Cell-vertex Finite Volume Method Research Articles

A penalty-based cell vertex finite volume method for two-dimensional contact problems

A penalty-based cell vertex finite volume method for two-dimensional contact problems

Three-dimensional thermoelastic analysis of functionally graded solids using a high-order staggered finite volume method

This paper has developed a 3D 2-order cell vertex finite volume method (CV-FVM) for thermoelastic problems of functionally graded materials (FGMs). The construction methods of 15-node triangular prismatic (P15) element and 20-node hexahedral (H20) element are proposed. The 10-node tetrahedral (T10) element is adopted together with P15 and H20 to deal with mixed element problems. Material variation is considered using the staggered grid technique by defining properties at the cell center while defining unknown variables at the node. This treatment makes the present method suitable for FGMs with arbitrary material distribution. The accuracy and capability of the 2-order CV-FVM are verified by numerical tests. The results show that all the T10 elements, P15 elements, H20 elements and 2-order mixed elements can provide accurate results. When the same mesh size is employed, 2-order CV-FVM could provide better numerical precision than the 1-order CV-FVM and H20 elements could provide more accurate results than T10 elements. Based on the same mesh size, the H20 element avoids numerical oscillations caused by the T10 element. The developed method exhibits advantages in analyzing materials with significant property changes, avoiding physically unrealistic stress distributions.

Thermoelastic analysis of functionally graded porous materials with temperature-dependent properties by a staggered finite volume method

The cell-vertex finite volume method (CV-FVM) is developed for the thermoelastic analysis of the functionally graded porous (FGP) structures with temperature-dependent material properties. The staggered grid technique is employed to incorporate property variation into the discretization of governing equations. And the iterative approach is used to solve the nonlinear heat conduction equation to obtain convergent and matched temperature solution. The force equilibrium equation is solved without iteration to obtain displacements. The accuracy of the present method for functionally graded materials with temperature-dependency has been validated. Then the thermoelastic performance of FGP plates with different material distribution is investigated. It is found that the thermal stress is much more sensitive to the material distribution than the temperature and the deformation. The combination of the material distribution and the temperature-dependency causes different locations of maximum magnitudes of the thermal stress and the normalized equivalent stress. In the present case, the existence of the porosity helps to reduce the thermal stress to some extent.

Thermo-mechanical Performance Of the Thermal Barrier Coated Piston

Analysis of the performance of thermal barrier coated (TBC) pistons is important for the design of adiabatic engine. The performance of the TBC piston with thermal load, mechanical load and thermo-mechanical load has been analysed using the cell-vertex finite volume method. Compared to the ordinary piston, TBC piston body withstands less thermal load and exists smaller stress and deformation, while the stress distribution is similar. The maximum stress exists in the TBC region, followed by the region around the pin boss. TBC isolated most heat to the piston body and bears large thermal load. The interface between the bonding layer and the ceramic layer has the most serious stress level. Due to the adoption of TBC, the stress on the piston fire shore arises, while the stress around the piston ring groove area decreases. The area around the pin boss exists the second stress level caused by both thermal and mechanical loads.

New Simplified Algorithm for the Multiple Rotating Frame Approach in Computational Fluid Dynamics

This paper deals with rotating effects simulation of steady flows in turbomachinery. To take into account the rotating nature of the flow, the frozen rotor approach is one of the widely used approaches. This technique, known in a more general context as a multiple rotating frame (MRF), consists on building axisymmetric interfaces around the rotating parts and solves for the flow in different frames (static and rotating). This paper aimed to revisit this technique and propose a new algorithm referred to it by a virtual multiple rotating frame (VMRF). The goal is to replace the geometrical interfaces (part of the computer-aided design (CAD)) that separate the rotating parts replaced by the virtual ones created at the solver level by a simple user input of few point locations and/or parameters of basic shapes. The new algorithm renders the MRF method easy to implement, especially for edge-based numerical schemes, and very simple to use. Moreover, it allows avoiding any remeshing (required by the MRF approach) when one needs to change the interface position, shape, or simply remove or add a new one, which frequently happened in practice. Consequently, the new algorithm sensibly reduces the overall computations cost of a simulation. This work is an extension of a first version published in an ASME conference, and the main new contributions are the detailed description of the new algorithm in the context of cell-vertex finite volume method and the validation of the method for viscous flows and the three-dimensional (3D) case which is of significant importance to the method to be attractive for real and industrial applications.

