Discretized pressure Poisson algorithm for the steady incompressible flow on a nonstaggered grid

In the aspect of numerical methods for incompressible flow problems, there are two different algorithms: semi-implicit method for pressure-linked equations (SIMPLE) series algorithms and the pressure Poisson algorithm. This paper introduced a new discretized pressure Poisson algorithm for the steady incompressible flow based on a nonstaggered grid. Compared with the SIMPLE series algorithms, this paper did not introduce three correction variables. So, there is no need to implement the guess-and-correct procedure for the calculation of pressure and velocity. Compared with the pressure Poisson algorithm, there is no need to calculate unsteady Navier–Stokes equations for steady problems in the new discretized pressure Poisson algorithm. Meanwhile, as the finite volume method and cell-centered grid are used, the governing equation for pressure is obtained from the continuity equation and the boundary conditions for pressure are easily obtained. This new discretized pressure Poisson algorithm was tested at the lid-driven cavity flow problem on a nonstaggered grid and the results are also reliable.     

PRESSURE BOUNDARY CONDITIONS FOR A SEGREGATED APPROACH TO SOLVING INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

It has been well accepted that Diricklet and Neumann boundary conditions for Ike pressure Poisson equation give the same solution. The purpose of this article is to reveal that the above statement is computationally acceptable but is not theoretically correct. Analytic proof as well as computational evidences are presented through examples in support of our observation. In this work we address that the mixed finite-element formulation for solving incompressible Navier-Stokes equations in primitive variables is equivalent to the formulation that involves solving the pressure Poisson equation, subject to Neumann boundary conditions, Uerativety with the momentum equations provided the velocity field is classified as having divergence-free and conservative properties.     

Thermal Buckling of Functionally Graded Plates According to a Four-Variable Refined Plate Theory

In this article, a four-variable refined plate theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. A power law distribution is used to describe the variation of volume fraction of material compositions. Equilibrium and stability equations are derived based on the present theory. The non-linear governing equations are solved for plates subjected to simply supported boundary conditions. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. The influences of many plate parameters on buckling temperature difference will be investigated. It is noticed that the present refined plate theory can predict accurately the critical temperatures of simply supported functionally graded plates.     

Implementation of boundary conditions and global mass conservation in pressure-based finite volume method on unstructured grids for fluid flow and heat transfer simulations

Proper boundary condition implementation can be critical for efficient, accurate numerical simulations when a pressure-based finite volume methodology is used to solve fluid flow and heat transfer problems, especially when an unstructured mesh is used for domain discretization. This paper systematically addresses the relationships between the flow boundary conditions for the discretized momentum equations and the corresponding conditions for the (Poisson type) pressure-linked equation in the context of a cell-centred finite volume formulation employing unstructured grids and a collocated variable arrangement. Special attention is paid to the treatment of the outflow boundary where flow conditions are either known to be fully developed (if a long enough channel is used) or unknown prior to solution (if a truncated channel is used). In the latter case, due to the singularity of the coefficient matrix of the pressure-linked equation, no pressure or pressure corrector solution can exist without explicitly enforcing global mass conservation (GMC) during each iteration. After evaluating published methods designed to ensure GMC, two new methods are proposed to correct for global mass imbalance. The validity of the overall methodology is demonstrated by solving the evolving flow between two parallel plates and the laminar flow over a backward-facing step on progressively truncated domains. In the latter case, our methodology is shown to handle situations where the outflow boundary passes through a recirculation zone.     

Thermal Buckling Response of Functionally Graded Plates with Clamped Boundary Conditions

In this research work, an exact analytical solution for thermal buckling analysis of functionally graded material (FGM) plates with clamped boundary condition subjected to uniform, linear, and non-linear temperature rises across the thickness direction is developed. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory accounts for parabolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. The material properties of FGM plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The governing equations are solved analytically for a plate with simply supported boundary conditions. Resulting equations are employed to obtain the closed-form solution for the thermal force resultant for each loading case. Numerical examples covering the effects of the plate aspect ratio, side-to-thickness ratio and gradient index on thermal force resultant are discussed.     

