Unstructured Finite Volume Method Research Articles

A parallel monolithic algorithm for the numerical simulation of large‐scale fluid structure interaction problems

SummaryA novel parallel monolithic algorithm has been developed for the numerical simulation of large‐scale fluid structure interaction problems. The governing incompressible Navier–Stokes equations for the fluid domain are discretized using the arbitrary Lagrangian–Eulerian formulation‐based side‐centered unstructured finite volume method. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant–Kirchhoff material, and the classical Galerkin finite element method is used to discretize the governing equations in a Lagrangian frame. A special attention is given to construct an algorithm with exact total fluid volume conservation while obeying both the global and the local discrete geometric conservation law. The resulting large‐scale algebraic nonlinear equations are multiplied with an upper triangular right preconditioner that results in a scaled discrete Laplacian instead of a zero block in the original system. Then, a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned sub‐domains is utilized for the modified system. The accuracy and performance of the proposed algorithm are verified for the several benchmark problems including a pressure pulse in a flexible circular tube, a flag interacting with an incompressible viscous flow, and so on. John Wiley & Sons, Ltd.

Strong transient effects of the flow around a harmonically plunging NACA0012 airfoil at low Reynolds numbers

The flow pattern around a NACA0012 airfoil undergoing harmonic plunging motion corresponding to the deflected wake phenomenon reported by Jones and Platzer (Exp Fluids 46:799–810, 2009) is investigated in detail using direct numerical simulations. An arbitrary Lagrangian–Eulerian formulation based on an unstructured side-centered finite volume method is utilized in order to solve the incompressible unsteady Navier–Stokes equations. The Reynolds number is chosen to be 252, and the reduced frequency of plunging motion (k = 2π fc/U ∞) and the plunge amplitude non-dimensionalized with respect to chord are set to be 12.3 and 0.12, respectively, as in the experimental study of Jones and Platzer (2009). The present numerical simulations reveal a highly persistent transient effect, and it takes two orders of magnitude larger duration than the heave period to reach the time-periodic state. In addition, the three-dimensional simulation reveals that the flow field is three-dimensional for the parameters used herein. The calculation reproduces the deflected wake and shows a good agreement with the experimental wake pattern. The instantaneous vorticity contours, finite-time Lyapunov exponent fields and particle traces are presented along with the aerodynamic parameters including the lift and thrust coefficients.

The numerical simulation of the wing kinematics effects on near wake topology and aerodynamic performance in hovering Drosophila flight

The parallel large-scale unstructured finite volume method based on an Arbitrary Lagrangian–Eulerian (ALE) formulation in Erzincanli and Sahin (2013) has been employed in order to investigate the effects of the wing kinematics on the near wake topology produced by a pair of flapping Drosophila wings in hover flight. The three-dimensional wing shape and the baseline wing kinematics are the same as those of the robotic fly by Dickinson et al. (1999). The simulations are used to quantify the important wing kinematic parameters for the wake topology and as well as their correlations with the force production. The angle-of-attack is shown to be very effective for producing lift during the wing translational motion. However, for larger values the angle-of-attack limits the angle during the stroke reversal and reduces the rotational lift. The angle-of-attack at which the maximum lift coefficient occurs is in excellent agreement with the earlier experiments in the literature. In addition, the numerical results confirm that the increase in the wing stroke amplitude leads to a prolonged attachment of the leading edge vortex (LEV) over a relatively large distance and increases force production. The variations in the heave angle include the figure-of-eight pattern, figure-of-O pattern and figure-of-U pattern. The constant heave angle and figure-of-eight pattern are found to have a more profound influence on the magnitude of force production. In addition, some asymmetric variations in the wing kinematics are introduced in order to assess the important parameters determining forward and backward flights. The paddling wing motion is shown to be very effective to initiate forward and backward acceleration.