A Grid-Insensitive LDA Method on Triangular Grids Solving the System of Euler Equations

The performance of the classic upwind-type residual distribution (RD) methods on skewed triangular grids are rigorously investigated in this paper. Based on an improved signals distribution, an improved second order RD method based on the LDA approach is proposed to faithfully replicate the flow physics on skewed triangular grids. It will be mathematically and numerically shown that the improved LDA method is found to have minimal accuracy variations when grids are skewed compared to classic RD and cell vertex finite volume methods on scalar equations and system of Euler equations.

Simulation of 2D fluid–structure interaction in inviscid compressible flows using a cell-vertex central difference finite volume method

In the present study, the applicability and accuracy of a cell-vertex finite volume method developed are assessed in simulating 2D fluid–structure interaction in inviscid compressible flows where the nonlinear phenomena exist in both the unsteady transonic fluid flows and the large nonlinear deformation of solid structures. The unsteady Euler equations are considered as the governing equations of the fluid flow in the arbitrary Lagrangian–Eulerian form and the large nonlinear deformation of the solid structure is considered to be governed by the Cauchy equations in the total Lagrangian form. Both the domains are discretized by a second-order central-difference cell-vertex finite volume method in space on arbitrary quadrilateral grids and a second-order implicit method in time. For each time step, the resulting set of coupled implicit nonlinear equations in either domains is solved iteratively in the pseudo-time. The fluidic and structural parts are coupled in a partitioned manner for the simulation of fluid–structure interaction problems where at the interface the grids of the two domains are coincident. The formulation for the structural dynamics applied here can accurately simulate different types of boundary conditions. It can also properly model the panel thickness and therefore the stress distribution within the structure can be reasonably obtained. Some benchmark test cases are studied and the results obtained are compared with the available experimental/numerical results. Simplicity of the implementation, similarity of the discretization and accuracy of the solution demonstrate the capability and robustness of the solution methodology proposed in simulating fluid–structure interaction problems in compressible flows.

Accuracy Variations in Residual Distribution and Finite Volume Methods on Triangular Grids

This paper presents an analytical and numerical approach in studying accuracy deterioration of residual distribution and cell-vertex finite volume methods on triangular grids. Results herein demonstrate that both methods preserve the order-of-accuracy reasonably well for uniformly skewed triangular grids and the $$L_2$$ errors of both second-order accurate methods behave similarly with values of the same magnitude. On the other hand, the first-order finite volume method has an $$L_2$$ error of about an order of magnitude higher than its residual distribution counterpart. Both first-order methods are unable to preserve the order-of-accuracy for high-frequency data when the grids are highly skewed although the residual distribution approach has a slightly better performance. Both second-order methods perform quite decently for high-frequency data on uniformly skewed grids. However, the order-of-accuracy of finite volume methods excessively deteriorate when the grids are skewed non-uniformly unlike the residual distribution methods which preserve the order-of-accuracy.