A Comparison of Fluid-Cell and Ghost-Cell Direct Forcing Immersed Boundary Method for Incompressible Flows with Heat Transfer

In direct-forcing immersed boundary methods, the forcing terms are generally applied to forcing points on either fluid side (fluid-cell forcing approach) or body side (ghost-cell forcing approach) of the immersed boundary. These direct forcing terms are added to the discretized equations over forcing points in order to implicitly enforce proper boundary conditions on the immersed boundary; hence they dictate the development of the computed flow field. Generally, different forcing approaches adapt different reconstruction models with different stencil supports to compute forcing terms. In this article, a general second-order accurate reconstruction model for solving incompressible Navier-Stokes equations with heat transfer is applied to both the fluid-cell forcing approach and the ghost-cell forcing approach. This allows a meaningful comparison of the solution accuracy and convergence between the two forcing approaches under various boundary conditions. In particular, a simple remedy to reduce the spurious oscillations in pressure force calculation over moving bodies for the ghost-cell forcing approach is devised and verified.     

COMPLETE PRESSURE CORRECTION ALGORITHM FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON A NONSTAGGERED GRID

Abstract

When Navier-Stokes equations for incompressible flow are solved on a nonstaggered grid, the problem of checkerboard prediction of pressure is encountered. So far, this problem has been cured either by evaluating the cell face velocities by the momentum interpolated principle III or by evaluating an effective pressure gradient in the nodal momentum equations [2] In this article it is shown that not only are these practices unnecessary, they can lead to spurious results when the true pressure variation departs considerably from linearity. What is required instead is afresh derivation of the pressure correction equation appropriate for a nonstaggered grid. The pressure correction determined from this equation comprises two components: a mass-conserving component and a smoothing component. The former corresponds to the pressure correction predicted by a staggered grid procedure, whereas the latter simply accounts for the difference between the point value of the pressure and the cell-averaged value of the pressure. The new pressure correction equation facilitates ( in a significant way) computer coding of programs written for three-dimensional geometries employing body-fitted curvilinear coordinate grids.     

A time marching strategy for solving parabolic and elliptic equations with Neumann boundary conditions

Abstract

In the community of computational fluid dynamics, pressure Poisson equation with Neumann boundary condition is usually encountered when solving the incompressible Navier–Stokes equations in a segregated approach such as SIMPLE, PISO, and projection methods. To deal with Neumann boundary conditions more naturally and to retain high order spatial accuracy as well, a sixth-order accurate combined compact difference scheme developed on staggered grids (NSCCD6) is adopted to solve the parabolic and elliptic equations subject to Neumann boundary conditions. The staggered grid system is usually used when solving the incompressible Navier–Stokes equations. By adopting the combined compact difference concept, there is no need to discretize Neumann boundary conditions with one-sided discretization scheme which is of lower accuracy order. The conventional Crank–Nicolson scheme is applied in this study for temporal discretization. For two-dimensional cases, D’yakonov alternating direction implicit scheme is adopted. A newly proposed time step changing strategy is adopted to improve convergence rate when solving the steady state solutions of the parabolic equation. High accuracy order of the currently proposed NSCCD6 scheme for one- and two-dimensional cases are shown in this article.     

Implementation of boundary conditions in the finite-volume pressure-based method—Part I: Segregated solvers

ABSTRACT

The paper deals with the formulation of a variety of boundary conditions for incompressible and compressible flows in the context of the segregated pressure-based unstructured finite volume method. The focus is on the derivation and the implementation of these boundary conditions and their relation to the various physical boundaries and geometric constraints. While a variety of boundary conditions apply at any of the physical boundaries (inlets, outlets, and walls), geometric constraints define the type of boundary condition to be used. The emphasis is on relating the mathematical derivation of the boundary conditions to the algebraic equations defined at each centroid of the boundary elements and their coefficients. All derived boundary conditions are validated through a set of test cases with comparison of computed results to available numerical and/or experimental data.     

