and robust solutions for the simulation of the complex interaction between the free surface flows and moving bodies.

Damping of three-dimensional waves on coating films dragged by moving substrates

Paints and coatings often feature interfacial defects due to disturbances during the deposition process, which, if they persist until solidification, worsens the product quality. In this article, we investigate the stability of a thin liquid film dragged by a vertical substrate moving against gravity, a fundamental flow configuration in various coating processes. The receptivity of the liquid film to three-dimensional disturbances is analyzed with Direct Numerical Simulations (DNS) and an in-house Integral Boundary Layer (IBL) film model. The latter was used for linear stability analysis and nonlinear wave propagation analysis. The numerical implementation of the IBL film model combines a finite volume formulation with a pseudo-spectral approach for the capillary terms that allows one to investigate non-periodic surface tension-dominated flows. The numerical model was successfully validated with DNS computations. The combination of these numerical tools allows one to describe the mechanisms of capillary and nonlinear damping and identify the instability threshold of the coating processes. The results show that transverse modulations can be beneficial for damping two-dimensional waves within the range of operational conditions considered in this study, which are relevant to air-knife and slot-die coating.

An efficient implementation of a conservative finite volume scheme with constant and linear reconstructions for solving the coagulation equation

Population balances provide an economic means of describing dense particulate phases with particle-particle interactions. Here, we focus on particle coagulation and address the evaluation of the corresponding integral source terms within the scope of a conservative finite volume formulation. Based on one kernel evaluation for every pair of parent cells, efficient analytical formulas are derived that obviate the explicit decomposition of the integration domains into elementary geometrical shapes and are valid for arbitrary volume-grids. Considering the volume-weighted cell averages as degrees of freedom, the formulas are presented for both cell-wise constant and linear reconstructions of the particle volume distribution. We find the latter to be effective at mitigating the unphysically heavy tails that are characteristic of solutions obtained with piecewise constant reconstructions. For Brownian coagulation, the computational expense is mainly caused by the kernel evaluations, while the cost of the parent-daughter pairing and integration algorithm is almost negligibly small.

MHD Effect on the Thermocapillary Silicon Melt Flow in Various Annular Enclosures

AbstractTo simulate thermocapillary magnetohydrodynamic (MHD) convection in different shallow annular enclosures filled with silicon melt, a 3D numerical approach using an implicit finite volume formulation is used. Radiation is emitted upward from the annular enclosure's free top surface, the bottom one heated vertically and is subjected to an external magnetic field. The steady and unsteady thermocapillary MHD flow in four annular enclosures (R = 0.5, 0.6, 0.7, and 0.8) field is studied. The effects of varying the annular gaps, R, on the hydrothermal wave number and azimuthal pattern are obtained. The effects of the magnetic field on azimuthal velocity, temperature disturbances, and the transition from oscillatory to steady flow are also depicted. The results reveal that: hydrothermal waves m = 4, m = 6 and m = 8 are observed in steady flow for R = 0.7, R = 0.6 and R = 0.5, respectively. Oscillatory flows dominated by a hydrothermal wave m = 3 are found also when R = 0.8. The azimuthal velocity and temperature fluctuations at the free surface decrease as the magnitude of the magnetic field (Ha) increases, and the oscillatory flow would turn steady when Ha exceeds a critical value.

Second-Law Considerations in Monte Carlo Ray-Trace and Discrete Green’s Function Analysis of Coupled Radiation and Conduction Heat Transfer

Abstract A generic Monte Carlo ray-trace (MCRT) engine for computing radiation distribution factors (RDFs), working in tandem with an efficient finite volume formulation based on discrete Green’s functions (DGFs), has been used to solve a massive (2636 nodes) coupled radiation/conduction problem. Distribution factors computed using the MCRT method are known to produce unconditionally stable solutions to pure radiation problems even though the RDFs themselves do not conform exactly to the reciprocity principle. However, when these same RFDs are introduced into time-dependent finite volume conduction formulations based on DGFs, the resulting model is found to be fundamentally unstable due to inherent violations of the second law of thermodynamics. A novel approach to addressing this instability is presented and demonstrated for the case of a telescope typical of those employed to monitor the planetary energy budget from low Earth orbit.