Computational study of compound angle film cooling flow field and aerodynamic losses using a parallel hybrid mesh Navier–Stokes code

Flow characteristics and aerodynamic losses of film cooling injection with compound angles of 30°, 60°, and 90° for velocity ratios of 0.5, 1.0, and 2.0 are numerically investigated using the high-quality hybrid meshes comprising tetrahedrons, prisms, and pyramids. The solutions are obtained by solving the compressible Reynolds-averaged Navier–Stokes equations, and the cell-vertex finite volume method is used to discretize these equations. The predicted results are firstly validated against the experimental data. It has been found that the variation tendency of mass average net total pressure loss coefficient with compound angles is reasonably predicted and in many cases well predicted. The analyses based on net total pressure losses show that the aerodynamic losses tend to increase with the increasing of compound angle regardless of the velocity ratio, and the tendency becomes more obvious when the velocity ratio is large. On the other hand, some typical cases are additionally calculated with different turbulence models and flux computation schemes for the comparisons. A latest proposed numerical flux computation method, simple low-dissipative AUSM (SLAU) scheme, is adopted in the predictions. The results indicate that SLAU scheme is slightly superior to Harten–Lax–van Leer–Einfeldt–Wada scheme in the prediction of compound angle film cooling injection.

Three Dimensional Calculation of Thermochemical Nonequilibrium Flowfield over Mars Entry Body

A three-dimensional computational fluid dynamics code is developed for predicting nonequilibrium flowfields over Mars entry probes. The numerical scheme is based on the cell-vertex finite volume method for a prismatic unstructured mesh system. Internal energy excitations and chemical reactions with finite rates are considered by introducing the two-temperature model of Park. Eight chemical species, C, N, O, CO, N2, NO, O2, and CO2, and nine chemical reactions are considered in the calculations. The developed code is verified in terms of prediction of heat flux to the body surface near the stagnation point of a Mars entry probe flying with the velocity of 6 km/s. The calculated heat flux reasonably agree with the calculated result in the past studies.

Application of the Staggered Cell-Vertex Finite Volume Method to Thermoelastic Analysis in Heterogeneous Materials

The two-dimensional (2D) staggered cell vertex finite volume method (CV-FVM) has been developed for unsteady thermoelastic problems in heterogeneous materials. The staggered grid technique is employed to incorporate property variation into the course of solution. The capabilities of methods for vibration analysis, unsteady thermoelastic analysis and elastic analysis with orthotropic materials have been validated. The applications of the method to functionally graded materials (FGMs), multi-phase materials with inclusions, and a thermal barrier coating (TBC) with a gradual FGMs layer or a graded microstructure layer have been accessed. The developed method provides accurate thermoelastic fields and avoids artificial discontinuous stress predictions in heterogeneous materials.

Thermoelastic analysis of functionally graded solids using a staggered finite volume method

A new approach based on cell vertex finite volume method (CV-FVM) has been developed by the introduction of the staggered grid technique for thermoelastic analysis in functionally graded materials (FGMs). The 4-node quadrilateral grid and the 3-node triangular grid are employed to deal with mixed-grid problems. The performance of the developed method has been demonstrated by four numerical tests. First, the consideration of the bilinear quadrilateral grid is found to be important for the elimination of artificial numerical oscillation in the displacement field. Second, the staggered grid technique is proved (within the considered test cases) to be efficient and reliable for thermoelastic analysis in FGMs without any undesirable discontinuous distribution. Third, the present method requires small computational and memory costs compared to the method adopting graded elements or higher-order theory.

A cell-vertex finite volume scheme for solute transport equations in open channel networks

A solute particle in a water flow behaves as a stochastic process, which is modeled by a stochastic differential equation. The solute transport equation governing macroscopic dynamics of solute concentration distribution in a locally one-dimensional open channel network is deduced from the Kolmogorov's forward equation associated to the stochastic differential equation. The cell-vertex finite volume method is applied for developing a computational scheme to numerically solve the solute transport equation. A computational domain is divided into a regular mesh, from which a dual mesh is generated. The exact solution to a local two-point boundary value problem is used for evaluating the flux at the interface of each pair of two dual cells. The scheme satisfies the total variation diminishing condition and consistently deals with any singular point such as junctions. The semi-implicit method is applied to temporal integration, and the stability condition for the time increment is presented. A series of test problems is examined in order to verify accuracy and conservative property of the scheme. Sufficiently accurate numerical solutions are obtained for test problems in a one-dimensional interval domain, while solute transport phenomena in an open channel network are correctly reproduced for cases with and without deposition of solute. It is concluded that the cell-vertex finite volume scheme is accurate, stable, and versatile in the numerical analysis of solute transport problems in open channel networks.