A CALCULATION PROCEDURE FOR SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON CURVILINEAR NON STAGGERED GRIDS

A recently proposed pressure-correction algorithm for solution of incompressible Navier-Stokes equations on nonstaggered grids introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure (9). The algorithm was derived for equations in Cartesian coordinates. In this article, the algorithm is extended to solution of Navier-Stokes equations in general curvilinear coordinates. By way of application, two cavity flow problems and two internal flow problems are solved. Comparisons with benchmark solutions or experimental data and (or) previous solutions employing staggered grids are made to validate the calculation procedure.     

A near-wall two-equation model for turbulent heat fluxes

A near-wall two-equation model for turbulent heat fluxes is derived from the temperature variance and its dissipation-rate equations and the assumption of gradient transport. Only incompressible flows with non-buoyant heat transfer are considered. The near-wall asymptotics of each term in the exact equations are examined and used to derive near-wall correction functions that render the modeled equations consistent with these behavior. Thus modeled, the equations are used to calculate fully-developed pipe and channel flows with heat transfer. It is found that the proposed two-equation model yields asymptotically correct near-wall behavior for the normal heat flux, the temperature variance and its near-wall budget and correct limiting wall values for these properties compared to direct simulation data and measurements obtained under different wall boundary conditions.     

Gaseous slip flow affected by inclined low magnetic field using second‐order boundary conditions

In this study, asymptotic solutions of a near continuum gaseous slip flow in two‐dimensional rectangular microchannels under the effect of electromagnetic force are presented. An inclined magnetic field was assumed in this study. Nondimensional equations were obtained that relate the pressure ratio, Mach number, magnetic Reynolds number, magnetic force number, and Reynolds number. The asymptotic solutions for the compressible, laminar, and steady flow were obtained by applying second‐order slip velocity and temperature jump wall boundary conditions. It was found that the electric and magnetic field with inclined angle had significant effects on the flow properties. The solutions obtained here using the second‐order boundary conditions result in tangible improvement over those obtained using first‐order boundary conditions. We compared our solutions against the numerical solutions that were provided in the literature and showed that our solutions were in good agreement with the numerical solution.     

Poiseuille Number Correlations for Gas Slip Flow in Micro-Tubes

Fundamental research in fluid flow characteristics in micro-tubes are required for designing microfluidic systems. In this study, Poiseuille number, the product of friction factor and Reynolds number (f · Re) for quasi-fully developed micro-tube flows, was obtained for slip flow regime. The numerical methodology was based on the Arbitrary-Lagrangian-Eulerian (ALE) method and the uncertainties of the results were assessed based on the grid convergence index (GCI). The numerical model was validated with the available experimental and numerical results. The compressible momentum and energy equations were solved for a wide range of Reynolds and Mach numbers with two thermal boundary conditions: constant wall temperature (CWT), and constant heat flux (CHF), respectively. The slip boundary conditions and their numerical implementation are appropriately documented. The tube diameter ranged from 3 to 10 μm and the tube aspect ratio was 200. The stagnation pressure was chosen in such a way that the exit Mach number ranged from 0.1–1.0. The outlet pressure was fixed at the atmospheric condition. It was found that for the case of compressible and slip flows, f · Re correlations are functions of the Mach and Knudsen numbers, and are different from the values obtained from the expression, 64/(1 + 8 Kn), available for the incompressible slip flow regime. The f · Re correlations obtained here are applicable to both no-slip and slip conditions, and for both incompressible and compressible flows. The results are in excellent agreement with the available experimental data.     

An unstructured SMAC algorithm for thermal non-equilibrium two-phase flows

The SMAC (Simplified Marker And Cell) algorithm is extended for an application to thermal non-equilibrium two-phase flows in light water nuclear reactors (LWRs). A two-fluid three-field model is adopted and a multi-dimensional unstructured grid is used for complicated geometries. The phase change and the time derivative terms appearing in the continuity equations are implemented implicitly in the pressure correction step. The energy equations are decoupled from the momentum equations for faster convergence. The verification of the present numerical method was carried out against a set of test problems which includes the single and the two-phase flows. The results are also compared to those of the semi-implicit ICE method, where the energy equations are coupled with the momentum equation for pressure correction.     