Comparison of structured- and unstructured-grid, compressible and incompressible methods using the vortex pairing problem

The accuracy, robustness, dissipation characteristics and efficiency of several structured and unstructured grid methods are investigated with reference to the low Mach double vortex pairing flow problem. The aim of the study is to shed light into the numerical advantages and disadvantages of different numerical discretizations, principally designed for shock-capturing, in low Mach vortical flows. The methods include structured and unstructured finite volume and Lagrange-Remap methods, with accuracy ranging from 2nd to 9th-order, with and without applying low-Mach corrections. Comparison of the schemes is presented for the vortex evolution, momentum thickness, as well as for their numerical dissipation versus the viscous and total dissipation. The study shows that the momentum thickness and large scale features of a basic vortical structure are well resolved even at the lowest grid resolution of 32×32 provided that the numerical schemes are of a high-order of accuracy or the numerical framework is sufficiently non-dissipative. The implementation of the finite volume methods in unstructured triangular meshes provides the best results even without low Mach number corrections provided that a higher-order advective discretization for the advective fluxes is employed. The compressible Lagrange-Remap framework is computationally the fastest one, although the numerical error for the momentum thickness does not reduce as fast as for other numerical schemes and computational frameworks, e.g., when higher-order schemes are utilized. It is also shown that the low-Mach number correction has a lesser effect on the results as the order of the spatial accuracy increases.

Application of a Riemann Solver Unstructured Finite Volume Method to Combustion Instabilities

Combustion instabilities are a common difficulty in the design of combustion systems for applications from power generation gas turbines to gas and liquid fuel rockets. Many low-order models have been developed for quick turnaround prediction of trends, while high-fidelity, large-eddy simulations are becoming a viable tool with the increasing availability of computing resources. In this paper, a three-dimensional linear thermoacoustic solver is developed to fill the void in design tools between the low-order modeling techniques and the expensive high-fidelity approach. The solver uses the finite volume method with a second-order Riemann solver flux calculation to solve the linearized Euler equations with unsteady heat release. The code is written for unstructured grids in order to easily accommodate the complex geometries of industrial combustors. Because the application requires the ability to resolve the effects of sharp gradients of mean density, a method for treating variable-coefficient hyperbolic systems is developed within the Riemann solver approach for unstructured grids. The code is tested with three verification cases with comparison to exact solution and three validation cases with comparison to experimental data. The code is shown to predict the frequencies and limit-cycle amplitudes of the validation experiments well, comparing favorably to lower-order models. In the final validation case, the code predicts the frequency and limit-cycle amplitude within 5% for a realistic gas turbine combustor design. More important, the code predicts the subsequent stability of the mode after a change in the operating conditions.

Unstructured Moving-Grid Finite-Volume Method for Incompressible Flows and Its Application to a Coupled Problem of Fluid-Dynamics and Body Motion

Unstructured Moving-Grid Finite-Volume Method for Incompressible Flows and Its Application to a Coupled Problem of Fluid-Dynamics and Body Motion

Numerical Simulation of PCM melting over a wavy surface

Purpose – The purpose of this paper is to conduct a numerical study to analyze the melting process along a vertical wavy surface with uniform surface temperature. Design/methodology/approach – The cavity horizontal walls are insulated while the left hot wavy wall and the right cold wall are maintained at temperatures, TH=38.3°C and TC=28.3°C, respectively. The enclosure was filled by solid Gallium initially at temperature TC. A numerical code is developed using an unstructured finite-volume method and an enthalpy porosity technique to solve for natural convection coupled to solid-liquid phase change. The validity of the numerical code used is ascertained by comparing the results with previously published results. Findings – The effect of number of wavy surface undulation and amplitude of the wavy surface on the flow structure and heat transfer characteristics is investigated in detail. The numerical results show that the enhanced total heat transfer rate seems to depend on the amplitude of the wavy surface. Originality/value – Flow and heat transfer from irregular surfaces are often encountered in many engineering applications to enhance heat transfer such as micro-electronic devices, flat plate solar collectors and flat-plate condensers in refrigerators, etc. Roughened surfaces could be used in latent storage systems where the wall heat flux is known. One of the reasons why a roughened surface is more efficient in heat transfer is its capability to promote fluid motion near the surface; in this way a complex wavy surface is expected to promote a larger heat transfer rate than a flat plate. This complex geometry will promote a correspondingly complicated motion in the fluid near the surface; this motion is described by the nonlinear boundary-layer equations.