Applying a new computational method for biological tissue optics based on the time-dependent two-dimensional radiative transfer equation

Optical tomography is a medical imaging technique based on light propagation in the near infrared (NIR) part of the spectrum. We present a new way of predicting the short-pulsed NIR light propagation using a time-dependent two-dimensional-global radiative transfer equation in an absorbing and strongly anisotropically scattering medium. A cell-vertex finite-volume method is proposed for the discretization of the spatial domain. The closure relation based on the exponential scheme and linear interpolations was applied for the first time in the context of time-dependent radiative heat transfer problems. Details are given about the application of the original method on unstructured triangular meshes. The angular space (4πSr) is uniformly subdivided into discrete directions and a finite-differences discretization of the time domain is used. Numerical simulations for media with physical properties analogous to healthy and metastatic human liver subjected to a collimated short-pulsed NIR light are presented and discussed. As expected, discrepancies between the two kinds of tissues were found. In particular, the level of light flux was found to be weaker (inside the medium and at boundaries) in the healthy medium than in the metastatic one.

An inverse design method for viscous flow in turbomachinery blading using a wall virtual movement

An inverse shape design method for turbomachinery blades based on a time-accurate solution of the viscous flow equations is presented. The design scheme is formulated such that either the blade pressure distributions on pressure and suction surfaces, or the blade pressure loading and its thickness distribution can be prescribed as design variables. The blade profile is modified using a virtual velocity distribution that would make the momentum flux on the blade surfaces equal to the design momentum flux. The flow is simulated by solving the Reynolds-averaged Navier–Stokes equations that are discretized using a cell vertex finite-volume method, where an arbitrary Lagrangian–Eulerian formulation is used to account for mesh movement. An algebraic Baldwin–Lomax model is used for turbulence closure. The inverse method is first validated for a transonic compressor cascade; it is then used to redesign a subsonic turbine and a transonic compressor. The results show that the design method is rather robust, flexible and useful in reshaping the blade geometry to achieve the prescribed design variables. They also indicate that by carefully tailoring the design target, significant improvement can be achieved in the blade aerodynamic performance.

The simulation of 3D unsteady incompressible flows with moving boundaries on unstructured meshes

The development of a computational model for the simulation of three-dimensional unsteady incompressible viscous fluid flows with moving boundaries is presented. The numerical model is based upon the solution of the Navier–Stokes equations on unstructured meshes using the artificial compressibility approach. An ALE formulation is adopted and the equations are discretized using a cell vertex finite volume method. The formulation ensures the satisfaction of the geometric conservation law when the mesh is allowed to move. An implicit time discretization is adopted and a dual time approach is employed. Explicit relaxation is used for the sub-iterations, with multigrid acceleration. For moving geometries, the mesh is deformed by adopting a spring analogy, combined with a wall distance function approach. The numerical procedure is validated on a standard problem and is then used for the simulation of flow over a flexible fish-like body.

Aerodynamic Inverse Design for Viscous Flow in Turbomachinery Blading

An inverse design method for turbomachinery blading based on a time-accurate solution of the compressible viscous flow equations on a time-varying geometry is presented. The blade pressure loading and thickness distributions are the prescribed design parameters. The blade profile is modified as it moves at a virtual velocity distribution that would make the momentum flux on the blade surfaces equal to the flux corresponding to the prescribed loading distribution. The unsteady flow due to the blade motion is simulated by solving the Reynolds-averaged Navier-Stokes equations that are discretized using a cell-vertex finite volume method in which an arbitrary Lagrangian-Eulerian formulation is used to account for the mesh movement and deformation during the design procedure. The method is first verified by inversely designing an existing blade using its loading distribution as the design target and starting from a profile that has a different camberline. The robustness, flexibility, and usefulness of this design method are demonstrated by redesigning a subsonic turbine and a transonic compressor blade for which, for the latter case, the conventional quasi-steady approach failed. The redesign cases demonstrate that the blade aerodynamic performance can be improved by carefully tailoring the target loading distribution.