Investigation of gas flow in micro-filters and modification of scaling law

In this study, important micro gas flow features including slip velocity, compressibility, and rarefaction effects for flow micro-filters are investigated. In this regard the compressible Navier–Stokes equations with wall slip and temperature jump boundary conditions are solved for a proper range of Knudsen numbers. For the filters, the ratio of the open orifice area to the total area is selected as 0.6 in order to compare the results with the literature. Considering that the filter holes repeat in a periodic fashion, gas flow is simulated through only one hole by imposing periodicity conditions. The simulation results are compared with the empirical scaling laws developed in the last works and a modified relation for the scaling law is presented. This modified relation predicts high-accuracy pressure drop through the micro-filters in the slip regime.     

Modelling of multi-component gas flows in capillaries and porous solids

The paper presents a derivation of the governing equations for multi-component convective-diffusive flow in capillaries and porous solids starting from a well-defined model and clear assumptions. The solution for the continuum regime is discussed in detail including a derivation of the diffusion slip boundary condition based on an improved momentum transfer theory. The Stefan–Maxwell species momentum equations are also re-examined and important distinctions made between the local and tube-averaged equations. An equation for the pressure gradient is derived and some examples of binary flows in capillaries are discussed. The theory for free-molecule flow is standard but the equations are recast into a form identical to the continuum equations which suggests an obvious method of interpolation for flow at arbitrary Knudsen number. There are no problems concerning viscous terms which have marred other derivations. The extension to flow in porous bodies is achieved by introducing a porosity–tortuosity factor but, unlike other treatments, this parameter is not absorbed into the gas diffusivities and flow permeability. It can then be eliminated from all but one of the equations and, with appropriate boundary conditions, the flux ratios can be obtained in terms of a mean pore radius only. The porosity–tortuosity parameter simply controls the absolute flux level and is best interpreted as a length scale-factor. The theory is applied with success to the prediction of some experimental data for helium–argon counter-diffusion and it is shown that, contrary to common belief, the mean pore radius is well-defined by flux ratio measurements if these are made with non-zero pressure differences.     

Free vibration analysis of a nonlocal thermoelastic hollow cylinder with diffusion

Abstract

In this article, the constitutive relations and the governing equations are derived for nonlocal thermoelastic solid in the presence of diffusion. The free vibration of a thermoelastic diffusive cylinder is investigated within the framework of the above newly derived model. Time-harmonic vibration is used to transform the governing equations into a system of ordinary differential equations. The frequency equation is taken under investigation for the survival of a range of possible modes in compact form for traction-free thermal boundary conditions: thermally insulated and isothermal boundary conditions. To explore the vibration analysis from frequency equations, we apply a numerical iteration technique for generating numerical data by taking assistance of the Matlab software. The numerically computed and simulated results for the frequency shift, natural frequency, and the thermoelastic damping are presented graphically. The effect of nonlocality on the above quantities is observed and shown graphically.     

高速齿轮箱迷宫密封流场和泄漏特性的数值研究

以某型大功率机车传动齿轮箱轴端迷宫密封为研究对象,建立错齿式迷宫密封的数值仿真模型。通过分析迷宫密封的密封机理和对泄漏量的影响因素,确定了数值模拟的边界条件。应用Fluent软件,求解N-5方程和能量方程,模拟迷宫密封腔内流场分布及泄漏特性,研究转速、压比和密封齿隙对泄漏特性的影响规律。研究结果表明：随着进出口压比的增加,泄漏系数先快速增加后趋于平缓;转速对泄漏系数影响很小;泄漏系数与齿隙呈分段线性关系。     

Thermal Response of Symmetric Cross-Ply Laminated Plates Subjected to Linear and Non-Linear Thermo-Mechanical Loads

The flexural response of symmetric cross-ply laminated plates subjected to uniformly distributed linear and non-linear thermo-mechanical loads is presented using trigonometric shear deformation theory. The in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress-free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Thermal stresses and displacements for three-layer symmetric square cross-ply laminated plates subjected to uniform linear and nonlinear and thermo-mechanical loads are obtained. The results of present theory are compared with those of classical plate theory, first-order shear deformation theory and higher-order shear deformation theory.