A modified dual time integration technique for aerodynamic quasi-static and dynamic stall hysteresis

Simulation of the aerodynamic stall phenomenon in both quasi-static and dynamic conditions requires expensive computational resources. The computations become even more costly for static stall hysteresis using an unsteady solver with very slow variation of angle of attack at low reduced frequencies. In an explicit time-marching solver that satisfies the low Courant number condition, that is, [Formula: see text], the computational cost for such simulations becomes prohibitive, especially at higher Reynolds numbers due to the presence of thin-stretched cells with large aspect ratio in the boundary layer. In this paper, a segregated solver method such as the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) is modified as a dual pseudo-time marching method so that the unsteady problem at each time step is reformulated as a steady state problem. The resulting system of equations in the discretized finite volume formulation is then reduced to zero or near-zero residuals using available convergence acceleration methods such as local time stepping, multi-grid acceleration and residual smoothing. The performance and accuracy of the implemented algorithm was tested for three different airfoils at low to moderate Reynolds numbers in both incompressible and compressible flow conditions covering both attached and separated flow regimes. The results obtained are in close agreement with the published experimental and computational results for both quasi-static and dynamic stall and have demonstrated significant savings in computational time.

Modelling mooring line snap loads using a high-order finite-volume approach

Wave energy converters’ mooring systems are subjected to highly dynamic motions responsible for complex loading mechanisms. Snap loads differ from other dynamic load types by the propagation of tension discontinuities along the cable, which can lead to premature mooring failures if not correctly predicted. Accurate prediction of these loads requires numerical models that can resolve non-smooth solutions. This work developed a (conservative) finite-volume formulation for solving the hyperbolic cable dynamics equation by applying the high-order non-oscillatory central-upwind scheme to calculate the numerical fluxes at the cell interfaces. The central-upwind scheme is a variant of the Godunov-type central finite-volume method and does not need a Riemann solver. This new numerical formulation was tested against analytical and experimental data for a single catenary model under snap-loading conditions. It is shown the method’s ability to simulate snap loads correctly, predicting motions and tensions. Furthermore, it exhibits second-order accuracy in space for smooth solutions. This approach is highly robust (with a numerical dissipation independent of the time step) and simpler to implement compared to other codes in the literature.

An immersed boundary-lattice Boltzmann flux solver for simulation of flows around structures with large deformation

In this paper, we present an immersed boundary-lattice Boltzmann flux solver (IB-LBFS) to simulate the interactions of viscous flow with deformable elastic structures, namely, two-dimensional (2D) and three-dimensional (3D) capsules formed by elastic membranes. The IB-LBFS is based on a finite-volume formulation and makes use of hydrodynamic conservation equations with fluxes computed by a kinetic approach; thus, it is more flexible and efficient than the standard immersed boundary-lattice Boltzmann methods. The membrane of the 2D capsule is represented by a set of discrete Lagrangian points, with in-plane and bending forces acting on the membrane obtained by a finite difference method. In contrast, the membrane of a 3D capsule is discretized into flat triangular elements with membrane forces calculated by an energy-based finite-element method. The IB-LBFS is first validated by studying the deformation of a circular capsule in a linear Newtonian and a power-law shear flow. Next, the deformation dynamics of a spherical, an oblate spheroidal, and a biconcave capsule in a simple shear flow are simulated. For an initially spherical capsule, the tank-treading motion of its membrane is reproduced at the steady state; while for oblate spheroidal and biconcave capsules, the swinging and tumbling motions are observed. Furthermore, under certain parameter settings, the transient mode from tumbling to swinging motions is also found, showing a rich and complex dynamic behavior of non-spherical capsules. These results indicate that the IB-LBFS can be employed in future studies concerning the dynamics of a capsule suspension in more realistic flows.