An axisymmetric unstructured finite volume method applied to the numerical modeling of an atmospheric pressure gas discharge

We develop a finite volume method based on unstructured triangular and/or quadrangular meshes for the numerical modeling of gas discharges at atmospheric pressure. The discretization of the computational domain is performed with median-dual control-volumes. Gradient calculations at the faces of the control-volumes are based on finite-element interpolation using shape functions. An FCT method was chosen for the species transport integration. The accuracy of the numerical scheme is verified for different types of mesh-elements in a plane-to-plane DBD geometry. We present comparisons with literature results based on the immersed boundary method in a point-to-plane geometry and investigate the need of a high-order scheme for an accurate model of a filamentary discharge. Finally, we apply our scheme in a study of a discharge between two hemispherical electrodes covered with dielectrics, while focusing on sufficient spatial resolution in the vicinity of the dielectric surface.

A coupled 1D–2D hydrodynamic model for flood simulation in flood detention basin

A coupled 1D–2D hydrodynamic model linking the channel and flood detention basin for flood simulation with complex topography and irregular boundary was presented in this paper. The 1D Saint-Venant equations were used for governing flows in channel, and the four-point implicit Preissmann scheme was used for discretization. To simulate floods in flood detention basin, the 2D shallow equations were adopted as governing equations, and a well-balanced, unstructured finite-volume method was proposed for the numerical resolution. The 2D model is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. The numerical fluxes are computed by HLLC algorithm, and the MUSCL–Hancock predictor–corrector scheme is used to achieve high-accuracy and high-resolution results. A simple and efficient method was proposed to reflect the coupled hydraulic connection between the channel and the flood detention basin. The novelties of the proposed model include (1) a robust method for wetting/drying treatment and (2) an efficient way to link the 1D and 2D models. The 1D, 2D, and coupled models are all tested through some benchmark cases, and numerical results validated the accuracy of the models. Furthermore, the coupled 1D–2D model was used for a real flood simulation in Jiakouwa flood detention basin, China. The flood-risk information including flood arrival time and maximal water depth was mapped using GIS. Those flood-risk maps can be used as an important decision-making basis of flood control and rescue for the flood control departments at all levels.

Large-scale parallel computation of the phonon Boltzmann Transport Equation

Non-equilibrium heat conduction, as occurring in modern-day sub-micron semiconductor devices, can be predicted effectively using the Boltzmann Transport Equation (BTE) for phonons. In this article, strategies and algorithms for large-scale parallel computation of the phonon BTE are presented. An unstructured finite volume method for spatial discretization is coupled with the control angle discrete ordinates method for angular discretization. The single-time relaxation approximation is used to treat phonon–phonon scattering. Both dispersion and polarization of the phonons are accounted for. Three different parallelization strategies are explored: (a) band-based, (b) direction-based, and (c) hybrid band/cell-based. Subsequent to validation studies in which silicon thin-film thermal conductivity was successfully predicted, transient simulations of non-equilibrium thermal transport were conducted in a three-dimensional device-like silicon structure, discretized using 604,054 tetrahedral cells. The angular space was discretized using 400 angles, and the spectral space was discretized into 40 spectral intervals (bands). This resulted in ∼9.7 × 109 unknowns, which are approximately 3 orders of magnitude larger than previously reported computations in this area. Studies showed that direction-based and hybrid band/cell-based parallelization strategies resulted in similar total computational time. However, the parallel efficiency of the hybrid band/cell-based strategy—about 88%—was found to be superior to that of the direction-based strategy, and is recommended as the preferred strategy for even larger scale computations.