Solution of depth‐averaged tidal currents in Persian Gulf on unstructured overlapping finite volumes

AbstractHydrodynamic simulation of tidal currents in Persian Gulf due to tidal fluctuations in Hurmoz strait and inflow for Arvand River is presented in this paper and tidal constituents of M2 and K1 are attained. The mathematical model utilized consists of depth‐averaged equations of continuity and motion in two‐dimensional horizontal plane which considers hydrostatic pressure distribution. The effect of evaporation is considered in the continuity equation and the effects of bed slope and friction, as well as the Coriolis effects are considers in two equations of motion. The cell vertex finite volume method is applied for converting the governing equations into discrete forms for unstructured overlapping control volumes. Using unstructured triangular meshes provides great flexibility for modelling of complex geometries in the real world flow domain. The accuracy of the developed flow solver is assessed using two test cases. Firstly, the results of the hydrodynamic model for fluctuating flow on the variable bed slope are compared with available analytical solution. Secondly, in order to simulate circulating flow patterns the numerical results are compared with the reported data in the literature for deep flow in a canal with sudden expansion. Finally, the performance of the computer model to simulate tidal flow in a geometrically complex domain is examined by simulation of tidal currents in the Persian Gulf and computing tidal constituents. Copyright © 2007 John Wiley & Sons, Ltd.

An inverse blade design method for subsonic and transonic viscous flow in compressors and turbines

An inverse blade design method applicable to two- and three-dimensional inviscid and viscous flow in turbomachinery cascades is presented and is applied to design cascades in two-dimensional viscous flow. The pressure distribution along the blade surfaces is prescribed and is reached by modifying the initial guess of the blade geometry. The geometry modification is computed from a virtual velocity distribution derived from the difference between the current and the target pressure along the blade surfaces. The inverse method is implemented into and is consistent with the unsteady Reynolds-averaged Navier-Stokes (RANS) equations where an arbitrary Lagrangian–Eulerian (ALE) formulation on a moving and deforming grid is used. The grid velocities are determined from the space conservation law (SCL), which ensures a fully conservative computational procedure. The unsteady RANS equations are discretized using a cell-vertex finite volume method and the time accuracy is achieved using a dual time stepping scheme. An algebraic Baldwin–Lomax model is used for turbulence closure. The design method is first validated, and then its robustness, flexibility and usefulness are demonstrated on the redesign of recent compressor and turbine blade geometries used in modern gas turbine engines.

Coupled solution of oil slick and depth averaged tidal currents on three-dimensional geometry of Persian Gulf

In this paper, simulation of oil spill due to tidal currents in Persian Gulf is performed by coupled solution of the hydrodynamics equations and an equation for convection and diffusion of the oil. The hydrodynamic equations utilized in this work consist of depth average equations of continuity and motion in two dimensional horizontal planes. The effect of evaporation is considered in the continuity equation and the effects of bed slope and friction, as well as the Coriolis effects are considered in two equations of motion. The overlapping cell vertex finite volume method is applied for solving the governing equations on triangular unstructured meshes. Using unstructured meshes provides great flexibility for modeling the flow in arbitrary and complex geometries, such as Persian Gulf flow domain. The results of the hydrodynamic model for tidal currents in Persian Gulf domain is examined by imposing tidal fluctuations to the main flow boundary during a limited period of time. Finally, the developed model is used to simulate an accidental oil spill from a point in Persian Gulf.