Agglomeration multigrid methods with implicit Runge–Kutta smoothers applied to aerodynamic simulations on unstructured grids

For unstructured finite volume methods an agglomeration multigrid with an implicit multistage Runge–Kutta method as a smoother is developed for solving the compressible Reynolds averaged Navier–Stokes (RANS) equations. The implicit Runge–Kutta method is interpreted as a preconditioned explicit Runge–Kutta method. The construction of the preconditioner is based on an approximate derivative. The linear systems are solved approximately with a symmetric Gauss–Seidel method. To significantly improve this solution method grid anisotropy is treated within the Gauss–Seidel iteration in such a way that the strong couplings in the linear system are resolved by tridiagonal systems constructed along these directions of strong coupling. The agglomeration strategy is adapted to this procedure by taking into account exactly these anisotropies in such a way that a directional coarsening is applied along these directions of strong coupling. Turbulence effects are included by a Spalart–Allmaras model, and the additional transport-type equation is approximately solved in a loosely coupled manner with the same method. For two-dimensional and three-dimensional numerical examples and a variety of differently generated meshes we show the wide range of applicability of the solution method. Finally, we exploit the GMRES method to determine approximate spectral information of the linearized RANS equations. This approximate spectral information is used to discuss and compare characteristics of multistage Runge–Kutta methods.

Unstructured finite volume method for water impact on a rigid body

A new method is presented for the water impact simulation, in which the air-water two phase flow is solved using the pressure-based computational fluid dynamics method. Theoretically, the air effects can be taken into account in the water structure interaction. The key point of this method is the air-water interface capture, which is treated as a physical discontinuity and can be captured by a well-designed high order scheme. According to a normalized variable diagram, a high order discrete scheme on unstructured grids is realised, so a numerical method for the free surface flow on a fixed grid can be established. This method is implemented using an in-house code, the General Transport Equation Analyzer, which is an unstructured grid finite volume solver. The method is verified with the wedge water and structure interaction problem.

A Parallel Unstructured Finite-Volume Method for All-Speed Flows

Parallel computational fluid dynamics is one of the key applications in the area of high-performance computations. The primary goals of this work are to develop a parallel unstructured finite-volume solver for all-speed flows based on a domain decomposition method, and to establish a comprehensive and intensive analysis method for the parallel performance. The numerical calculations of several typical flows using hundreds of CPU cores validate the accuracy and parallel performance of the parallel solver. The analysis of the decomposed efficiencies reveals the key factors that limit the parallel performance. This work is quite generic and can be extended to the large-scale parallel calculation and performance analysis of complex fluid flow and heat transfer problems.

E225 非構造格子を用いたSMAC法による非圧縮性流体解析における流出境界条件

E225 非構造格子を用いたSMAC法による非圧縮性流体解析における流出境界条件

Numerical Study on Transient Heat Transfer of a Quartz Lamp Heating System

Thermal ground testing is an accepted and frequently used method for simulating the aerodynamic heating of high speed flight vehicles. A numerical method based on a finite volume method for a quartz lamp heating system, used in thermal testing, is proposed. In this study, the unstructured finite-volume method (UFVM) for radiation has been formulated and implemented in a fluid flow solver GTEA on unstructured grids. For comparison and validation of the proposed method, a 2D furnace with cooling pipes was chosen. The results obtained from the proposed FVM agreed well with the exact solutions. Numerical results show that the quartz lamp can be simplified as a slat with the same temperature radiation source, and a simplified 2D thermal testing case was then simulated with the coupling effects of radiation, convection, and conduction heat transfer. Different temperature loading curves and ratios of intervals between the lamps and lamp length (l/s) were studied using the proposed method. The radiation heat flux on the metal surface was a wave-shaped curve. Comparing the different interval ratios, we found that the smaller the interval ratio, the larger the maximum value and the smaller the difference between the maximum and minimum heat flux.

Analysis of the Properties of Adjoint Equations and Accuracy Verification of Adjoint Model Based on FVM

There are two different approaches on how to formulate adjoint numerical model (ANM). Aiming at the disputes arising from the construction methods of ANM, the differences between nonlinear shallow water equation and its adjoint equation are analyzed; the hyperbolicity and homogeneity of the adjoint equation are discussed. Then, based on unstructured meshes and finite volume method, a new adjoint model was advanced by getting numerical model of the adjoint equations directly. Using a gradient check, the correctness of the adjoint model was verified. The results of twin experiments to invert the bottom friction coefficient (Manning’s roughness coefficient) indicate that the adjoint model can extract the observation information and produce good quality inversion. The reason of disputes about construction methods of ANM is also discussed in the paper.

Melting over a wavy surface in a rectangular cavity heated from below

The current numerical study is conducted to analyze melting in a rectangular closed enclosure by subjecting the bottom wavy surface to a uniform temperature. The cavity vertical walls and the top wall are insulated while the bottom wall is maintained at temperature TB = 38.3 °C. The enclosure was filled by solid Gallium initially at temperature Ti = 28.3 °C. A numerical code is developed using an unstructured finite-volume method and an enthalpy porosity technique to solve for natural convection coupled to solid–liquid phase change. The validity of the numerical code used is ascertained by comparing our results with previously published results. The effect of the amplitude of the wavy surface on the flow structure and heat transfer characteristics is investigated in detail. It is found that the rate of the melting increases with the elevation in the magnitude of the amplitude value of the wavy surface.

An unstructured finite volume method for large-scale shallow flows using the fourth-order Adams scheme

In this paper, we introduce a new upwind finite volume method using unstructured grids for large-scale shallow flows. This method uses a high-order upwind scheme for the calculation of the numerical flux, and the fourth-order Adams method with a splitting approach for time integration. The process includes three stages: in the first and third steps the Coriolis term is integrated analytically, and in the second step the flux term is integrated numerically. Most upwind schemes perform well for gravity waves but they lead to a high level of damping or numerical oscillations for Rossby waves. The proposed method presents the advantage that it performs well for both gravity and Rossby waves. The use of fourth-order Adams method without any iteration on the corrector is enough to suppress the short-wave numerical noise without damping the long waves that are essential in the transport of energy Rossby waves, in large-scale oceanic and atmospheric flows.

DDES and Acoustic Prediction of Rudimentary Landing Gear Experiment Using Unstructured Finite Volume Methods

The far-field noise produced by a Rudimentary Landing Gear (RLG) was investigated using Delayed Detached Eddy Simulation (DDES) coupled with a Ffowcs-Williams Hawkings (FWH) integral. Computational results are presented for the RLG geometry using three unstructured grids of quasi-nested refinement. The geometry consists of a square post attached to an axle supporting four wheels, which is entirely suspended from an inviscid flat plate. The Reynolds number was fixed at 106 based on the wheel diameter and freestream properties. The freestream Mach number was fixed at 0.115. The effect of grid resolution using a low dissipation scheme is examined in terms of both aerodynamic forces as well as acoustic spectra. Flow visualizations reveal rich three-dimensional unsteady content and the pattern of wall-pressure fluctuations are consistent with experimental observations. In general the FWH equation predictions of the far-field noise are in very good agreement with acoustic microphone measurements. From these preliminary results it appears that DDES has the potential to become a viable tool for evaluating low noise gear designs during the early design phases before wind tunnel and flight testing have commenced. Efforts are underway to extend the approach to more complex landing gear geometries.

An arbitrary Lagrangian–Eulerian formulation for solving moving boundary problems with large displacements and rotations

An Arbitrary Lagrangian–Eulerian (ALE) formulation based on the unstructured finite volume method is proposed for solving moving boundary problems with large displacements and rotations. The numerical method is based on the side-centered arrangement of the primitive variables that does not require any ad-hoc modifications in order to enhance pressure coupling. The continuity equation is satisfied within each element at machine precision and the summation of the continuity equations can be exactly reduced to the domain boundary, which is important for the global mass conservation. A special attention is given to construct an ALE algorithm obeying the discrete geometric conservation law (DGCL). The mesh deformation algorithm is based on the indirect Radial Basis Function (RBF) algorithm at each time level while avoiding remeshing in order to enhance numerical robustness. For the parallel solution of resulting large-scale algebraic equations in a fully coupled form, a matrix factorization is introduced similar to that of the projection method for the whole system and the parallel algebraic multigrid solver BoomerAMG is used for the scaled discrete Laplacian provided by the HYPRE library which we access through the PETSc library. The present numerical algorithm is initially validated for the decaying Taylor–Green vortex flow, the flow past an oscillating circular cylinder in a channel and the flow induced by an oscillating sphere in a cubic cavity. Then the numerical algorithm is applied to the numerical simulation of flow field around a pair of flapping Drosophila wings in hover flight. The time variation of the Eulerian coherent structures in the near wake is shown along with the aerodynamic loads.